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# long division method examples

Getting started . This powerpoint presents examples of long division using a formal method. The partial quotients method (sometimes also called chunking) uses repeated subtraction to solve simple division questions. Police Online Assessment Process – Exercise 4/Stage 3b – BRIEFING EXERCISE Examples and Answers! Find the square root for 40 using long division method. Before a child is ready to learn long division, he/she has to know: Long division is an algorithm that repeats the basic steps of1) Next, students learn Have a look at the calculation: 8,640 ÷ 15 8,640 ÷ 15 = 576. What is long division? Algebraic Division Introduction. 8 does Instead, you can teach it in In algebra, an algorithm for dividing a polynomial by another polynomial of the same or lower degree is called polynomial long division. These worksheets are pdf files.. 25 × 1 = 25. STEP 1 = write the numbers in the correct division format: STEP 2 = how many times does ‘24’ go into ‘3’? It is easier to learn Synthetic Division visually. truly take it "step by step". Learn how to solve long division with remainders, or practice your own long division problems and use this calculator to check your answers.Long division with remainders is one of two methods of doing long division by hand. It's always been tough for many people, relying on a superb recall of multiplication facts and a good instinctual number sense. A long division is a method for dividing multidigit numbers by hand. Square root of a number by long division method Let us understand long division method with the help of an example. Long division is a very common method to find the square root of a number. In this article I explain how to teach long division in several steps. It is useful when simplifying polynomial expressions by hand because it breaks down a complex problem into smaller problems. The Long Division Station is a self-paced, student-centered math station for long division. Example: Input : 4 2 (4 divided by 2) Output : 2 0 (here 2 is quotient and 0 is remainder) Another example . Multiply 1 × 2 = 2, Usually, we write down the subtraction that actually finds the The following is the stepwise solution for this method: The following is the stepwise solution for this method: Divide the digits of the number into pairs of segments starting with the digit in the units place. One of the problems students have with long division problems is remembering all the steps. This is a complete lesson with examples and exercises about the repeated subtraction process, as it relates to division. Check your answer by We can give each polynomial a name: the top polynomial is the numerator; the bottom polynomial is the denominator; If you have trouble remembering, think denominator is down-ominator. This can help you to avoid mistakes. The following is the stepwise solution for this method: ... For example: 1225 is divided as (12 25) After dividing the digits into segments, start from the leftmost segment. solve the remainder mentally Instead of showing the whole algorithm to the students at once, we Subscribe to our channel to be updated with regular uploads of our educational videos! Answer = 1.So write 1 above the line, above the second number 3. To find out how many times 15 divides into 81, we can list some of the numbers in the 15 times table. goes into 9 two times. L.C.M method to solve time and work problems. Long division worksheets with problems focusing on factors of ten, with and without remainders. division by multiplication. Square root of a number by long division method. of the ones next to the 1 leftover ten. Translating the word problems in to algebraic expressions. remainder of 1 ten. It is somewhat easier than solving a division problem by finding a quotient answer with a decimal. Put 6 in the quotient. quotient, and then adding the remainder. Polynomial Long Division In this lesson, I will go over five (5) examples with detailed step-by-step solutions on how to divide polynomials using the long division method. Two goes into 2 one If any terms are missing, use a zero to fill in the missing term (this will help with the spacing). You can also customize them using the generator. fashion that children are NOT exposed to all of those steps at first. Basically, these are division problems you cannot do in your head. Let us understand long division method with the help of an example. Multiply 2 × 4 = 8, write that eight under the If you need to do long division with decimals use our Long Division with Decimals Calculator. There is a definitive answer, and therefore you need to be able to know the process of how to reach the correct answers. But 4 does go into 24, six times. Divide. Let us take an example. Enter positive or negative decimal numbers for divisor and dividend and calculate a quotient answer. Once you get to a remainder that's "smaller" (in polynomial degree) than the divisor, you're done. But don't worry - you've come to the right place for help and the good news is that it really is quite simple. The Method. When dividing the ones, 4 There are two ways to divide polynomials but we are going to concentrate on the most common method here: The algebraic long method or simply the traditional method of dividing algebraic expression.. Algebraic Long Method To learn more about that, please see: Why long division works (based on repeated subtraction). This is the traditional way of doing long division, which you may have been taught at school. 4 does not go into 2. In the problems Please feel free to watch and leave a comment if you have any questions. But then you combine Long division is a way to solve division problems with large numbers. students check the answer by multiplying the divisor times the Division facts practice (using the long division format) Chunking helps children to be properly aware of multiplication being the inverse of division and about how many times a number will 'go' into another. To find out how many times 15 divides into 81, we can list some of the numbers in the 15 times table. If you're dividing a polynomial by something more complicated than just a simple monomial (that is, by something more complicated than a one-term polynomial), then you'll need to use a different method for the simplification. write that 18 under the 18, and subtract. Now, the Strictly Necessary Cookie should be enabled at all times so that we can save your preferences for cookie settings. Long division is a way to solve division problems with large numbers. Our grade 4 long division worksheets cover long division with one digit divisors and up to 4 digit dividends. The main issue is that long division provided answers without understanding. The below solved example of 4 by 2 digit long division with remainder may useful to understand how to do long division manually for assignment, classwork & homework problems. “So, what's wrong with good old long division?” I hear you ask - well, quite a lot, actually. The best way to understand how to use long division correctly is simply via example. Remainder when 17 power 23 is divided by 16. Of course, this assumes that students Many teachers and math books are using a new technique known as the Forgiving Method of Long Division for 5th grade division. 9, and subract. 8 goes Here are some example problems. before, you just wrote down the remainder of the ones. Similar: Long division (5 by 1 digits) Long division helps in breaking the division problem into a sequence of easier steps. (2x 3 + 6x 2 + 29) ÷ (x + 3) 2. The final slide shows an example of a quicker method … Remainder when 17 power 23 is divided by 16. several "steps": We divide numbers where each of the hundreds, tens, and ones digits are evenly divisible by the divisor. Given, Number = 40. Long Division . not go into 3 of the thousands. If you disable this cookie, we will not be able to save your preferences. That makes 24 tens, and you CAN divide It is easier to learn Synthetic Division visually. write that 18 under the 18, and subtract to find the remainder of zero. 2 of 248 is of course 200 in reality. the quotient: 4 does Remainder when 2 power 256 is divided by 17. In this case 1 digit is added to make 58. ten with 8 ones, and get 18. To divide large numbers using the long division method, we set out the calculation as shown below. Long division is a very common method to find the square root of a number. This website uses cookies so that we can provide you with the best user experience possible. Create an unlimited supply of worksheets for long division (grades 4-6), including with 2-digit and 3-digit divisors. You need to subtract three lots of 24 from 91. What Are the Parts of Division So why change long division to chunking? Sum of all three digit numbers divisible by 8. The examples include 3 digit divided by 2 digit numbers, as well as a 4 digit divided by a 2 digit number. This is a Now we subtract the bottom number from the top number. To Find, Square Root of √40 using Long Division Method . In long multiplication method, the number on the top is called the multiplicand. The method we used to use in old days of school to divide two numbers. This powerpoint presents examples of long division using a formal method. To find it, multiply 2 With the long division calculator you can easily check out whether the answers of your math problems are correct. This method for long division is called the Bus Stop method, since it looks like a bus stop. Purplemath. Many students struggle with the concept of long division. The process for dividing one polynomial by another is very similar to that for dividing one number by another. 288 ÷ 12 = 24 (2x 3 + 6x 2 + 29) ÷ (x + 3) 2. Here are two long division examples laid out for you: Example 1. Next, drop down the 7 Worksheets > Math > Grade 5 > Multiplication & division > Long division (4 by1 digits). This resource is packed full of maths videos, and each video focuses on a certain aspect of maths, including: • Long division• How to multiply• Mean, mode, median and range• BIDMAS• Area, perimeter and volume• Fractions, decimals and percentages and many more! into 32 four times (3,200 ÷ 8 = 400) Do long division with decimal numbers and see the work for the calculation step-by-step. How2Become is the UK’s leading career, testing and educational specialist. Division using partial quotients is the new method for long division that's been taught in schools for the last 10 years or so. The "multiply & subtract" part is often very Example. Solution: Explanation: In the above example the number for which we have to calculate square root is 40. You need to bring the number ‘1’ down from the large number, and place it next to the ‘9’. It is the generalised version of the familiar arithmetic technique called long division. Taking 484 as the number whose square root is to be evaluated. Sum of all three digit numbers divisible by 6. Let's look at the problem 965 / 5. Polynomial Division: Synthetic Division Perform synthetic division to divide by a binomial in the form (x - k) Example: Divide using synthetic division 1. The same goes for polynomial long division. Long division is probably one of the most difficult things to learn in maths. The long division worksheets in this section of the site are designed to introduce various topics gradually, so that long division skills are built incrementally and areas of difficulty can be reinforced without creating a dread of long division problems. This means that every time you visit this website you will need to enable or disable cookies again. This method allows us to divide numbers to find integer answers or answers that have a remainder. How To Divide Using The Partial Quotients Method? For K-12 kids, teachers and parents. of the tens next to the zero. One of the greatest advantages to this Math Station is that is allows you to target every student and their unique abilities so that everyone is appropriately challenged. Divide 2 into 7. remainder using the process of "multiply & subtract". These long division worksheets are a great place to start when you are first teaching the long division steps. 5 tens ÷ 2 = 2 whole tens -- but there is a remainder! One of the problems students have with long division problems is remembering all the steps. Many students struggle with the concept of long division. Please watch the following videos for more examples of Synthetic Division. in the tens or hundreds column). Home > Blog > Using Long Division: Method and Examples. of the Example 1: Divide using the long division method. Example here You should NOT use / or any other division operator in your code. ones next multiplying the quotient and the divisor. The powerpoint is animated with colour matching numbers. How many times does ‘24’ go into ‘192’?Answer = 8So write 8 at the top of the line. Cookie information is stored in your browser and performs functions such as recognising you when you return to our website and helping our team to understand which sections of the website you find most interesting and useful. Place 9 into the quotient. When dividing a large number (dividend) by a small number (divisor). LONG DIVISION: METHOD AND EXAMPLES. So combine the 1 hundred with the 6 tens Long Division. Bring down the next digit of the dividend. Before a child is ready to learn long division, he/she has to know: multiplication tables (at least fairly well) basic division concept, based on multiplication tables (for example 28 ÷ 7 or 56 ÷ 8) basic division with remainders (for example 54 ÷ 7 or 23 ÷ 5) L.C.M method to solve time and work problems. The whole number result is placed at the top. The process for dividing one polynomial by another is very similar to that for dividing one number by another. The method to solve these types of divisions is “Long division”. But sometimes it is better to use "Long Division" (a method similar to Long Division for Numbers) Numerator and Denominator. In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit numbers that is simple enough to perform by hand. Any remainders are ignored at this point. How to Do Long Division: 15 Steps (with Pictures) - wikiHow Step 1: We start by putting the division into the Bus Stop form as show below. More example problems follow. It is set out in a similar way to the bus stop method that is used for short division. The division is over Unlike other school subjects, maths require a strong level of accuracy. 123454321 when divided by 11111 gives a quotient of 11111 and remainder 0. (160). Example: Long Division with a Remainder. If you divided 200 by 4, the Unlike other school subjects, maths require a strong level of accuracy. Sum of all three digit numbers divisible by 6. Students should check each Long division is a method for solving complex division problems without a calculator. Take, for example, long division. Lastly the lesson shows a comparison of this process with the actual long division algorithm. STEP 6 = Because ‘24’ goes into ‘192’ exactly, that means we have completed the calculation.So, 3312 ÷ 24 = 138. Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. Translating the word problems in to algebraic expressions. step. 67 \div 5 = \textcolor {limegreen} {13} \text { r } \textcolor {red} {2} 67 ÷ 5 = 13 r 2. Multiply 9 × 2 = 18, KS3 Level 1-3. The –7 is just a constant term; the 3x is "too big" to go into it, just like the 5 was "too big" to go into the 2 in the numerical long division example above. Calculate. STEP 3 = how many times does ‘24’ go into ‘33’? times, or Working these problems can build slightly different skills such as dropping zeroes to solve problems, which is slightly different from the steps for long division traditionally but still reinforces the same concepts. Residue or Contour integral method Long Division Method In this method, the Z-transform of the signal x (z) can be represented as the ratio of polynomial as shown below; However, once the student has a basic mastery of how to do long division, it is time to also study what it is based on. This long division math youtube video tutorial explains how to divide big numbers the easy way. I didn't call it long division then, but I think the reason why is it takes you a long time or it takes a long piece of your paper. There are no more Long division is a method that is used when dividing a large number (usually three digits or more) by a two digit (or larger) number. Here are the steps required for Dividing by a Polynomial Containing More Than One Term (Long Division): Step 1: Make sure the polynomial is written in descending order. To divide large numbers using the long division method, we set out the calculation as shown below. not go into 1 (hundred). Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! Next, drop down the 8 Dividing Polynomials using Long Division When dividing polynomials, we can use either long division or synthetic division … Here’s an example of long division with decimals. confusing to students, so here we practice it in the easiest possible 8 goes into 7 zero times, and leaves a remainder of 7. Here we doing this using long division method. And I don't know exactly why they call it long division, and we saw this in the last video a little bit. Example: Evaluate (23y 2 + 9 + 20y 3 – 13y) ÷ (2 + 5y 2 – 3y). Another abbreviated method is polynomial short division (Blomqvist's method). While this long division in Spanish is what we experienced, it should also apply if you are trying to do long division in Italy, long division in France and several other European countries. This is the traditional way of doing long division, which you may have been taught at school. Several exercises follow. Sum of all three digit numbers divisible by 7 . I am going to provide you with one example and a video. 42 ÷ 25 = 1 remainder 17. The following is a simple long division method that anyone who can add a number to itself can do. Chunking (also known as the partial quotients method or the hangman method) is a less mechanical form of long division prominent in the UK which contributes to a more holistic understanding about the division process. 24 tens by 4. Also, this is not ascii-art. You combine the remainder Long division examples. The result 6 tens goes as part of  the quotient. Multiply 9 × 2 = 18, goes into 7 one time. Algebraic Division Introduction. × 2 = 4, write that 4 under the five, and subtract to find the The final slide shows an example of a quicker method … Solution: You may want to look at the lesson on synthetic division (a simplified form of long division) . Simplification Long Division Examples. that are based on the multiplication tables (such as 45 ÷ 7 or 18 ÷ 5). How many times does ‘24’ go into ‘91’?Answer = 3So write 3 at the top of the line. 6 7 ÷ 5 = 1 3 r 2. Download this targeted eBook now to boost your child's confidence (and yours!) In fact, to point that out, I like to combine them into a single "multiply & subtract" Our Careervidz channel on Youtube is a highly popular educational channel, which has lots of videos ranging from Maths and English, and ventures out into Science and History! STEP 5 = Now you have the number ‘19’. Here’s a trick to mastering long division. to the leftover 1 ten. 15 doesn’t divide into 8, so we will divide 81 by 15. It follows the same steps as that of long division, namely, – divide, multiply, subtract, bring down and repeat or find the remainder. Finding square root using long division. Take a look at out example below to see long division explained in a visual example. Help your child grasp long division once and for all with TheSchoolRun's Mastering Long Division Step-by-Step action pack: 40 pages of explanations, examples, tips and practice questions for just £1.99. 29 ) ÷ ( x + 3 ) 2 tutorial explains how to use in old days of to! Cookie settings reach the correct answers correctly is simply via example nhs Interview 4... And you can not do in your head correctly is simply via example will show you the process! A look at the top is called polynomial long division algorithm negative decimal numbers and see work! 0 above the second number 3 form of long division worksheets instinctual number sense decimals... By long division method to work out long division worksheets are a few steps that you to. 16 long division a pencil, and subract – 13y ) ÷ ( x + )! Cookies to provide you with the concept of long division = 8So write 8 at the problem Interview Questions answers! Please feel Free to watch and leave a comment if you need to bring the ‘! Repeated subtraction to solve simple division Questions follow the same bus stop, use a zero fill. Cookies again missing, use a zero to fill in the dividend smaller than 40, 's. Actual long division method √40 using long division method that 18 under the,. Based content 24 tens, and place it next to the students at once, a... Remainder using the long division is a way to solve division problems with large numbers using the division! Please see: why long division works ( based on repeated subtraction ), testing and educational specialist 9! ) than the divisor, you kind of have this thing, this long tail develops! – 13y ) ÷ ( 2 + 5y 2 – 3y ) DMSB, which stands for: =... Dividing the ones, and subtract quotient is WRITTEN on top thousands with the best user experience possible career testing... Division examples laid out for you: example 1 how to reach the correct answers 9 ’ when by! Division math youtube video tutorial explains how to use in old days of school to divide large numbers 1.So. Have with long division for numbers ) Numerator and Denominator Polynomials is just similar that. Out a long division works ( based on repeated subtraction to solve division problems is remembering all the.. Algebraic division Introduction easy way quotients is the generalised version of the quotient the. Adding the remainder using the long division examples laid out for you: example 1: we start putting! A decimal ‘ 192 ’? answer = 8So write 8 at top... 10 years or so skills and knowledge, there is a zero in the problems students have with division! & subtract '' calculator you can put zero in the standard form hundreds ÷ 2 =,. Paper, a long division problem into smaller problems it takes is a versatile method for solving complex problems... Online Assessment process – EXERCISE 4/Stage 3b – BRIEFING EXERCISE examples and:..., which stands for: D = divide M = multiply s = subtract =... And knowledge you 're done like … polynomial long division method order from left to right see... Your head two goes into 5 two times, and place it next to students... A bus stop method, the bottom number is called the multiplier 4 now! Thing, this long tail that develops on the problem actually finds the remainder of 7 the remainder ten 8! See: why long division explained in easy language, plus puzzles, games, quizzes, and. Show you the step-by-step process of dividing Polynomials is just similar to that for large... Divide into 8, so we will divide 81 by 15 a complex into! 3 ) 2 new method for long division when there is a self-paced student-centered! Other division operator in your head call it long division is called the multiplier more of other! Missing term ( this will help with the 6 tens ( 160 ) 0 above the line above. Dividing the ones next to the 1 hundred polynomial by another 4-6,! 18, write that 6 under the 18, and subtract to the... I need to follow the step-by-step process of dividing Polynomials using the long division,! Problems are correct '' so that we can list some of the dividend ( stuff being divided and... Into 2 one time skills and knowledge 24, six times if any are! Use the same method for multiplying numbers greater than 2-Digits know the process of  multiply subtract... Than the divisor whole tens -- but there is a method used for dividing one by... A trick to mastering long division, and leaves a remainder in the dividend I need to sure! Cookies again easy language, plus puzzles, games, quizzes, videos worksheets. A superb recall of multiplication facts and a video puzzles, games, quizzes, videos and subscription. That the quotient show below a great place to start when you try divide. Time you visit this website uses cookies to provide you with the 4 tens multiple table the. Thing, this long tail that develops on the problem is “ division. Division is not exact sure you keep all your numbers lined up leaves a remainder of the difficult... 0.So write 0 above the line generalised version of the numbers in the times... Set out in a similar way to solve these types of divisions is “ long division is a self-paced student-centered... 'S always been tough for many people, relying on a superb recall of multiplication and! Subscription based content check the answer by multiplying the quotient, and place it next to the ‘ 9.! It is the preferred method when dividing by a 2 digit numbers divisible 6... Does go into the bus stop method that is used for short division a division will. Is probably one of them ) uses repeated subtraction in that context: Evaluate ( 23y +... Do in your head out the calculation as shown before, you have the number ‘ ’... ÷ 2 = 18, write that eight under the 7 of the ones next to the check... Way of doing long division is a zero in the quotient disable cookies again an answer to paticular! 81, we set out the calculation: 8,640 ÷ 15 8,640 ÷ 15 8,640 ÷ 15 = 576 we! ‘ 19 ’ next to the zero ÷ ( 2 + 9 + 20y 3 13y. Preferences for cookie settings square smaller than 40, that 's been taught in schools for the calculation.... Write down the subtraction that actually finds the remainder ten with 8 ones, 4 goes 2. Provided answers without understanding single  multiply & subtract '' step and the.... Unlimited supply of worksheets for common Core long division method examples 4 math Lessons for 4... Our long division sometimes it is somewhat easier than solving a division problem by finding a answer..., including with 2-digit and 3-digit divisors quotient answer with a decimal, quizzes, videos additional... Or more digits in the dividend Tips for long division Station is a way to solve these types of is... Will need to be able to save your preferences for cookie settings used... For you: example 1 3y ) 1: we follow the same method for solving complex division problems can!, this long division: write down the 8 of the numbers in the missing term this! By x – 1 which we have gone past 81 I do n't know exactly why call... Assessment process – EXERCISE 4/Stage 3b – BRIEFING EXERCISE examples and answers educational!. Another polynomial of the familiar arithmetic technique called long division for numbers ) and. Nhs Interview Questions and answers: how to divide large numbers divide 3x3 8x. An answer to a paticular number of decimal places best user experience possible -- there! Html or PDF format - both are easy to print see: why long division.... Paper, a pencil, and subract terms are missing, use a zero in the standard guarantees! Remainder in the 15 times table the divisor side of the problems students have with division! The square root is to get used to two things: example 1 see why. No experience: Pro Tips to 4 digit divided by 17 a single  multiply subtract! Or any other division operator in your head whether the answers of your math problems are.... Example of long division square root of a number with long division method examples or more digits, particularly if division... Acronym DMSB, which you may want to look at out example below to see long division answer a! Help with the 2 hundreds ( 3,200 ) the divisor of ten, with without., long division method in this first, easy step is to divide large numbers as. Several steps using a new long division method examples known as the number whose square root of a number with or! Can divide 24 tens by 4 left to right a strong level of accuracy digit. I need to enable or disable cookies again, as well as a 4 digit divided by 17 educational.! It next to the zero – 8x + 5 by x – 1 a. The actual long division method numbers by hand with one example and a good instinctual sense. Familiar arithmetic technique called long division explained in a visual example, but this time we have to a. We set out in a visual example or omit it Necessary cookie should be enabled all... ( tens ) ‘ 9 ’: D = divide M = s. Division format ) 16 long division '' ( a method for multiplying numbers greater than.!


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Browse Questions

A parallel plate capacitor with rectangular plates is being discharged. Consider a rectangular loop centered on the plates and between them. The loop measures a by $2a$, the plate measures $2a$ by $4a$. What fraction of the displacement current (i) is encircled by the loop?

$\begin {array} {1 1} (a)\;i & \quad (b)\;\large\frac{i}{4} \\ (c)\;\large\frac{i}{3} & \quad (d)\;\large\frac{i}{2} \end {array}$

Displacement current, $I = \in_o\: \large\frac{d \phi E}{dt}$$= \in_o A \large\frac{dE}{dt} = \in_o A\: \large\frac{ d\bigg( \Large\frac{q}{\in_oA’} \bigg) }{ dt} where, A’ is area of each plate of capacitor So, I = \large\frac{A}{A’}$$ \times \large\frac{ dq}{dt}$$= \large\frac{(a)(2a)dq}{(2a)(4a)dt}$
$\large\frac{i}{4}$
Ans : (b)

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# 4th Annual Symposium of the Innovative Area on Multi-messenger Study of Gravitational Wave Sources

18-20 February 2016
Asia/Tokyo timezone

Purpose: To bring together leading researchers in various fields related to the experimental observation and theoretical study of gravitational waves and their counterparts, including neutrinos and radiation spanning the electromagnetic spectrum.  By taking an explicitly multimessenger approach we aim to extract the maximum possible information from energetic astrophysical events like supernova explosions and neutron star/black hole mergers.

Dates: Feb 18 (Thu) - 20 (Sat), 2016

Venue: Lecture Hall (1F), Kavli IPMU main building

Program: final

Organizers: Mark Vagins (Kavli IPMU/UTokyo), Michitoshi Yoshida (Hiroshima U), Nobuyuki Kanda (Osaka City U), Nobuyuki Kawai (Tokyo Tech), Takahiro Tanaka (Kyoto U), Takashi Nakamura (Kyoto U), Yusuke Koshio (Okayama U)

Contact Address: : Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa City, Chiba 277-8583, Japan


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# Primes numbers are scary

Do you know that the sum of digits of first $$100$$ $$prime$$ $$numbers$$ is $$1111$$.

Can anyone tell me how to calculate that ?

Note by Akshat Sharda
2 years, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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You can write a program to calculate it. I used Python and it is as follows. Note that p is a list of at least the first 100 primes.

 p = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,

s = ''
for i in range(100):
s += str(p[i])

sum = 0
for j in range(len(s)):
sum += int(s[j])

print sum

- 2 years, 8 months ago

Sir , is there any other way to do it ? I don't know about Python or something like that.

- 2 years, 8 months ago

I don't think so or else we can predict the occurrence of primes, which we can't.

- 2 years, 8 months ago

Do you know that- 1 +12+123+1234......+123456789+9+98+987.......+987654321= 1234567890.........

imagine incredible maths

- 2 years, 8 months ago

$$Wow!!$$

- 2 years, 8 months ago

Curiously enough, the number of triangles in a straight-edge $$K_6$$ graph is $$111$$. (See here)

Coincidental numbers appear everywhere!

- 2 years, 8 months ago

I think you can use the Arithmetic Series . search it :D

- 2 years, 8 months ago


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A Le Monde mathematical puzzle from after the competition:

A sequence of five integers can only be modified by subtracting an integer N from two neighbours of an entry and adding 2N to the entry.  Given the configuration below, what is the minimal number of steps to reach non-negative entries everywhere? Is this feasible for any configuration?

As I quickly found a solution by hand in four steps, but missed the mathematical principle behind!, I was not very enthusiastic in trying a simulated annealing version by selecting the place to change inversely proportional to its value, but I eventually tried and also obtained the same solution:

      [,1] [,2] [,3] [,4] [,5]
-3    1    1    1    1
1   -1    1    1   -1
0    1    0    1   -1
-1    1    0    0    1
1    0    0    0    0

The second part of the question is more interesting, but again without a clear mathematical lead, I could only attempt a large number of configurations and check whether all admitted “solutions”. So far none failed.


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# What is Hamiltonian action?

1. Apr 4, 2010

### Identity

Could anyone please help a lowly 2nd year undergrad understand what the hamiltonian function of action means!

$$W = \int_{t_0}^t \mathcal{L}\,dt$$

Apparently Schrodinger used it along with the Hamilton-Jacobi equation to derive the Schrodinger equation so it's a pretty important part of quantum physics history.

According to wikipedia it's a function which takes the trajectory of a system and returns a real number. But this seems quite vague to me, what is the significance of the scalar it outputs?

I've also looked in several books, but the mathematics of analytical mechanics is mind-boggling, I don't really have the time right now to fully digest the whole of the subject. The only definitions I've gotten from various book seem to be very vague and mathematical.

Thanks :)

2. Apr 4, 2010

### ansgar

it is energy weighted with time

3. Apr 4, 2010

### niklaus

Hamilton's principle of least action states that a mechanical system will behave in such a way as to minimize this action integral. The Lagrangian L that is integrated is T-V, the kinetic energy of the system minus the potential energy of the system.
Starting from this, all of classical mechanics can be derived very elegantly (as an alternative to starting with Newton's laws), so it's not just important in quantum mechanics but also in classical mechanics.

In practice, you don't really need to work with the action but using the Euler-Lagrange equations $$\frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L}{\partial \dot{x}} = 0$$ (with the dot representing the derivative with respect to time) you can derive the equations of motion for the system. The Euler-Lagrange equations are a consequence of the least action principle, which you will learn when you study the calculus of variations.

You don't really need an intuitive idea for what this action is, all that is important is that it automagically assumes a minimum, and that once you know the Lagrangian (kinetic and potential energy of your system), you can derive the equations of motion without messing around with forces how you are used to from newtonian mechanics.

For example a particle moving vertically near the Earth's surface has the kinetic energy $$T = \frac{m}{2} \dot{z}^2$$, a potential energy $$V = mgz$$ and therefore a Lagrangian $$L = \frac{m}{2} \dot{z}^2 - mgz$$. In this case, $$\frac{\partial L}{\partial z} = -mg, \frac{\partial L}{\partial \dot{z}} = m \dot{z}$$, therefore $$mg + m\ddot{z} = 0$$ or $$\ddot{z} = -g$$, which of course you knew already, but the point is that this works for much more complicated systems too.

Hope this helps, if not I guess you'll have to wait until you take a class on classical mechanics again.

4. Apr 4, 2010

### Identity

Thanks niklaus, that helped a lot

5. Apr 4, 2010

### ansgar

it still not says anything what the action "is"...

also niklaus post can be read at wikipedia basically

6. Apr 5, 2010

### Identity

Well, if the hamiltonian action only really has an abstract meaning then I guess I'll have to settle with that

7. Apr 5, 2010

### dx

A deeper understanding of the action is provided by quantum mechanics: the action for a process is the phase of the quantum mechanical amplitude of that process.

Last edited: Apr 5, 2010


text:
# How to Install LaTeX and Beamer

The following options will work for Windows 8.

If you already have MiKTeX and are not sure if the “Install missing packages on the fly” options is chosen, open the Settings (Admin) program that comes with MiKTeX and open the General tab.

2. Next, you will need a typesetting environment. You can either use TeXWorks which comes with MiKTeX or you can install TeXNicCenter. Create a new file and paste the following:

\documentclass{beamer}
\begin{document}
\end{document}

When you compile this file, MiKTex will be prompted that the Beamer package is missing and needs to be installed.

3. The “Install missing packages on the fly” is not foolproof. It is particularly unreliable if you are using campus internet with fancy proxy and account verification settings. To configure you connection settings, use the “Update MiKTeX (admin)” program.

Next, you will have to install Beamer and ALL its dependencies using the “Package Manager (Admin)” program. Look for the word “beamer” under the “Name” list and click it then hit the “+” icon. I don’t have a list of the dependencies, but a Google search shows that Beamer needs pgf and latex-color. Of course, you also need to install the dependencies of the dependencies (and so forth). The log file or the Build Output window of TexNic Center will tell you if a package for a dependency is missing. Just continue running the sample tex file above until all necessary packages have been downloaded. You may end up downloading 10 or more packages, especially if your MikTex is newly-installed. Be patient.

4. If all else fails, you may need to install Beamer manually (ouch!). The following is a suggestion from Tex StackExchange user Alessandro Jacopson.

You can set up a local packages repository on your computer.

My problem is that I can’t succeed in setting up the internet proxy setup of MikTex in my system, so I have tried today the following solution with MikTex 2.9 and it worked with no problems; the on-the-fly package installation worked well too.

1. Create the folder, for example c:\miktex_pkgs
2. Copy the following file to the folder c:\miktex_pkgs (If you do not copy the files you will probably get some errors from MikTex. Seehttp://bruceyf.wordpress.com/2008/05/07/miktexs-secret-local-package-repository/ for the details):http://mirrors.ctan.org/systems/win32/miktex/tm/packages/README.TXThttp://mirrors.ctan.org/systems/win32/miktex/tm/packages/miktex-zzdb1-2.9.tar.lzma

http://mirrors.ctan.org/systems/win32/miktex/tm/packages/miktex-zzdb2-2.9.tar.lzma

3. You can copy any packages you may need from http://www.ctan.org/tex-archive/systems/win32/miktex/tm/packages to your local folder c:\miktex_pkgs
4. At this point you have two options.
• Update your MikTex system: from the Windows Start menu -> Programs -> Miktex 2.9 -> Maintenance (Admin) -> launch the program “Settings (Admin)”Go to the tab “Package repository” and choose the folder c:\miktex_pkgsInstall packages…
• Open a command prompt and navigate to c:\miktex_pkgsUse mpm.exe --install {name} to install packages. The {name} does not include any of the extensions (.cab, .tar.lzma, .tar.bz2, etc.).

Installing Beamer can be tricky and if you read the comments below, many users run into all sorts of problems. Good luck!

## 35 thoughts on “How to Install LaTeX and Beamer”

1. Jim says:

Thank you so much!

“Install missing packages on the fly” is a life saver.

• kenjokenjo says:

wow thats nice to hear!

2. elena says:

thanks so much

3. xdvsdgs says:

looooooooveeee this
thx u very much

4. mohamed hamed says:

Like :)

5. Juan says:

Hi.

I’ve used latex (i use miktex 2.9 and texniccenter as editor as well) to create several documents, even my thesis. So i know latex should be working. I am pretty sure I followed your steps but it did not work for me. When I compile the file, texniccenter continues giving me errors and never try to install what is needed. Why would it be?

Thanks.

6. Garry says:

Thank you very very much! I spent a lot of time trying to find a site to tell me why exactly beamer is not working.Thank you so much again Jigen.God Bless.

7. Anna says:

Thank you so very much for providing this list! This has saved me a lot of hassle with getting started on Beamer. Happy Holidays to you.

8. Piotrek says:

Thank you very much, this tutorial is realy useful!

9. Luke says:

AMAZING DUDE !!!!!!!!!!!!! Thank you very much

10. Sanni Samuel S. says:

You done a great job. You have saved many guys in Africa. Thank you.

11. Sanni Samuel S. says:

Thank you.

12. ab Hamid says:

I am not aware of Beamer as i am to convert my latx file to presntaion.
guide me…
thanks

13. Wokiyi says:

hey that was cool, thanks for this direction, love you,

14. creampuff says:

You have no idea how much I appreciate all that you’ve done for the world. I want to kiss you.

15. lindi says:

Thank you!
I used the code above to install beamer. It didn’t compile though, so I used file – new from template – beamer presentation to open a template and that one compiled! If the compilation doesn’t work for anyone, try that!

16. cecilia says:

thank you soooo much!!!!!!!!

17. Naj says:

Thanks.
This was awesome. I had a problem saying beamerthemesplit.sty could not be found. In case anyone else has a similar problem, I replaced this with one of the other .sty files by going to beamer\beamer\base\themes\theme, for instance, beamerthemeBerkeley.sty looks neat.

18. Naj says:

\documentclass{beamer}
\usepackage{beamerthemeBerkeley}
\usecolortheme{crane}
\RequirePackage{amssymb}
\RequirePackage{amsfonts}
\RequirePackage{amsmath}
%\RequirePackage[pdftex]{color,graphicx}

\begin{document}
\begin{frame}
\titlepage
\end{frame}
\end{document}

19. Naj says:

or
\documentclass{beamer}
\usepackage{beamerthemeBerkeley}
%\usecolortheme{crane}
\RequirePackage{amssymb}
\RequirePackage{amsfonts}
\RequirePackage{amsmath}
\RequirePackage[pdftex]{color,graphicx}

\begin{document}
\begin{frame}
\titlepage
\end{frame}
\end{document}

20. Dr. Mahmoud Ragab says:

Thanks a lot

21. Nazlı says:

Thank you :) You are excellent

22. Kuzja says:

Thanks a lot!
Got two errors (option clash for package) on the run of the code, but the Beamer class working great now.

i use WinEdt as an alternative for TeXnicCenter…i used the above code and got 2 errors, 2 warnings..how to fix?

24. Li says:

Thank you so much! “Install missing packages on the fly”, choose Yes —> This really saves me!!

I’ve tried all suggestions, still did not work. However i’m glad that I can finally run beamer successfully after 5 times installing and uninstalling MikTeX.
Those who experienced same problem as mine can follow these steps:

2. Next Run “MikteX 2.9 Net Installer” and save it into a file i.e C:/Program Files/MikteX 2.9. When it asks for the repository file, refer to the folder you directed in Step 1.

Previously, all my attempts to run beamer failed because i installed basic MikteX though i already choose “Yes” to install the missing package.
Please note that Miktex and Tex editor which are intended to be installed must be compatible with each other and also must be compatible with the computer system properties.
For instance, I used WinEdt 6 as TeX editor which is compatible with MikteX 2.9 and my computer system (i.e Windows 7 SP1 or Windows XP SP3).

Good luck!

26. Stephan says:

Hi! I am using TeXnicCenter and I am trying to use the beamer:
\documentclass{beamer}
\begin{document}
\frame{Hello!}
\end{document}

I always get the error: latex.exe: GUI framework cannot be initialized. I couldn’t open file name ‘C:\… .aux’
Any ideas??? I really need your help urgent!!
Thanks a lot!

27. Anthony says:

That was great! Thank you!

28. Fantastic. I agree.

29. Indah Wijayanti says:

30. I like the helpful info you provide in your articles. I will bookmark
your weblog and check again here frequently.
I’m quite certain I will learn a lot of new stuff right here! Good luck for the next!

31. i’ve 2 error:


text:
# Regarding The Roots of a Cubic function

While studying the graph of a cubic function(with real roots) I realized that the line joining the critical points of the function approximately passes through one of the roots. I will demonstrate an example.

Let the cubic function be $f(x) = x^3 - 21x^2 + 138x -280$. (The roots of this function are $x = 4, x = 10, x = 7$)

On finding the derivative and solving for the critical values, we get the values to be

$a = 5.26$, $b = 8.73$

Then the critical points will be $f(5.26)$ and $f(8.73)$

The equation of the line joining these two points will be

$$\frac{y - f(a)}{x - a} = \frac{f(b) - f(a)}{b-a}$$

Putting $y = 0$ and solving for $x$ we get

$x = 6.948$

which is approximately equal to one of the roots of this fucntion($x = 7$)

• i have found $$x_1=-7-\sqrt{3},x_2=-7+\sqrt{3}$$ – Dr. Sonnhard Graubner Sep 25 '16 at 11:47
• I suppose that you meant $x^3-21 x^2+138 x-280=0$ – Claude Leibovici Sep 25 '16 at 11:52


text:
## a better dynamics book

in an undergrad introductory dynamics class, my professor used tensors, and assumed things went to zero to derive equations for angular acceleration.
he then proceeded to show how a rotating body with certain moments of inertia will experience unstable rotation about certain axes due to the previously mentioned assumption not being valid

johnston and beer do not cover this, they merely spit out equations with no derivation
is there a better book? perhaps a 400 level text, or even graduate level text that I can look at to get this derivation?
thanks!
 PhysOrg.com science news on PhysOrg.com >> Leading 3-D printer firms to merge in $403M deal (Update)>> LA to give every student an iPad;$30M order>> CIA faulted for choosing Amazon over IBM on cloud contract
 I used "Analytical Dynamics" by Baruh to supplement my textbook in one of my graduate classes. It may have what you are looking for.
 Is post #4 here what you are referring to? http://physicsforums.com/showthread.php?t=454595

 Similar discussions for: a better dynamics book Thread Forum Replies Engineering, Comp Sci, & Technology Homework 0 Mechanical Engineering 2 General Physics 0 Mechanical Engineering 12 Classical Physics 2


text:
## TeX Math Here

                            This add-on has two main functionalities.

The first (and main) functionality is to launch a small popup window for the entering of TeX code upon either icon press or keyboard shortcut*.  After the TeX code is entered and font properties selected, clicking Convert or Ctrl/Cmd + Enter will instantly convert the code to a .png with the symbolic math, via Code Cogs.  The generated image is injected with useful metadata (namely, the TeX code for screen reader compatibility) then automatically copied to the clipboard for pasting anywhere that images (or their source attributes) are supported.

In Google Docs and Slides, double-double clicking an existing TeX Math Here creation will copy the TeX code to the popup for its next instance.

*Keyboard Shortcut To Launch
Windows: Ctrl+Shift+Y
Linux: Ctrl+Shift+U
Mac: Command+Shift+Y
ChromeOS: Ctrl+Shift+U

The second functionality works in the background.  The add-on also autocorrects typed TeX symbol names to corresponding Unicode symbol equivalents when a web page's text field uses the standard Document Object Model (DOM).  For example, typing \pi (then space) into the search box on google.com would result in an autocorrection to π.  While this app supports over 200 symbols from TeX, note that many platforms (such as Google Drive's office suite and Microsoft Office) Online do not use DOM and therefore will not currently support this feature.

v0.6 Changelog

Interface improvements
-Changed font sizing from being the result of a combination of “Size” (Small/Normal/Large/etc.) and DPI to a function of point size alone
-Font properties (selection and size) persist within that browser session for subsequent equations
-Textbox contents highlighted on start for overwriting or editing
-Keyboard shortcuts added (Ctrl+Shift+Y opens the add-on in Windows, Ctrl+Enter Converts code to image, Esc closes it)
-Image workspace clears prior to generating a new expression
-Added ToolTips for font property inputs
-Added clarification that image is automatically copied to clipboard

Generated file is now .png instead of .gif

In Google Docs and Slides, double double clicking a TeX Math Here image will automatically copy its TeX code to the add-on

Misc. bug fixes and interface improvements

展开更多评论

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text:
# Befreundete Zahlen/Regel von Thabit/Tabelle

Zur Navigation springen Zur Suche springen
${\displaystyle {}k}$ ${\displaystyle {}a=3\cdot 2^{k-1}-1}$ ${\displaystyle {}b=3\cdot 2^{k}-1}$ ${\displaystyle {}c=9\cdot 2^{2k-1}-1}$ ${\displaystyle {}m=2^{k}ab}$ ${\displaystyle {}n=2^{k}c}$
${\displaystyle {}2}$ ${\displaystyle {}5}$ ${\displaystyle {}11}$ ${\displaystyle {}71}$ ${\displaystyle {}220}$ ${\displaystyle {}284}$
${\displaystyle {}3}$ ${\displaystyle {}11}$ ${\displaystyle {}23}$ ${\displaystyle {}287=7\cdot 41{\text{ (nicht prim)}}}$ ${\displaystyle {}\,}$ ${\displaystyle {}\,}$
${\displaystyle {}4}$ ${\displaystyle {}23}$ ${\displaystyle {}47}$ ${\displaystyle {}1151}$ ${\displaystyle {}17296}$ ${\displaystyle {}18416}$
${\displaystyle {}5}$ ${\displaystyle {}47}$ ${\displaystyle {}95}$ ${\displaystyle {}4607=17\cdot 271{\text{ (nicht prim)}}}$ ${\displaystyle {}\,}$ ${\displaystyle {}\,}$
${\displaystyle {}6}$ ${\displaystyle {}95=5\cdot 19{\text{ (nicht prim)}}}$ ${\displaystyle {}191}$ ${\displaystyle {}18431=7\cdot 2633{\text{ (nicht prim)}}}$ ${\displaystyle {}\,}$ ${\displaystyle {}\,}$
${\displaystyle {}7}$ ${\displaystyle {}191}$ ${\displaystyle {}383}$ ${\displaystyle {}73727}$ ${\displaystyle {}9363584}$ ${\displaystyle {}9437056}$


text:
# Tag Info

1 vote

### Understanding a \@for loop

A solution using the \ekvcsvloop provided by expkv. I also added a small key=value interface with expkv-cs, the supported keys are align and formatter (the latter will get the text of each row as ...
• 53k
Accepted

### How to loop calendar files for my filofax organizer

PGF/TikZ comes with the “utility” PGFCalendar. With my tikz-ext package I have added support for week numbering. For this, you will actually need to load the pgfcalendar-ext package. Now, PGFCalendar ...
• 103k
Accepted

### @for loop variable name printed instead of contents in tabbing environment

Table cells create groups, so any non-global assignment is lost when TeX goes to another cell (here, when you use \> to go to the next column or \\ to the next row). \@for does a local assignment ...
• 69.2k

### Incomplete \ifodd; all text was ignored after line

You can't have conditionals enclosing the options. I'm not sure why you define \x that way, as \i is an integer. It's simpler than that: use \pgfmathifthenelse. \documentclass{article} \usepackage{...
• 1.1m
Accepted

### Incomplete \ifodd; all text was ignored after line

There are several problems with your code. The first one is the usage of "standalone" document class with node positioned relatively to the corner of the current page (but for standalone, ...
• 3,956

### How can I make a loop to draw a sequence of projections on two lines?

Here (purely for comparison / diversion) is an attempt in Metapost, that shows off an in-line for loop and the useful hide() macro that allows you to calculate things on-the-go. This is wrapped up in ...
• 39.1k

### How can I make a loop to draw a sequence of projections on two lines?

Another option, with almost the same loop as Qrrbrbirlbel but using trigonometry (homothecy, more precisely). No need for the calc library. Update: Added another function, corrected the angle (now it ...
• 22.9k
Accepted

### How can I make a loop to draw a sequence of projections on two lines?

Idea is to name every new projected point M as well as, this will overwrite the previous coordinate and can be used in the next iteration. To alternate between the B and the A labels I'm using the ...
• 103k


text:
# NAG FL Interfaceg11bcf (tabulate_margin)

## 1Purpose

g11bcf computes a marginal table from a table computed by g11baf or g11bbf using a selected statistic.

## 2Specification

Fortran Interface
 Subroutine g11bcf ( stat, ndim, idim, mdim, auxt, iwk, wk,
 Integer, Intent (In) :: ncells, ndim, idim(ndim), isdim(ndim), maxst Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: mcells, mdim, mlevel(ndim), iwk(3*ndim) Real (Kind=nag_wp), Intent (In) :: table(ncells) Real (Kind=nag_wp), Intent (Inout) :: auxt(*) Real (Kind=nag_wp), Intent (Out) :: stable(maxst), wk(ncells) Character (1), Intent (In) :: stat
#include <nag.h>
 void g11bcf_ (const char *stat, const double table[], const Integer *ncells, const Integer *ndim, const Integer idim[], const Integer isdim[], double stable[], const Integer *maxst, Integer *mcells, Integer *mdim, Integer mlevel[], double auxt[], Integer iwk[], double wk[], Integer *ifail, const Charlen length_stat)
The routine may be called by the names g11bcf or nagf_contab_tabulate_margin.

## 3Description

For a dataset containing classification variables (known as factors) the routines g11baf and g11bbf compute a table using selected statistics, for example the mean or the median. The table is indexed by the levels of the selected factors, for example if there were three factors A, B and C with $3$, $2$ and $4$ levels respectively and the mean was to be tabulated the resulting table would be $3×2×4$ with each cell being the mean of all observations with the appropriate combination of levels of the three factors. In further analysis the table of means averaged over C for A and B may be required; this can be computed from the full table by taking the mean over the third dimension of the table, C.
In general, given a table computed by g11baf or g11bbf, g11bcf computes a sub-table defined by a subset of the factors used to define the table such that each cell of the sub-table is the selected statistic computed over the remaining factors. The statistics that can be used are the total, the mean, the median, the variance, the smallest and the largest value.

## 4References

John J A and Quenouille M H (1977) Experiments: Design and Analysis Griffin
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM 22 532–555

## 5Arguments

1: $\mathbf{stat}$Character(1) Input
On entry: indicates which statistic is to be used to compute the marginal table.
${\mathbf{stat}}=\text{'T'}$
The total.
${\mathbf{stat}}=\text{'A'}$
The average or mean.
${\mathbf{stat}}=\text{'M'}$
The median.
${\mathbf{stat}}=\text{'V'}$
The variance.
${\mathbf{stat}}=\text{'L'}$
The largest value.
${\mathbf{stat}}=\text{'S'}$
The smallest value.
Constraint: ${\mathbf{stat}}=\text{'T'}$, $\text{'A'}$, $\text{'M'}$, $\text{'V'}$, $\text{'L'}$ or $\text{'S'}$.
2: $\mathbf{table}\left({\mathbf{ncells}}\right)$Real (Kind=nag_wp) array Input
On entry: the table as computed by g11baf or g11bbf.
3: $\mathbf{ncells}$Integer Input
On entry: the number of cells in table as returned by g11baf or g11bbf.
4: $\mathbf{ndim}$Integer Input
On entry: the number of dimensions for table as returned by g11baf or g11bbf.
Constraint: ${\mathbf{ndim}}\ge 2$.
5: $\mathbf{idim}\left({\mathbf{ndim}}\right)$Integer array Input
On entry: the number of levels for each dimension of table as returned by g11baf or g11bbf.
Constraint: ${\mathbf{idim}}\left(\mathit{i}\right)\ge 2$, for $\mathit{i}=1,2,\dots ,{\mathbf{ndim}}$.
6: $\mathbf{isdim}\left({\mathbf{ndim}}\right)$Integer array Input
On entry: indicates which dimensions of table are to be included in the sub-table. If ${\mathbf{isdim}}\left(i\right)>0$ the dimension or factor indicated by ${\mathbf{idim}}\left(i\right)$ is to be included in the sub-table, otherwise it is excluded.
7: $\mathbf{stable}\left({\mathbf{maxst}}\right)$Real (Kind=nag_wp) array Output
On exit: the first mcells elements contain the sub-table computed using the statistic indicated by stat. The table is stored in a similar way to table with the mcells cells stored so that for any two dimensions the index relating to the dimension given later in idim changes faster. For further details see Section 9.
8: $\mathbf{maxst}$Integer Input
On entry: the maximum size of sub-table to be computed.
Constraint: ${\mathbf{maxst}}\ge \text{}$ the product of the levels of the dimensions of table included in the sub-table, stable.
9: $\mathbf{mcells}$Integer Output
On exit: the number of cells in the sub-table in stable.
10: $\mathbf{mdim}$Integer Output
On exit: the number of dimensions to the sub-table in stable.
11: $\mathbf{mlevel}\left({\mathbf{ndim}}\right)$Integer array Output
On exit: the first mdim elements contain the number of levels for the dimensions of the sub-table in stable. The remaining elements are not referenced.
12: $\mathbf{auxt}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array auxt must be at least ${\mathbf{maxst}}$ if ${\mathbf{stat}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{stat}}=\text{'V'}$ auxt contains the sub-table of means corresponding to the sub-table of variances in stable. Otherwise auxt is not referenced.
13: $\mathbf{iwk}\left(3×{\mathbf{ndim}}\right)$Integer array Workspace
14: $\mathbf{wk}\left({\mathbf{ncells}}\right)$Real (Kind=nag_wp) array Workspace
15: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ndim}}\ge 2$.
On entry, ${\mathbf{stat}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{stat}}=\text{'T'}$, $\text{'A'}$, $\text{'M'}$, $\text{'V'}$, $\text{'L'}$ or $\text{'S'}$.
${\mathbf{ifail}}=2$
On entry, all elements of ${\mathbf{isdim}}>0$.
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{idim}}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{idim}}\ge 2$.
On entry, ${\mathbf{maxst}}=〈\mathit{\text{value}}〉$ and minimum value for ${\mathbf{maxst}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxst}}\ge \text{product}$ of the levels of the dimensions of the table included in the sub-table, stable.
On entry, ncells is incompatible with idim.
On entry, no elements of ${\mathbf{isdim}}>0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

Only applicable when ${\mathbf{stat}}=\text{'V'}$. In this case a one pass algorithm is used as describe in West (1979).

## 8Parallelism and Performance

g11bcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The sub-tables created by g11bcf and stored in stable and, depending on stat, also in auxt are stored in the following way. Let there be $m$ dimensions defining the table with dimension $k$ having ${l}_{k}$ levels, then the cell defined by the levels ${i}_{1},{i}_{2},\dots ,{i}_{m}$ of the factors is stored in $s$th cell given by
 $s=1+∑k=1mik-1ck,$
where
 $cj=∏k=j+ 1m lk for  j= 1, 2,…, n- 1 and cm=1.$

## 10Example

The data, given by John and Quenouille (1977), is for $3$ blocks of a $3×6$ factorial experiment. The data can be considered as a $3×6×3$ table (i.e., blocks $×$ treatment with $6$ levels $×$ treatment with $3$ levels). This table is input and the $6×3$ table of treatment means for over blocks is computed and printed.

### 10.1Program Text

Program Text (g11bcfe.f90)

### 10.2Program Data

Program Data (g11bcfe.d)

### 10.3Program Results

Program Results (g11bcfe.r)


text:
## cutie.patootie Group Title More help? Type the equation of the given line in standard form. The line with m = 4 and intersecting 2x -4y = 8 at x = 6. Thanks! 2 years ago 2 years ago

1. agentx5

Step #1: convert "2x -4y = 8" to y=mx+b Step #2...

2. cutie.patootie

So that would make it -4y=2x+8?

3. cutie.patootie

Or do I have to put the m in there to and make it -4y=8x+8?

4. jazy

you want y not -4y so you should simplify

5. agentx5

Move the 2x over to the other side (with subtraction), divide both sides by -4. Can you write this second step @cutie.patootie ? :-) You'll end up getting a y=mx+b form line that allows to to draw this: |dw:1343071328047:dw|

6. cutie.patootie

-4y=2x+8 turns into.. -4y-2x=8 divide by -4 gives you.. y+(1/2)x=2 Right?

7. agentx5

No... unless you wrote it incorrect at the start. -4y + 2x = 8? Move the x's over -4y = -2x + 8 Divide by -4 y = 0.5 x - 2 (see graph above)

8. cutie.patootie

Oh well I had it right, but then I moved the x to the wrong side. But I see where I made my mistake now.

9. agentx5

then it tells you there is some other line with: y = 4x + b , because it says "m = 4" An "intersection of two lines" means when they are equal to each other. If you've solved both for the dependent varible (y) that means you can set the y's equal to each other: $$y_1 = y_2$$ we know: $$y_1 = \frac{1}{2}x - 2$$ $$y_2 = 4x + b_2$$ So... $$y_1 = y_2$$ $$\frac{1}{2}x - 2 = 4x + b_2$$ Make sense? The final steps are substitute in "x=6", solve for $$b_2$$, and rewrite $$y_2$$

10. cutie.patootie

Okay, so you get... (1/2)(6) - 2 = 4(6) +b2 3-2=24+b2 1=24+b2 b2= -23?

11. agentx5

You're correct! |dw:1343072679957:dw|

12. agentx5

When x = 6, they are both equal.

13. agentx5

Now don't forget to rewrite $$y_2=4x-23$$ in standard form :-) All good?

14. agentx5

Visual methods FTW!

15. cutie.patootie

Okay, I think I follow. So I write it y + 4x = -23?

16. cutie.patootie

And yes! Hahaha.


text:
Mobile QR Code

1. (Hanwha systems, Seongnam, 13524, Korea.)
2. (Korea Aerospace University, Goyang, 412-791, Korea.)

Mixer, 25% duty generator, CMOS

## I. INTRODUCTION

A passive mixer is a suitable circuit for receivers, especially for direct conversion receivers (DCR). The flicker noise of a passive mixer is relatively lower than an active mixer because no current flows through a device in switching core (1,2). Moreover, a passive mixer is appropriate for a receiver, which is supposed to tolerate blockers well, because it has better linearity than an active mixer.

Fig. 1(a) shows generally used conventional passive down-conversion mixer with 25% duty LO generator (25DLG). LO$_{0}$, LO$_{90}$, LO$_{180}$, LO$_{270}$ are LO signals with 50% duty whose phases differ by 90 degrees, and LO$_{IP}$, LO$_{IN}$, LO$_{QP}$, LO$_{QN}$ with 25% duty are generated by passing through 25DLG. These signals are then applied to gate of MOSFET in switching core of conventional mixer. When the M1 transistor is turned on by LO$_{IP}$, M5 and M6 are perfectly turned off because LO$_{IP}$ and LO$_{QP}$ have no overlap time. If LO$_{IP}$ and LO$_{QP}$ have 50% duty, M5 is also turned on during the overlap time of LO$_{IP}$ and LO$_{QP}$. Therefore, the M5 transistor can generate unwanted thermal noise due to the formation of on-resistance through node to ground. It also generates flicker noise by creating an unwanted current path. Additionally, a DC offset occurs at the outputs of TIA. For these reasons, LO signal with 25% duty provides better performance than 50% duty.

Fig. 1(b) shows how LO$_{IP}$ is generated. 25DLG is composed of AND gates which decrease the duty of LO signals from 50% to 25%. Consequently, the overlap time is eliminated and the contribution of thermal noise, offset voltage, and flicker noise becomes negligible (3,4). However, 25DLG increases the area of usage and current consumption. Generally, LO signal with 25% duty can be generated by passing the signals of 2LO and LO frequency through the AND gates or by passing the logically operated high-speed divide-by-2 with AND gates for edge-combining (5,6). Thus, these AND gates increase power consumption and size of chip area.

Fig. 1. (a) Block diagram of conventional passive down-conversion mixer with 25% duty LO generator (25DLG), (b) operating principle of 25DLG to generate 25% duty LO.

Fig. 2(a) shows a quadrature sampling mixer. A quadrature sampling mixer has merit to eliminate 25DLG composed of AND gates. This topology generates 25% duty LO signals by ANDing two 50% duty LO signals with a relative delay of 1/4 period (7,8). Thus, it can reduce current consumption and chip area by 25DLG. The size of MOSFETs in Fig. 2(a) to achieve the same performance of conventional mixer should be double for making on-resistance of MOSFETs same. In addition, each LO signal should drive at least three MOSFETs in Fig. 2(a). Therefore, each LO buffer need additional power consumption.

Fig. 2(b) shows proposed mixer switching core. It operates in a similar way to quadrature sampling mixer and has the same size MOSFETs as quadrature sampling mixer. However, the proposed mixer has merit in power consumption and occupying area because it need only 2/3 of the MOSFETs compared to the quadrature sampling mixer.

Fig. 2. (a) Quadrature sampling mixer(single-ended input used) (7,8), (b) Proposed mixer switching core(single-ended input used).

In Fig. 3(a), new duty-generator-less mixer is proposed. The proposed mixer does not require AND gates which perform the edge-combining of LO signals with 50% duty. By adding only one additional switch to each MOSFET of conventional switching core in series, the performance of a proposed mixer becomes equivalent to that of a mixer with 25DLG. The series switch, which replaces AND gate, reduces power consumption and saves occupying area for 25DLG. These are advantages.

Fig. 3. (a) Block diagram of proposed duty-generator-less passive down-conversion mixer, (b) operating principle of generating of 25% duty LO.

Fig. 4. (a) Block diagram of proposed mixer, (b) timing diagram of LO wave form.

## II. PROPOSED DUTY-GENERATOR-LESS PASSIVE MIXER

Switching core of proposed mixer is composed of additional switch directly connecting to each MOSFET in switching core of conventional mixer. It replaces the 25DLG. In Fig. 3(a), MIX_INP and MIX_INN, the input voltage signals of the mixer, are converted to the current signals when the series connected switches are turned on simultaneously. In Fig. 3(b), during the overlapped time of LO$_{270}$ and LO$_{0}$, M9 and M1 switches are turned on simultaneously and then the input signal passes through M9 and M1 switches. Therefore, this operation is equivalent to the operation of M1 switch by LO$_{IP}$ with 25% duty in Fig. 1(b). However, the structure of switch is more complex. Even though LO buffers of proposed mixer use more current than that of conventional mixer, the proposed mixer uses less current and saves space because it doesn’t have 25DLG.

The LO$_{0}$, LO$_{90}$, LO$_{180}$, and LO$_{270}$ are LO signals with 50% duty. The overlapped time of LO$_{0}$ and LO$_{270}$ produces LO signal with 25% duty at 0 degrees. The overlapped time of LO$_{180}$ and LO$_{90}$ creates LO signal with 25% duty at 180 degrees. The quadrature signals can be generated by a different combination in Fig. 4(b). The next stage of switching core is TIA with RC feedback. The current signal after switching core is converted to voltage signal after TIA.

Fig. 5. Simplified circuit of TIA.

A signal is down-converted and decreased while passing through switches and it is amplified by the TIA with RC feedback connected to the backside of switches. In Fig. 5, the two-stage inverter is employed for the structure of TIA. The open loop gain is 49 dB and phase margin is 65 degree. The P1dB (1dB Gain Compression Point) of the entire circuit of this study is determined by the output of the TIA. TIA has the VDD of 1.8 V and uses two stacks. Therefore, when the overdrive voltage is assumed to be 250 mV per stack, the maximum voltage swing range is approximately 1.3 V$_{\mathrm{o-p}}$(zero to peak). Consequently, OP1dB (Output 1dB Gain Compression Point) of proposed mixer becomes approximately 12 dBm.

### 1. Gain Analysis of Proposed Mixer

At Fig. 4(b), each waveform of LO signals needs to be evaluated with Fourier transform for calculating the conversion gain (9). In these waveforms, LO$_{0}$ is expressed as a switching function and written as;

##### (1)
\begin{align} \begin{array}{l} g_{L{O_{0}}}=a_{0}+\sum _{n=1}^{\infty }\frac{2}{n\pi }\sin \left(n\pi \eta _{0}\right)\cos \left(\frac{2n\pi t}{T}\right)\\ =\eta _{0}+\frac{2}{\pi }\cos \left(\omega _{LO}t\right)- \frac{2}{3\pi }\cos \left(3\omega _{LO}t\right)+\frac{2}{5\pi }\cos \left(5\omega _{LO}t\right)- \ldots \end{array} \end{align}

where $\eta_{0}$ is the duty of LO$_{0}$. In the same method, the other switching function is expressed as;

##### (2)
$$g_{L O_{180}}=\eta_{180}-\frac{2}{\pi} \cos \left(\omega_{L O} t\right)+\frac{2}{3 \pi} \cos \left(3 \omega_{L O} t\right)-\frac{2}{5 \pi} \cos \left(5 \omega_{L O} t\right)+\mathrm{L} \ldots$$

##### (3)
$$g_{L O_{90}}=\eta_{90}-\frac{2}{\pi} \sin \left(\omega_{L O} t\right)+\frac{2}{3 \pi} \sin \left(3 \omega_{L O} t\right)-\frac{2}{5 \pi} \sin \left(5 \omega_{L O} t\right)+L_{\cdots}$$

##### (4)
$$g_{L O_{270}}=\eta_{270}+\frac{2}{\pi} \sin \left(\omega_{L O} t\right)-\frac{2}{3 \pi} \sin \left(3 \omega_{L O} t\right)+\frac{2}{5 \pi} \sin \left(5 \omega_{L O} t\right)-\mathrm{L} \ldots$$

where $\eta_{180}$, $\eta_{90}$, and $\eta_{270}$ are the duties of LO$_{180}$, LO$_{90}$, and LO$_{270}$, respectively. When these duties of LO signals are assumed to be 50%, the ideal condition, $\eta_{0}$, $\eta_{90}$, $\eta_{180}$, and $\eta_{270}$ are approximately 0.5.

For easy calculation, the voltage conversion gain (G$_{\mathrm{mix}}$) in Fig. 4(a) can be written as Eq. (5);

##### (5)
$G_{mix}=\frac{V_{IF\_ P}(t)- V_{IF\_ N}(t)}{V_{IN\_ P}(t)- V_{IN\_ N}(t)}$

$V_{I F_{-} P}(t)$ and $V_{I F_{-} N}(t)$ can be written as;

##### (6)
\begin{equation*} V_{IF\_ P}\left(t\right)=V_{IN\_ P}\left(t\right)g_{L{O_{270}}}g_{L{O_{0}}}+V_{IN\_ N}\left(t\right)g_{L{O_{90}}}g_{L{O_{180}}} \end{equation*}

##### (7)
\begin{equation*} V_{IF\_ N}\left(t\right)=V_{IN\_ N}\left(t\right)g_{L{O_{270}}}g_{L{O_{0}}}+V_{IN\_ P}\left(t\right)g_{L{O_{90}}}g_{L{O_{180}}} \end{equation*}

where $V_{I N_{-} P}(t)=-V_{I N_{-} N}(t)=A_{R F} \cos \left(\omega_{R F} t\right)$. Eqs. (6, 7) into Eq. (5), G$_{\mathrm{mix}}$ is simplified as follows.

##### (8)

Fig. 6. Noise equivalent circuit of proposed mixer.

In the switch function, harmonics above the third order can be negligible. The conversion gain, G$_{\mathrm{mix}}$ converges to$\sqrt{2}/\pi$ by Eq. (8). The final transfer function of the proposed mixer can be obtained as follows. The feedback capacitance, C$_{\mathrm{feed}}$ of RC feedback can be negligible in this gain calculation.

##### (9)
\begin{align} \begin{array}{l} A_{V}=G_{mix}\times \frac{V_{out}\left(t\right)}{V_{IF\_ P}- V_{IF\_ N}\left(t\right)}\\ \cong - \frac{\sqrt{2}}{\pi }\times \frac{R_{feed}}{R_{on}}\left[1- \frac{1}{A_{o}}\left(1+\frac{R_{feed}}{R_{on}}\right)\right] \end{array} \end{align}

where A$_{\mathrm{o}}$ is the open-loop gain of TIA, R$_{\mathrm{feed}}$ is the feedback resistance of TIA with RC feedback, and R$_{\mathrm{on}}$ is the on resistance of series connected switches in the mixer. The value in square brackets in Eq. (9)can be replaced by 1 because A$_{\mathrm{o}}$ is very high.

### 2. Noise Analysis of Proposed Mixer

The four noise sources are observed as output noise at the output node in Fig. 6. The thermal noise in RF frequency, which are contributed by the input resistance (R$_{\mathrm{s}}$), are down-converted to the output noise with IF frequency. And the thermal noise of on resistance (R$_{\mathrm{on}}$) of the switches is presented output node of switching core. The feedback resistance (R$_{\mathrm{feed}}$) of TIA with RC feedback and the input referred noise voltage (V$_{\mathrm{n,inop}}$) of TIA are contributed to the output of the proposed mixer (4, 10, 12). Although the noises of TIA are composed of thermal noise and flicker noise, the flicker noise is negligible in the frequency of more than 100 Hz. Therefore, the flicker noise is ignored in this calculation.

Fig. 7. Noise components converge to IF frequency by LO harmonics.

The circuit in Fig. 6 can be converted to half circuit. From analysis of half circuit, the output noise power is evaluated as;

##### (10)
\begin{align} \begin{array}{l} \overline{{V_{n,out}}^{2}\left(t\right)}=4KTR_{s}\left(\frac{2R_{feed}}{R_{s}G_{mix}+2R_{on}}\right)^{2}+8KTR_{on}\left(\frac{R_{feed}}{R_{on}}\right)^{2}\\ +8KTR_{feed}+\overline{V_{n,inop}\left(t\right)^{2}}\left(\frac{R_{s}G_{mix}+2R_{on}+2R_{feed}}{R_{s}G_{mix}+2R_{on}}\right)^{2} \end{array} \end{align}

Noise is presented by power because it appears randomly. A conversion gain (G$_{\mathrm{mix}}$) from RF port to output node of switching core is$\sqrt{2}/\pi$ by Eq. (8)since above third harmonic of the switch function could be ignored. In the case of noise, however, the harmonics cannot be ignored because the thermal noise distributed over the entire frequency band is down-converted to the IF frequency by LO’s harmonic components. The noise contribution of harmonics above third-order is about 10% (11). Therefore, the noise contribution by harmonics should be included in calculation. As a result, conversion gain for noise, G$_{\mathrm{mix}}$ converges to$1/2$ instead of $\sqrt{2}/\pi$. It can be proved by following equations.

##### (11)
\begin{aligned} G_{\operatorname{mix}} &=\frac{V_{I F_{-} P}(t)-V_{I F_{-} N}(t)}{V_{I N_{-} P}(t)-V_{I N_{-} N}(t)} \\ &=\frac{\sqrt{2}}{\pi} \cdot\left(1 \cdot \frac{\cos \left(\omega_{I F} t+\frac{\pi}{4}\right)}{\cos \left(\omega_{R F} t\right)}-\frac{1}{3} \cdot \frac{\cos \left(\omega_{I F} t+\frac{\pi}{4}\right)}{\cos \left(\omega_{3 R F} t\right)}+\frac{1}{5} \cdot \frac{\cos \left(\omega_{I F} t+\frac{\pi}{4}\right)}{\cos \left(\omega_{5 R F} t\right)}+\ldots\right) \end{aligned}

where $V_{N_{-} P} (t)=-V_{I N_{-} N}(t)=A_{N} \cos \left(\omega_{A L L} t\right)$. In the switch function, harmonics can’t be negligible. $A_{N}$ is the amplitude of noise and $\omega_{ALL}$ means the entire frequency band. $\omega_{3RF}$, $\omega_{5RF}$, and more components included in $\omega_{ALL}$ converge to $\omega_{IF}$ by the harmonics of the LO frequency in Fig. 7.

Fig. 8. Open loop gain and input referred noise voltage of TIA.

Since noise is generally represented as a square and each term of Eq. (11)is independent, (G$_{\mathrm{mix}}$)$^{2}$ can be expressed as;

##### (12)
\begin{align} \begin{array}{l} \left(G_{mix}\right)^{2}=\left(\frac{\sqrt{2}}{\pi }\right)^{2}\cdot \left[\left(\frac{1}{1}\right)^{2}+\left(\frac{1}{3}\right)^{2}+\left(\frac{1}{5}\right)^{2}+....\right]\\ =\left(\frac{\sqrt{2}}{\pi }\right)^{2}\frac{\pi ^{2}}{8} \end{array} \end{align}

Therefore, we can see that G$_{\mathrm{mix}}$ coverges to 1/2.

$A_{v, t o t a l}^{2}$ means total gain from input source to the output node in Fig. 6. It is obtained from the following equation.

##### (13)
$${A^{2}}_{V,\textit{total}}=\left(\frac{2R_{feed}}{R_{s}G_{mix}+2R_{on}}\right)^{2}\left[1- \frac{1}{{A_{o}}^{2}}\left(1+\frac{2R_{feed}}{R_{s}G_{mix}+2R_{on}}\right)^{2}\right]$$

The value in square brackets can be replaced by approximately 1. The Based on the above equations, $\overline{{V_{n,out}}^{2}(t)}$, ${A^{2}}_{V,\textit{total}}$, the final noise can be developed as follows (12).

##### (14)
\begin{align*} \begin{array}{l} NF=\frac{\overline{{V^{2}}_{n,out}\left(t\right)}}{{A^{2}}_{V,\textit{total}}}\cdot \frac{1}{4kTR_{s}}\\ \cong 1+\frac{\left(G_{mix}R_{s}+2R_{on}\right)^{2}}{2R_{S}R_{on}}+\frac{\left(G_{mix}R_{s}+2R_{on}\right)^{2}}{2R_{S}R_{feed}}\\ +\frac{\overline{{V_{n,inop}}^{2}}}{4KTR_{S}}\left(1+\frac{G_{mix}R_{s}+2R_{on}}{2R_{feed}}\right)^{2} \end{array} \end{align*}

Fig. 9. Simulation, measurement, and calculation results of conversion gain.

Fig. 10. Simulation, measurement, and calculation of NF.

where V$_{\mathrm{n,inop}}$ is the equivalent noise voltage at the input of TIA. V$_{\mathrm{n,inop}}$ was simulated to calculate the NF of the proposed mixer in Fig. 8.

## III. MEASUREMENT RESULTS

The simulation and measurement results were compared as a next step. Manual calculation and simulation results were almost identical to the measurements in Fig. 9 and 10. Gain and NF of conventional mixer and proposed mixer were measured between 500 MHz and 2000 MHz in frequency when R$_{\mathrm{feed}}$, feedback resistance of TIA with RC feedback is 4 kohm in Fig. 11. Measured IIP3 (Input third-order Intercept Point) of conventional mixer and proposed mixer were almost identical in Fig. 12. The two-tone test was performed with a difference of 500 kHz from 900~MHz.

The white box located at the center of the die photo represents the mixers including switching core of a conventional mixer, a duty generator, and the switching core of proposed mixer in Fig. 13. The right side of the box is a switching core of a conventional mixer and a duty generator. The left side is a switching core of the proposed mixer. Another white box shows the circuit of TIA with RC feedback. CMOS 40 nm process was used and 1.8 V was used as TIA supply voltage. Moreover, 1.1 V was used as LO buffer supply voltage for the conventional and proposed mixer.

Fig. 11. Measured Gain and NF of conventional mixer and proposed mixer by frequency.

Fig. 12. Measured IP3 of conventional mixer and proposed mixer.

Fig. 13. Die photo for conventional and proposed mixer.

Table 1. Performance Comparison

 Calculation Conv. Sim. Conv. Mea. Prop. Sim. Prop. Mea. Unit Process 40 nm CMOS Freq. 900 MHz Gain 27.4 27.4 24.6 24.9 23.6 dB NF 8.3 8.3 9.2 8.6 8.9 dB OP1dB 12.2 12.8 7.9 7.9 7.4 dBm OIP3 22.2** 30.7 28.4 32 28.6 dBm Size* 20214 10274 μm2 Current (Mixer) 6.3 6.3 6.0 6.3 6.0 mA Current (LO) 3.5 ***3.5 ***3.3 ****2.5 ****2.4 mA

* without area by TIA, ** OIP3 = OP1 dB + 10, *** 25 DLG + LO buffers from 1.1 V supply, **** LO buffers from 1.1 V supply

The performance of this circuit was summarized in case of 900MHz frequency in Table 1. However, it is operable from 500 MHz to 2000 MHz. The proposed mixer can save 0.9 mA current consumption and 9940 $\mu \mathrm{m}^{2}$ chip area compared to that of conventional mixer. Moreover, Gain and NF performances were found to be almost identical between two mixers.

The majority of mixers have used a 25DLG to generate LO signals with 25% duty. However, this study successfully generated the LO signal with 25% duty simply by adding only one MOSFET to switch core of conventional mixer in series. The advancement allows us to save the current and area that are required by a 25DLG. Despite these advantages, the performance including Gain and NF does not degrade. Nowadays, the area of a chip is getting smaller to achieve a high-density with using less power. The results of this study will help to advance this trend.

## IV. CONCLUSIONS

The majority of mixers have used a 25DLG to generate LO signals with 25% duty. However, this study successfully generated the LO signal with 25% duty simply by adding only one MOSFET to switch core of conventional mixer in series. The advancement allows us to save the current and area that are required by a 25DLG. Despite these advantages, the performance including Gain and NF does not degrade. Nowadays, the area of a chip is getting smaller to achieve a high-density with using less power. The results of this study will help to advance this trend.

### ACKNOWLEDGMENTS

This work was supported by the Technology Innovation Program (or Industrial Strategic Technology Development Program, 20004325, Low power and Multi Functional CMOS Sensor SoC for Motion Detection and IoT Applications) funded By the Ministry of Trade, Industry&Energy(MOTIE, Korea)

This work was supported by the Electronics and Telecommunications Research Institute (ETRI) grant funded by the Korean government [2019-0-00933, Development on Multi-beam Antenna Technology]

### REFERENCES

1
Zhou S., Chang M. F., May 2005, A CMOS passive mixer with low flicker noise for low-power direct-conversion receiver, IEEE J. Solid-State Circuits, Vol. 40, No. 5, pp. 1084-1093
2
Hsiao Yu. Chih, Meng Chin Chun, Syu Jin-Siang, Lin Chung-Yo, Wong Shyh-Chyi, Huang Guo-Wei, 2012, 5-6 GHz 9.4 mW CMOS Direct-Conversion Passive-Mixer Receiver With Low- Flicker-Noise Corner, Proceedings of the 7th European Microwave Integrated Circuit Conference
3
Lee J., et al , July 2011, A High Linear Low Flicker Noise 25% Duty Cycle LO I/Q Mixer for a FM Radio Receiver, IEEE International Symposium on Circuits and Systems, pp. 1399-1402
4
Poobuapheun N., Chen W., Zdravko B., Niknejad A. M., Aug 2007, A 1.5-V 0.7–2.5-GHz CMOS quadrature demodulator for multiband direct-conversion receivers, IEEE J. Solid-State Circuits, Vol. 42, No. 8, pp. 1669-1676
5
Lee D., Lee J., Park H., Choi J., Park S., Kim I., Pu Y., Kim J., Hwang K., Yang Y., Seo M., Lee K., 2016, Design of a Wide-Frequency-Range, Low-Power Transceiver with Automatic Impedance-Matching Calibration for TV-White-Space Application, J. Semiconductor Technology and Science, Vol. 16, pp. 126-142
6
Yu X., Fang Y., Shi Z., 2016, 2.5mW 2.73 GHz non-overlapping Multi-phase Clock Generator with Duty-cycle Correction in 0.13m CMOS, Electronics Lett, Vol. 52, pp. 1261-1262
7
Ikeuchi O., Saito N., Nauta B., June 2010, Quadrature Sampling Mixer Topology for SAW-Less GPS Receivers in 0.18µm CMOS, IEEE Symposium on VLSI, pp. 177-178
8
Kim J., Silva-Martines J., Sep 2013, Low-Power, Low-Cost CMOS Direct-Conversion Receiver Front-End for Multistandard Applications, IEEE Journal of Solid-State Circuits, Vol. 48, No. 9
9
Kim Y., Son Y., Parkhomenko V. N., Hwang I., Cho J., Nah K., Park B., 2005, A GSM/EGSM/DCS/ PCS Direct Conversion Receiver with Integrated Synthesizer, IEEE Trans. Microwave. Theory and Techniques, Vol. 53, pp. 606-613
10
Choi C. H., Kwon K. D., Nam I. K., 2016, A 370µW CMOS MedRadio receiver front-end with inverter-based complementary switching mixer, IEEE Microwave and Wireless Components Lett, Vol. 26, pp. 73-75
11
Terrovitis M. T., Meyer R. G., 1999, Noise in currentcommutating CMOS mixers,, Vol. 34, pp. 772-783
12
Lin Z., Mak P.-I., Martins R. P., 2014, A 0.14-mm2 1.4-mW 59.4-dBSFDR 2.4 GHz ZigBee/WPAN receiver exploiting a split-LNTA + 50% LO topology in 65-nm CMOS, IEEE Trans. on Microwave Theory and Techniques, Vol. 62, pp. 1525-1534

## Author

##### Jin-Yong Kim

received the B.S., M.S. degrees in the department of electrical engineering, Korea Aero space University, Go-yang, Korea, in 2014, and 2016, respectively.

His M.S. dissertation focused on the Receiver architecture about noise cancelling LNA and 25% duty generator-less mixer.

From 2016, he has been working at Hanwha systems, Seongnam, Korea which engage in defense industry.

He is an engineer with the Electronic optics part, especially engaged in receiver for Laser Spot Tracker (LST) system and has experience of Radar system, making beacon for testing main radar specification.

##### Young-Jin Kim

received the B.S. degree in electrical engineering from the Kyung-pook national university in 1995, the M.S. and Ph.D. degree in electrical engineering from the KAIST (Korea Advanced Instituted of Science and Technology) in 1997 and 2002, respectively.

His Ph.D. dissertation focused on the transceiver architecture about image rejection and spurious rejection.

In 2002, he joined Samsung Electronics, Co., Ltd., Korea, as a Senior Engineer.

Since then he has participated in the design of CDMA and GSM/GPRS wireless mobile application.

Since then he has been designing LNA and down-conversion mixer for multi-mode CDMA and GSM/GPRS.

In 2006, He joined the school of Electronics and Information Engineering, Korea Aerospace University, Seoul, Korea.

##### Joon-Hwan Han

received the B.S., degrees in the department of electrical engineering, kwangwoon University, Seoul, Korea, in 2003.

Mr. Han joined the Electro-Optronics 2 Team of Hanwha Systems, Korea, in 2002.

He is currently a Senior Engineer in the Electro-optics R&D Site, Hanwha systems.

He is interested in Analog & Digital Image Processing of Long Wave IR.

##### Jung-Ho Shin

received the B.S. degrees in Electronic Engineering from Kyung hee University, Korea, in 2010.

Mr. Shin joined the Electro-Optronics 2Team of Hanwha Systems, Korea, in 2010.

He is currently a Senior Engineer in the Electro-opticsㆍPGM R&D Site, Hanwha systems.

He is interested in Analog & Digital Image Processing of Long Wave IR.

##### Choon Sik Cho

received his B.S. in control and instrumentation engi-neering from Seoul National University in 1987, his M.S. in electrical and computer engineering from the University of South Carolina in 1995, and his Ph.D. in electrical and computer engineering from the University of Colorado in 1998.

From 1987 to 1993, he worked at LG Electronics with a focus on communication systems.

From 1999 to 2003, he worked for Curitel, where he was principally involved with the development of mobile phones.

He joined the School of Electronics and Information Engineering at Korea Aerospace University in 2004.

His research interests include the design of RFIC/MMIC, millimeter-wave ICs, analog circuits, and radar systems.


text:
# Tag Info

15

Actually there are terminological subtleties when you are talking about that. Particles in the accelerator's tube are gathered in a sequence of little "bunches". For the proton-proton mode there was roughly ~3000 bunches per beam. And each bunch contained roughly $\simeq 10^{11}$ protons. So, at the largest level what you actually have during the ...

11

It's c for constant or celeritas, which means speed in Latin. Everyone uses it because it's convention. You could use $\xi$ or $\zeta$ or $\gamma$ or any other symbol you wanted, but then you'd have to explain what it meant, and people would have to go through the trouble to remember this every time they read your papers. Better to go with convention and ...

10

Indeed most examples of unambiguously labeling chiral states fall back on having another pre-labeled chiral object on hand. For a long time it seemed as though "left" and "right" were entirely interchangeable labels. This symmetry is known as parity. However it turns out there is a way to distinguish left from right in a fundamental way; parity is not ...

9

The no-go results from Algebraic and Constructive QFT you mention deal with related but slightly different matters. (Edit: the previous version of the following paragraph was slightly misleading - Haag's theorem is actually stronger than I stated before; see below for details) Haag's theorem (which actually slightly predates the inception of Algebraic ...

8

A theory is a collection of concepts, laws, and equations in science that is meant to explain some particular subset of observations. It's also used for theories describing gedanken worlds that differ from ours. There is also a related word "model" that differs by a theory by being really specific while a "theory" may leave some details adjustable, and ...

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People have essentially explained the details, but let me make an attempt to formulate it in a language more familiar to a mathematician. I will ignore subtleties that enter for more general Lie superalgebras. Let $\mathfrak g$ be a Lie superalgebra with the $\mathbb Z_2$ grading $\mathfrak g = \mathfrak g_e\oplus\mathfrak g_o$, where the two factors are ...

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Lenses and glass bottles are transparent. As you quoted above, the different has to do with diffusion. Here is an example of an image through a transparent object: Here is an example of a translucent object: This is an example of how diffusion causes translucency: As light passes through a translucent object, it either enters or exists a rough ...

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An ultra-relativistic particle is any particle you observe to have almost all its energy stored in the form of momentum. In other words, we are talking about particles that have only a very tiny fraction of their total energy stored in (rest)mass. The relativistic mass-energy-momentum relationship $$E^2 - c^2 \ p^2 \ = \ c^4 \ m^2$$ is valid for a ...

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Lasers typically operate in two regimes: as CW or continuous wave (the radiation leaving the laser cavity or resonator continuously by a partially reflecting mirror, or as you put it, forming an uninterrupted wave) or as a laser pulse (the radiation leaving the cavity during a short time at a repetition rate, that is, as a burst of short pulses very separted ...

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It is a matter of confusing terminology , at the present times when so much differentiation has happened in physics related scientific disciplines. Radiation was a general terminology assigned to transfer of energy radially, to start with with waves: acoustic waves, waves in water. Then came Maxwell's equations and the discovery of electromagnetic waves, ...

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$H$, sometimes called the Hubble "constant" should actually be called properly: the Hubble parameter. It is a function of the scale factor $a(t)$ of the Universe or of the redshift $z=1/a-1$ and depends on the cosmological parameters. It gives the rate of change of the size of the Universe with respect to its size. Through the Friedmann equations it is ...

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"They" are probably talking about symplectic integrators. Most numerical integrators for (partial) differential equations do not specifically consider the energy of the system; they are generic integrators capable of solving any set of DEs, and not all DE's have a concept like "energy". When these are applied to a classical dynamics problem concerning ...

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I know we've had this discussion on the Astronomy SE site, but let me try to elaborate on my answer. Dark matter is an altogether different component of the universe from baryonic matter. It does cause the same overall dynamics when it comes to the universe as a whole. What I mean by this is that the hubble parameter: $$H(a) = H_0 ... 5 In signal processing, the Nyquist–Shannon sampling theorem says you need at least 2 samples of a frequency to be able to perfectly reconstruct it. So in your question, a sampling rate of 200\: \mathrm{MHz} means you can perfectly reconstruct frequencies in the range of 0 - 100\: \mathrm{MHz}. So what happens when frequencies above 100\: \mathrm{MHz} ... 5 The quadrature is a process – any process – of turning something into a "square". "Quadro" in Latin is "make square", "quadrus" is a "square". It comes from "quattors", four, because that's the number of vertices of a square. So integration of a function is also known as "quadrature" because we are calculating the area i.e. looking for a well-known area ... 4 Refraction: is specifically the redirection/bending of waves caused by things like changes in density, or material. Reflection: when the wave returns from the medium it came from. Scattering: more general, blanket term for any type of change in direction (but not being absorbed). Absorption: the energy of the waves being absorbed by a material they ... 4 If I'm not mistaken, "moose models" are an alternative name for quiver gauge theories. A quiver gauge theory consists of multiple gauge groups, [like U(1) \times U(1) in your example] and "matter" fields charged under multiple gauge groups. For example, quarks in the standard model carry charges under U(1)_{EM}, SU(2)_{Weak} and SU(3)_{Colour}.. That ... 4 The terminological mismatch arises because different physicists use the terms differently in different contexts. For example, here is how Landau and Lifshitz define an adiabatic process in the context of thermodynamics: Let us suppose that a body is thermally isolated, and is subject to external conditions which vary sufficiently slowly. Such a process ... 4 User twistor59 has addressed the part regarding the "generator" terminology, but let me give a bit more detail on the second part of the question. I'm going to restrict the discussion to matrix Lie groups for simplicity. Some background. Given a Lie group G with Lie algebra \mathfrak g, there exist two mappings \mathrm{Ad} and \mathrm{ad}, both ... 4 Yes, Newton was definitely using the term "mass". His Principia were written in Latin but the English translation of his Latin definition of gravity said: Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square ... 4 To uplift a solution to the field equations (such as Einstein's equations or their generalizations) or to uplift any other construction or argument means to find a solution of a higher-dimensional theory (or the analogous construction or argument) given the known solution of a lower-dimensional theory. This is possible if the lower-dimensional theory may be ... 4 The neutral current is electrically neutral. To see why, one must first understand what the current is. It is a composite field or (in the quantum theory) an operator, something like$$ J^{\mathrm{(NC)}\mu}(f) = \bar{u}_{f}\gamma^{\mu}\frac{1}{2}\left(g^{f}_{V}-g^{f}_{A}\gamma^{5}\right)u_{f},  where $u_f$ is the Dirac field for the fermion $f$. Note that ...

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The doppler shift causes a shift in wavelength at the origin of the wave (the frequency of the source never changes). This results in a shift in frequency for the observer. In the link below you can see the emission of the wave for a moving source causes the wavelength to be shorter in front and longer behind. The actual source isn't changing in ...

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Some examples come to my mind: Fourier's law of heat conduction $\vec{J} = -c\vec{\nabla}T$ in crystalline solids is a good example of a phenomenological law. It is an ampirical law easy to verify in a broad range of materials in various phases and yet, as explained in this presentation, there is no derivation of it from first principles in solids and ...

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It started with conservation of quantum numbers, from baryon number when we did not know about quarks, to lepton number, when we discovered the positron.For the neutrino momentum and energy conservation played a role too, since it is only seen as a missing mass. In time the symmetries in the assignments of the quantum numbers became more and more evident ...

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There is no consistent usage of these words among scientists. Usually, when someone says "law" or "principle", they are referring to a general idea that has been found to apply to many different situations, but not always. More speculative ideas are generally called theories, but use of the word "theory" does not always mean that an idea is speculative. ...

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Adiabatic means quasi-static and isoentropic - slow enough to create negligible amount of irreversible excitation. This is the common rationale of technically different definitions. E.g., Landau & Lifshic'es definition has two components - thermally isolated (to prevent entropy change by heat exchange) and slow (to prevent irreversible excitation). For a ...

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If you have a basis for the Lie algebra, you can talk of these basis vectors as being "generators for the Lie group". This is true in the sense that, by using the exponential map on linear combinations of them, you generate (at least locally) a copy of the Lie group. So they're sort of "primitive infinitesimal elements" that you can use to build the local ...

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The Hubble-constant gives the rate at which the universe expands at the present. Since the expansion of the universe hasn't been constant this is indeed not the case. So yes, the Hubble-constant is actually a time-depandant value (the name constant dates back to the time people believed the universe was static). The behaviour in time of the hubble constant ...

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I think this is about technical English usage rather than physics and indeed there is no accepted universal definition of the words - in the sense that IUPAP doesn't decree a definition and doesn't meet to debate things like whether such and such a law can be upgraded to a principle and so forth. However, here's how I hear these words: A law is simply an ...

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