Discussion 3 - ShanghaiTechsist.shanghaitech.edu.cn/faculty/zhoupq/Teaching/Fall15/Discussion... · How to use superposition? Discussion 3 4 Problem 1 Use superposition to find i.

Electric Circuits (Fall 2015) Pingqiang Zhou

Discussion 3

10/21/2015

Discussion 3 1


Outline

• Linearity property

• Superposition

• Source transformation

• Thevenin and Norton’s theorems

• Examples of applications

Discussion 3 2


Linear Circuit

Linearity: homogeneity and additivity

Linear circuit: consists of only linear elements and linear

sources.

Linear resistors, capacitors, inductors

Linear sources: DC or AC?

Question: The current through a branch in a linear network is 2 A when the input source

voltage is 10V. If the voltage is reduced to 1V and the polarity is revised, the current

through the branch is?

Answer: -0.2 A

3Discussion 3


Superposition

What is superposition?

When to use superposition?

How to use superposition?

Discussion 3 4

Problem 1 Use superposition to find i.

Answer: 1.875 A


Applying Superposition

Discussion 3 5

Problem 2 Use superposition to find 𝑖0 and 𝑣0.

Do we turn off the dependent source?

Can we turn off the voltage source by open circuit?

Answer: 17.99 V, 1.799A

No

No


Source Transformation

Discussion 3 6

equivalent circuit: has identical terminal I-V characteristics

If the sources are turned off, resistance at the terminals

are both R

If the terminals are short circuited, the currents need to

be the same.

What principle can we get?

R

viRiv ssss or


Source Transformation

Discussion 3 7

Problem 3 Use source transformation to find 𝑖0.

Answer: 1/9 A


Dependent Sources

Source transformation also applies to dependent sources

The same relationship between the voltage and current

holds here:

R

viRiv ssss or

Discussion 3 8


Dependent Sources

Discussion 3 9

Problem 4 Use source transformation to find 𝑖𝑥.

Answer: 12/47.1 = 254.8 mA


New problem

• Reanalyze the circuit for each change in the load?

Discussion 3 10

Problem 5 Find the voltage drop between 𝑅𝐿 when

𝑅𝐿=10 Ω, 20Ω, 30 Ω, and 40 Ω.


Thevenin’s Theorem and Norton’s Theorem

A linear two terminal circuit may be replaced with an

equivalent circuits:

Discussion 3 11

a voltage source

in series with a resistor

(Thevenin’s theorem)

a current source

in parallel with a resistor

(Norton’s theorem)


How to find the Thevenin/Norton equivalents?

Find the equivalent source 𝑽𝑻𝒉 / 𝑰𝑵Thevenin equivalent: 𝑽𝑻𝒉 = terminal open circuit voltage

Norton equivalent: 𝑰𝑵 = terminal short circuit current

Discussion 3 12


How to find the Thevenin/Norton equivalents?

Find Thevenin/Norton equivalent resistance 𝑹𝑻𝒉/𝑹𝑵

In fact, 𝑅𝑇ℎ = 𝑅𝑁 (source transformation)

Step 1: turn off all the independent sources;

Step 2: apply a voltage source 𝑉𝑜 to the terminals and

determine the current 𝐼𝑜 , then 𝑅𝑇ℎ = 𝑅𝑁 = 𝑉𝑜/𝐼𝑜.

Can we apply a current source 𝐼𝑜 and determine 𝑉𝑜?

If there are no dependent source, can step 2 be simplified?

Discussion 3 13


Practice

Discussion 3 14

Answer: 20 Ω, -49.2 V

Problem 6 Obtain the Thevenin equivalent circuit at

terminals a-b of the circuit.


Practice

Discussion 3 15

Problem 7 Find the Thevenin equivalent as seen by R.

Answer:


Practice

Discussion 3 16

Problem 8 Use Norton’s theorem to find 𝑉𝑜.

Answer: 𝑰𝑵 = −𝟎. 𝟑 𝒎𝑨, 𝑽𝒐 = −𝟐𝟖𝟓. 𝟕𝒎𝑽


Practice

Discussion 3 17

Problem 9 Find the Thevenin and Norton equivalent circuit

across terminal a and b.

Answer:


Maximum Power Transfer

• VTH and RTH are fixed. By varying the load resistance RL, the

power delivered to the load varies.

• Maximum power transferred is when RL=RTH.

–+

VTh

RTh

RL

iL+

vL

–

Thé venin equivalent circuit

Discussion 3 18


Practice

Discussion 3 19

Answer: 21.48 mW (𝑹𝑻𝒉 = 𝟏𝟎𝟒. 𝟕𝟓 𝛀, 𝑽𝑻𝒉 = 𝟑𝑽)

Problem 10 Determine the maximum power delivered to

the variable resistor R in the circuit.


Application of Equivalent Theorem

• Resistance Measurement

• Source Modeling

𝑅𝑥 to be measured.

Discussion 3 20


Summary

• Superposition Leave one independent source on at a time

Sum over all responses

Voltage off SC

Current off OC

• Source Transforms: Voltage sources in series with a resistance can be converted to current

source in parallel with a resistance.

• Thevenin and Norton Equivalent Circuits Solve for OC voltage

Solve for SC current

Discussion 3 21


End

Discussion 3 22

Discussion 3 - ShanghaiTechsist.shanghaitech.edu.cn/faculty/zhoupq/Teaching/Fall15/Discussion... · How to use superposition? Discussion 3 4 Problem 1 Use superposition to find i.

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