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Circuits Lecture 7: Equivalence 李宏毅 Hung-yi Lee Textbook • Chapter 2.1, 2.3 Outline • Concept of Equivalent networks • Equivalent networks only with resistors • Equivalent networks with independent sources • Equivalent networks with controlled sources Outline • Concept of Equivalent networks • Equivalent networks only with resistors • Equivalent networks with independent sources • Equivalent networks with controlled sources Network Network: Part of a complete circuit Two-terminal network Equivalent Network i-v characteristics i i v Network A v i v Network B Network A and B are equivalent. Equivalent networks in circuits are just like functions in programming! Circuit Design is like Programming int main () { // create array of grades double quiz1_grade[50] = …… double total = 0; for (int i=0; i<50; i++){ total += quiz1_grade[i]; } double avg = total / 50; print avg; return 0; } (the program for computing the average score of the first quiz) The complete circuit is the “main”. Put everything in “main” is not a good idea. Circuit Design is like Programming • Use function int main () { Define IO of a function: input: double array (scores) output: double (average score) // create array of grades double quiz1_grade[50] = …… print avg_score(quiz1_grade); } We do not care what happens in the function. Understandable name Easy to read! function avg_score(double* score){ double total = 0; for (int i=0; i<50; i++){ total += score[i]; } return total/50; } Can be re-used! Circuit Design is like Programming Complete Circuit (Main) Network (function) Function Name??? 6k i-v characteristics i Simpler equivalent network (function name) 6k v (IO of function) Benefit of Equivalent Network • 1. Simplify the complete circuit • Easier to analyze • 2. Useful network can be reused just like elements • Voltage amplifier (refer to lecture 5) • Current source (later) • Negative Resistor (later) Outline • Concept of Equivalent networks • Equivalent networks only with resistors • Equivalent networks with sources • Equivalent networks with controlled sources i-v curse Series 1 slope R1 R2 v v1 v2 R1i R2i ( R1 R2 ) i v i R1 R2 i-v curse Parallel 1 1 slope R1 R2 v v i i1 i2 R1 R2 1 1 v R1 R2 1 R par 1 1 R1 R2 R1 R2 R1 R2 Example 2.4 4k 5k 6k 15k • Find i 20k || 20k 10k 10k || 15k 6k i 40 / 8k 5mA Beyond Series and Parallel Req = What is Req? Find i-v characteristics Way1: Add voltage source find current Way2: Add current source find voltage Cubic Puzzle Req = What is Req? A and B are the two terminals of the network. All resistors have resistance R. Hint: consider i-v characteristics Cubic Puzzle 2 (solution is at the end of the slides) Infinity Puzzle = Req Infinite resistors What is Req? (solution is at the end of the slides) Outline • Concept of Equivalent networks • Equivalent networks only with resistors • Equivalent networks with sources • Equivalent networks with controlled sources Sources i-v curse Voltage v s Sources i-v curse Current is Sources Source Transformation i vs R vs R 1 slope R vs vs Ri v 1 1 i v vs R R v Source Transformation vs R is R vs Ri v 1 1 i v vs R R vs R R and is R v is i R 1 i v is R The two circuits are equivalent. Source Transformation R vs vs is R R Source Transformation R v s is R is R Why Source Transformation? Parallel Why Source Transformation? Series Be careful about the directions of voltage and current sources Example • Find vo Simply the networks by their equivalent networks Example • Find vo Network A Network B Example • Find vo How about …. Bad idea…… Do not put the target in the network to be simplified. Example • Find vo Two-terminal Network Example • Find equivalent network Typical Network 3 Example • Find equivalent network 3 3 Example 2 3 6 || 3 2 4V vo 3.2V Outline • Concept of Equivalent networks • Equivalent networks only with resistors • Equivalent networks with sources • Equivalent networks with controlled sources Equivalent Network with Controlled Source – Example 2.8 What is the equivalent network? Find the i-v characteristics i iR iC v g mv R 1 gm R v R Resistor gm R 2 v R Req i 1 gm R Req R Negative Resistor! Source Transformation for Controlled Sources vc ic vc ic R vc ic R Equivalent Network with Controlled Source – Example 2.8 i v R vc i iR iC v g mv R 1 gm R v R vc ic R g m vR v iR vc iR g m vR Remind 1 • When computing i-v characteristics, we need reference direction of v and i • Without reference direction, we cannot really answer the i-v characteristics Network without sources i Load v i Network Network with sources i Source v i Network The current goes into the terminal with high potential. The current goes out the terminal with high potential. Remind 2 • Select good networks • Put a controlled source and its control variable in the same network Three-terminal Network Three-terminal network Chapter 4.6 (out of the scope) Four-terminal Network Chapter 14 Problem • 2.32, 2.36 Thank you! Beyond Series and Parallel Req = What is Req? Find i-v characteristics: Way1: Add voltage source find current Way2: Add current source find voltage Ans:1.1K Cubic Puzzle Req = What is Req? All resistors have resistance R. A and B are the two terminals of the network. 1 1 1 v iR iR iR 3 6 3 5 Req R 6 Cubic Puzzle 2 1.5 http://e2e.ti.com/blogs_/archives/b/thesignal/archive/2013/03/18/resistorpuzzle-solution-and-a-rant-on-schematics.aspx?DCMP=scblog&HQS=hpa-paopamp-thehub-20140129-thesignal-20130318-en Infinity Puzzle = Req Infinite resistors What is Req? Req 1 3 R https://www.youtube.com/watch?v=MgN7h1z5bMQ (the answer in the video is not correct) Problem - Answer • 2.32 • −6 • 2.36 • (a) What if = 1? • (b) (5 – 0.5)K • (c) = 12 Brain Teaser 1V 1 1A 1 http://rochester.ieee.org/files/2014/03/Newsletter_4-2014.pdf Acknowledgement • 感謝 范廷瀚 (b02) • 糾正 Infinity Puzzle 的錯誤答案