linearity and superposition.pdf

8/10/2019 linearity and superposition.pdf

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4: Linearity and Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources

Single Unknown Source Superposition and Power

Proportionality

Summary

E1.1 Analysis of Circuits (2014-4121) Linearity and Superposition: 4 1 / 9


2/55

Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary


Suppose we use variables instead of fixed values for all of the independent

voltage and current sources. We can then use nodal analysis to find all

node voltages in terms of the source values.


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Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary





(1) Label all the nodes


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Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






(2) KCL equations

XU1

2 + X

1 + XY

3 = 0YX

3 + (U2) = 0


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Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






(2) KCL equations

XU1

2 + X

1 + XY

3 = 0YX

3 + (U2) = 0

(3) Solve for the node voltagesX= 13 U1+

23 U2, Y =

13 U1+

113 U2


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Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






(2) KCL equations

XU1

2 + X

1 + XY

3 = 0YX

3 + (U2) = 0


23 U2, Y =

13 U1+

113 U2

Steps (2) and (3) never involve multiplying two source values together, so:


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Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






(2) KCL equations

XU1

2 + X

1 + XY

3 = 0YX

3 + (U2) = 0


23 U2, Y =

13 U1+

113 U2


Linearity Theorem:For any circuit containing resistors and independent

voltage and current sources, every node voltage and branch current is a

linear function of the source values and has the form

aiUi where theUiare the source values and the ai are suitably dimensioned constants.


8/55

Linearity Theorem

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






(2) KCL equations

XU1

2 + X

1 + XY

3 = 0YX

3 + (U2) = 0


23 U2, Y =

13 U1+

113 U2


Linearity Theorem:For any circuit containing resistors and independent

voltage and current sources, every node voltage and branch current is a

linear function of the source values and has the form

aiUi where theUiare the source values and the ai are suitably dimensioned constants.

Also true for a circuit containingdependentsources providing their values

are sums of multiples of other voltages and/or currents in the circuit.


9/55

Zero-value sources

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary


Azero-valuedvoltage source has zero volts

between its terminals for any current. It is

equivalent to ashort-circuitor piece of wire

or resistor of0 (or S).


10/55

Zero-value sources

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary







11/55

Zero-value sources

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






Azero-valuedcurrent source has no current

flowing between its terminals. It is equivalentto anopen-circuitor a broken wire or a

resistor of (or0 S).


12/55

Zero-value sources

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary






Azero-valuedcurrent source has no current

flowing between its terminals. It is equivalentto anopen-circuitor a broken wire or a

resistor of (or0 S).


13/55

Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary


From the linearity theorem, we know thatX=a1U1+a2U2 so all weneed to do is find the values of a1anda2.


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources


Proportionality

Summary



If we setU2 = 0thenX=a1U1. ForU

2= 0the current source becomes anopen

circuit


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources

Single Unknown Source

Superposition and Power

Proportionality

Summary





circuit


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





circuitand now the3 kresistor plays no partin the circuit.


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



If we setU2 = 0thenX=a1U1. ForU2 = 0the current source becomes anopencircuitand now the3 kresistor plays no partin the circuit.


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




2 k and1 kform a potential divider and soa1 =

1 k2 k+1 k =

13 .


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .

If we setU1 = 0thenX=a2U2. ForU1 = 0the voltage source becomes ashortcircuit


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .

If we setU1 = 0thenX=a2U2. ForU1 = 0the voltage source becomes ashortcircuit


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .

If we setU1 = 0thenX=a2U2. ForU1 = 0the voltage source becomes ashortcircuitand the2 kand1 kare in parallel.

2 k||1 k = 2 k1 k

2 k+1 k

= 2

3

k.


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .


2 k||1 k = 2 k1 k

2 k+1 k

= 2

3

k.

NowX= 23 U2 and soa2= 23 .


23/55

Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .


2 k||1 k = 2 k1 k

2 k+1 k

= 2

3

k.


Combining these two givesX=a1U1+a2U2 = 13 U1+

23 U2.


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Superposition

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





1 k2 k+1 k =

13 .


2 k||1 k = 2 k1 k

2 k+1 k

= 2

3

k.


Combining these two givesX=a1U1+a2U2 = 13 U1+

23 U2.

Superposition:Any voltage or current in a circuit may be found by adding

up the values due to each of the independent sources in the circuit whilesetting all the otherindependent sources to zero.


25/55

Superposition and dependent sources

4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


Adependent sourceis one that is determined by the voltage and/or current

elsewhere in the circuit. HereV Y X.


26/55


4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




Step 1:Pretend all sources are independent

and use superposition to find expressions for

the node voltages:


27/55


4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1

Y = 2U1


28/55


4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2

Y = 2U1+ 6U2


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4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V


30/55


4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V

Step 2:Express the dependent source values in terms of node voltages:V =Y X


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4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V


Step 3:Eliminate the dependent source values from the node voltage

equations:

X= 103 U1+ 2U2+ 16(Y X)Y = 2U1+ 6U2+

12(Y X))


32/55


4: Linearity and

Superposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V



equations:

X= 103 U1+ 2U2+ 16(Y X) 76 X 16 Y = 103 U1+ 2U2Y = 2U1+ 6U2+

12(Y X))


33/55


4: Linearity andSuperposition

Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V



equations:

X= 103 U1+ 2U2+ 16(Y X) 76 X 16 Y = 103 U1+ 2U2Y = 2U1+ 6U2+

12(Y X))

12 X+

12 Y = 2U1+ 6U2


34/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V



equations:

X= 103 U1+ 2U2+ 16(Y X) 76 X 16 Y = 103 U1+ 2U2Y = 2U1+ 6U2+

12(Y X))

12 X+

12 Y = 2U1+ 6U2

X= 3U1+ 3U2

Y =U1+ 9U2


35/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary






the node voltages:

X= 103 U1+ 2U2+ 16 V

Y = 2U1+ 6U2+ 12 V



equations:

X= 103 U1+ 2U2+ 16(Y X) 76 X 16 Y = 103 U1+ 2U2Y = 2U1+ 6U2+

12(Y X))

12 X+

12 Y = 2U1+ 6U2

X= 3U1+ 3U2

Y =U1+ 9U2

Note:This is analternativeto nodal anlysis: you get the same answer.


36/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


Any current or voltage can be writtenX=a1U1+a2U2+a3U3+. . ..


37/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



Suppose we knowU2 = 6 mA.


38/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary





39/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




ThenX= 13 U1+ 23 U2 =

13 U1+ 4.


40/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




ThenX= 13 U1+ 23 U2 =

13 U1+ 4.

If all the independent sources except for U1have known fixed values, then

X=a1U1+b

whereb =a2U2+a3U3+. . ..


41/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




ThenX= 13 U1+ 23 U2 =

13 U1+ 4.

If all the independent sources except for U1have known fixed values, then

X=a1U1+b

whereb =a2U2+a3U3+. . ..

This has a straight line graph.

-2 -1 0 1 23

3.5

4

4.5

5

U1


42/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


The power absorbed (ordissipated) by a component always equalsV Iwhere the measurement directions ofV andIfollow the passive sign

convention.

For a resistorV I= V2

R =I2R.


43/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



convention.


R =I2R.

Power in resistor isP = (U1+U2)2

10 = 6.4 W


44/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



convention.


R =I2R.


10 = 6.4 W

Power due toU1 alone isP1 = U

2

1

10 = 0.9 W


45/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



convention.


R =I2R.


10 = 6.4 W


2

1

10 = 0.9 W

Power due toU2 alone isP2 = U2

2

10 = 2.5 W


46/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



convention.


R =I2R.


10 = 6.4 W


2

1

10 = 0.9 W


2

10 = 2.5 W

P=P1+P2 Power does not obey superposition.


47/55



Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary



convention.


R =I2R.


10 = 6.4 W


2

1

10 = 0.9 W


2

10 = 2.5 W

P=P1+P2 Power does not obey superposition.

You must use superposition to calculate the total V and/or the totalIandthen calculate the power.


48/55

Proportionality


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


From the linearity theorem, all voltages and currents have the form

aiUiwhere theUi are the values of the independent sources.

If you multiplyallthe independent sources by the same factor,k, then all

voltages and currents in the circuit will be multiplied byk.

The power dissipated in any component will be multiplied by k2.


49/55

Proportionality


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


From the linearity theorem, all voltages and currents have the form

aiUiwhere theUi are the values of the independent sources.

If you multiplyallthe independent sources by the same factor,k, then all

voltages and currents in the circuit will be multiplied byk.

The power dissipated in any component will be multiplied by k2.

Special Case:

If there is only one independent source,U, then all voltages and currents

are proportional toUand all power dissipations are proportional toU2.


50/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary


Linearity Theorem: X=

iaiUi over all independent sourcesUi


51/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary




Superposition:sometimes simpler than nodal analysis, often moreinsight.

Zero-value voltage and current sources

Dependent sources - treat as independent and add dependencyas an extra equation


52/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary







If all sources are fixed except forU1 then all voltages and currents inthe circuit have the formaU1+b.


53/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary








Powerdoes not obeysuperposition.


54/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary









Proportionality:multiplying all sources byk multiplies all voltages andcurrents byk and all powers byk2.


55/55

Summary


Linearity Theorem

Zero-value sources

Superposition

Superposition and

dependent sources



Proportionality

Summary








Proportionality:multiplying all sources byk multiplies all voltages andcurrents byk and all powers byk2.

For further details see Irwin & Nelms Chapter 5.

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linearity and superposition.pdf