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Superposition, Thévenin and Norton

Reading: Chapter 3 of A&L

6.002 CIRCUITS AND ELECTRONICS

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Review Circuit Analysis Methods

l Circuit composition rules l Node method – the workhorse of 6.002

KCL at nodes using V ’s referenced from ground KVL implicit in pattern

l KVL: KCL: VI

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l  Introduction to linear circuits

l  Properties of linearity

l  The superposition tool for your toolkit

l  The Thévenin method

l  The Norton method

Overview

Let’s start by introducing linearity

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Consider Linearity

Write node equations –

1R

2R+"–"

g

e

VI

5

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Linearity Write node equations --

Rearrange --

021

=−+− I

Re

RVe

linear in IVe ,,

1R

2R+"–"

g

VI

e

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1R

2R+"–"

g

VI

eLinearity 0

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=−+− I

Re

RVe

IRVe

RR+=⎥

⎦

⎤⎢⎣

⎡+

121

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Linearity Homogeneity Superposition ⇒

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Linearity Homogeneity Superposition

Homogeneity

. . .

⇒

. . .

⇓

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Linearity Homogeneity Superposition

Superposition ⇒

. . . . . .

⇓

. . .

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Specific superposition example:

Method 4: Superposition method

1. Find the responses of the circuit to each source acting alone 2. Sum the individual respones

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i

+"–"+

- v

i

short

+

- v

i

g

+

- v

Each source acting alone means this

i

open

+

- v

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Back to the example

1R

2R+"–"

g

e

VI

Use superposition method 13

1R

2R+"–"

g

VI

eBack to the example Use superposition method

V acting alone V

0=I2R+"–"

1R

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Back to the example Use superposition method

1R

2R+"–"

g

VI

e

I acting alone 0=V

I1R

2R g

Ie

VRR

ReV21

2

+=

Voilà !

By superposition, sum the two partial voltages

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output shows superposition

Demo

?

salt water

Low freq sinusoid

+"–"

High freq triangular wave

+"–"

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Consider Yet another method…

resistors

g

mVnI

Arbitrary network N

+

- v g

i

iSuppose I want to determine v

By superposition

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RiIVv nn

nmm

m ++= ∑∑ βαresistors

g

mVnI

Arbitrary network N

+

- v g

i

i

Independent of external excitation and behaves like a voltage.

THv

also independent of external excitement and behaves like a resistor. Let’s call it “RTH”

Let’s call it “ ”

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resistors

g

mVnI

Arbitrary network N

+

- v g

i

i

RiIVv nn

nmm

m ++= ∑∑ βα

In other words, as far as the external world is concerned (for the purpose of the

i-v relation), “arbitrary network N” is indistinguishable from:

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iRvv THTH +=

i +"–"

THR

THv g

+

-

v

Thévenin equivalent network

Nresistors

g

mVnI

Arbitrary network N

+

- v g

i

i

THRTHv

Resistance of network seen from port ( ’s, ’s set to 0) mV nI

Open circuit voltage seen at terminal pair (aka port)

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Method 4: The Thévenin Method

E g

+"–"+"–"

i

+

- v

N

+"–"

THR

THv+

-

v

i

E Thévenin equivalent

1. Replace network N

with its Thévenin

equivalent

2. Then solve with

external network E.

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Example: Find i1 1R

V+"–"

1i

Step 1: Replace N with Thévenin equivalent

Network N

2R Ig

1R

V+"–"

1i

Network E

Network E

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:THV

:THR

Network N

2R Ig

Step 1: Replace with Thévenin equivalent

open circuit voltage of network N

turn off all independent sources and measure resistance of N

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1R

V+"–"

1i

THR

THV +"–"

Step 2: Solve with external network E

Network E

1R

V+"–"

1iN

2R Ig

E

2IRVTH =2RRTH =

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Graphically, iRvv THTH +=

i

v

+"–"

THR

THv+ v

i

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Method 5: The Norton Method

g

+"–"+"–"

i

+

- v

Norton equivalent

NTH RR =NI g

THR Resistance of network seen from port ( ’s, ’s set to 0) mV nI

Short circuit current seen at port

mVnI

NI

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+"–"

THR

THv+ v

i

Norton equivalent

NTH RR =NI g

Thevenin equivalent

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Summary Discretize matter by agreeing to

observe the lumped matter discipline

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Circuit analysis methods

KVL, KCL, I — V Combination rules Node method Superposition Thévenin Norton

Summary

Next – Nonlinear networks