Understanding Equivalent Networks in Circuits

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Explore the concept of equivalent networks in circuits, detailing how they simplify complex circuits, along with comparisons to programming functions. Delve into the parallels between circuit design and programming, highlighting the importance of structuring circuits like functions. Discover the benefits of equivalent networks and how they aid in simplifying circuit analysis.

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Circuits Lecture 7: Equivalence 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 i Network A Network B v v v 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; The complete circuit is the main . return 0; } Put everything in main is not a good idea. (the program for computing the average score of the first quiz)

Circuit Design is like Programming Define IO of a function: input: double array (scores) output: double (average score) Use function int main () { function avg_score(double* score){ // create array of grades double quiz1_grade[50] = double total = 0; for (int i=0; i<50; i++){ total += score[i]; } return total/50; print avg_score(quiz1_grade); We do not care what happens in the function. } } Understandable name Easy to read! Can be re-used!

Circuit Design is like Programming Complete Circuit (Main) Network (function) Function Name??? 6 k i-v characteristics i Simpler equivalent network (function name) v 6 k (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 R R 1 2 = + = + v v ( v R i R i 1 2 1 2 = + ) R R i 1 2 v + = i R R 1 2

i-v curse Parallel 1 R 1 R = + slope 1 2 v v 1 = + = + = i i i R 1 2 1 R 1 R R R par + 1 2 1 1 1 R 2 = + v R = 1 + 2 R R R 1 2 R 1 2

Example 2.4 + + = 4 5 6 15 k k k k Find i = 20 || 20 10 k k k = = 40 / 8 mA 5 i k = k 6 10 || 15 k k

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. Hint: consider i-v characteristics All resistors have resistance R.

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 Sources sv i-v curse Current Sources si

Source Transformation i R sv 1 vs = slope R R v sv = + vs Ri v 1 1 = + i v sv R R

Source Transformation R sv si R v = + = + 1 vs Ri v is i + R v 1 1 = + i v sv = i si R R R v The two circuits are equivalent. s= = s i R R and R

Source Transformation R v s= s i sv R R

Source Transformation R si = v i R s s 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 B Network A

Example How about . Find vo 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 = vo V 2 3 4 V = 2 6 || 3

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 v R = i i i =1 = R Resistor R C eq i g R v m = g v m R 1 = = Req R 2 gm R g R = m v R Negative Resistor!

Source Transformation for Controlled Sources vc ic v c= c i c= v i R c R

Equivalent Network with Controlled Source Example 2.8 i + R v cv = = = i i i v i R g vR R C c c m v = g v m = + R 1 v = iR + v c g R iR g vR = m v m R

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 Network with sources i i + v + v Load Network Source Network i i 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 Ans:1.1K Way2: Add current source find voltage

Cubic Puzzle Req = What is Req? 1 1 1 All resistors have resistance R. = + + v iR iR iR 3 6 3 A and B are the two terminals of the network. 5 = R R eq 6

Cubic Puzzle 2 5 . 1 http://e2e.ti.com/blogs_/archives/b/thesignal/archive/2013/03/18/resistor- puzzle-solution-and-a-rant-on-schematics.aspx?DCMP=scblog&HQS=hpa-pa- opamp-thehub-20140129-thesignal-20130318-en

Infinity Puzzle = Req Infinite resistors What is Req? ( 1+ )R = 3 Req https://www.youtube.com/watch?v=MgN7h1z5bMQ (the answer in the video is not correct)