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Superposition

Another way to analyze a DC circuit withmultiple loops or meshes

Lets us analyze a circuit with more thanone independent source by finding thecontribution of each source separately

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Superposition

The voltages across (or the currentthrough) an element is the algebraic sum

of the voltages across (or the currentsthrough) that element due to eachindependent source acting alone

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Superposition

If we are analyzing one source, we mustturn off every other source in the circuit inthe following ways:

Every voltage source must be replacedwith 0V ± a short circuit

Every current source must be replacedwith 0A ± an open circuit

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Superposition

Superposition is most often used in ACcircuit analysis

± Used for AC sources with differentfrequencies

± Used for AC signals that have a DC offset

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Steps to Apply Superposition

1. Turn off all independent sources except for one. Find the output (voltage or current)due to that active source using basic lawsor mesh or nodal analysis.

2. Repeat step one for each of the other independent sources.

3. Find the total contribution by addingalgebraically all the contributions due to theindependent sources

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Example 4.3Use the superposition theorem to find v in the

circuit below.

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Example 4.3Use the superposition theorem to find v in the

circuit below.

There are two sources,so we first say that:

v 1 is the voltage acrossthe 4- resistor due to

the 6-V source

v 2 is the voltage acrossthe 4- resistor due tothe 3-A source

21 vvv !

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Example 4.3Use the superposition theorem to find v in the

circuit below.To find v 1, we turn off thecurrent source and makeit an open circuit.

Now we can find v 1 usingKirchoff¶s Voltage Lawand/or voltage division:

V V v 284

46

1!

;;

;!

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Example 4.3Use the superposition theorem to find v in the

circuit below.

Now we find v 2 byeliminating the

voltage source.

Start finding v 2 byfinding R

eq.

;!;;

;;! 6667.2

84

84eq R

V A Riv eq s86667.23

2!;!!

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Example 4.3Use the superposition theorem to find v in the

circuit below.

So now we canapply superposition:

V V V vvv 108221

!!!

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Pr actice Prob lem 4.3Using the superposition theorem, find v o in

the circuit below.

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Pr actice Prob lem 4.3Using the superposition theorem, find v o in

the circuit below.

There are two sources,so we first say that:

v 1 is the voltage acrossthe 2- resistor due to

the 8-A source

v 2 is the voltage acrossthe 2- resistor due tothe 20-V source

21 vvvo !

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Pr actice Prob lem 4.3Using the superposition theorem, find v o in

the circuit below.To find v 1, we turn off thevoltage source and makeit a short circuit.

Now we can find v 1 usingcurrent division andOhm¶s Law:

V Av 8241 !;!

A Ai 455

58

1!

;;

;!

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Pr actice Prob lem 4.3Using the superposition theorem, find v o in

the circuit below.

Now we find v 2 byeliminating the

current source.

Find v 2 by voltagedivision

V V v 4532

220

2!

;;;

;!

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Pr actice Prob lem 4.3Using the superposition theorem, find v o in

the circuit below.

So now we canapply superposition:

V V V vvv 124821

!!!

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Source Transformation

We¶ve already used transformations of seriesand parallel resistors to simplify circuit analysis

We can also use source transformation ± It is sometimes helpful to have all voltage sources or

current sources in a circuit, so we want a way toswitch back and forth

± A voltage source in series with a resistor is the sameas a current source in parallel with the same resistor

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Source Transformation

R I S S !

R I S

S !

These two are equivalent as long as they are related by Ohm¶s Law

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Source Transformation

Equi valence means that two circuits haveidentical v- i characteristics

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Example 4.6Use source transformation to find v o in the

circuit below.

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Example 4.6Use source transformation to find v o in the

circuit below.Replace the 3-A current source with itsequivalent voltage source in series with the 4-resistor.

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Example 4.6Use source transformation to find v o in the

circuit below.Combine the 2- and 4- resistors in series, toget an equivalent resistance of 6 .Transform the 12-V source and 6 resistor intoa 2A current source.

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Example 4.6Use source transformation to find v o in the

circuit below.Transform the 12-V voltage source into a currentsource in parallel with a 3 resistor.

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Example 4.6Use source transformation to find v o in the

circuit below.The 2A and 4A current sources can becombined to a 2A current sourceThe three resistors can also be combined toform one resistor.

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Example 4.6Use source transformation to find v o in

the circuit below.

Just calculate using Ohm¶s Law

V A R I vo2.36.12 !;!!