### Circuits Lecture 2: Equivalence

```Circuits
Lecture 7: Equivalence

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 () {
double total = 0;
for (int i=0; i<50; 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)
}
We do not care what
happens in the function.
Understandable name
function avg_score(double* score){
double total = 0;
for (int i=0; i<50; i++){
total += score[i];
}
}
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
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

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
(the answer in the video is not correct)
• 2.32
• −6
• 2.36
• (a) What if  = 1?
• (b) (5 – 0.5)K
• (c)  = 12
Brain Teaser
1V
1
1A
1