Lab. Lab. 4 4 중첩정리중첩정리, , TheveninThevenin 및및 Norton Norton 정리정리

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이이 실험의실험의 목표목표이이 실험의실험의 목표목표

Superposition Theorem 이해 Superposition Theorem 이해 What is Linearity of circuit systems?

Thevenin’s Theorem 이해 Thevenin s Theorem 이해

Norton’s Theorem 이해

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Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

The Superposition theorem is The Superposition theorem is very helpful in determining the voltage across an element or

current through a branch when the circuit contains multiple b f lt tnumber of voltage or current sources

One big advantage is that we do not have to use Cramer’s rule or complicated that we do not have to use Cramer s rule or complicated

mathematical operations but simply algebraically adding solutions obtained from analyzing the network with one source activated at a timesource activated at a time

The Superposition theorem states that “Th t th h lt l t i li “The current through, or voltage across an element in any linear

bilateral network is equal to the algebraic sum of the currents or voltages produced independently by each source”

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Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

in other words, this theorem allows us to find a solution for a o e o ds, s eo e a o s us o d a so u o o acurrent or voltage using only one source at a time

once we have a solution for each source we can combine the results to obtain the total solutionresults to obtain the total solution

LinearityLi t h th t th t if d( ) ( ) ( ) ( ) Linear system have the property that if and , then

Consider the square-law system of

1 1( ) ( )x t y t 2 2( ) ( )x t y t

1 2 1 2( ) ( ) ( ) ( ) ( ) ( )x t x t x t y t y t y t 2( ) ( )y t x t Linear system?q y

1( )x t 1( )y t( )w t

1( )x t( )y t( )x t

( ) ( )y y

2 ( )x t 2 ( )y t 2 ( )x t( )

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( )w t ( )y twill equal when the system is linear !

Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

Analyzing procedure: To consider the effects of each Analyzing procedure: To consider the effects of each source, the other sources have to be removed when removing a voltage source from a network schematic,

replace it with a direct connection (short-circuit) of zero ohms when removing a current source from a network schematic,

replace it by an open circuit of infinite ohmsreplace it by an open circuit of infinite ohms any internal resistance associated with the removed source(s)

must remain in the network

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Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

Since the effect of each source will be determined S ce e e ec o eac sou ce be de e edindependently, the number of networks to be analyzed will equal the number of sources

For Superposition All dependent sources must be left intact!! For Superposition, All dependent sources must be left intact!! You can’t apply O/C and S/C on dependent sources

Superposition cannot be applied to power

1 2TI I I

2 2 2 2 21 2 1 2 1 2 1 2 1 2

delivered to the circuits are ( ) 2T

The powerP I I R I R I R I I R P P I R I R 6

Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

Example: Find v in the circuit shown Example: Find v in the circuit shown

3A is discarded by i itopen-circuit

6V is discarded by 6V is discarded by short-circuit

Th i 10 VThe answer is v = 10 V7

Superposition TheoremSuperposition TheoremSuperposition TheoremSuperposition Theorem

Example: Find vx in the circuit shown Example: Find vx in the circuit shown

The answer is vx = 12.5 V

2A is discarded 2A is discarded by open-circuit

20 v120 v2

10V is discarded by open-circuit

Dependant source keep unchanged

4 10 V+

0.1v14

2 A0.1v24 0.1v2

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(a) (b)

Thevenin’sThevenin’s TheoremTheoremThevenin sThevenin s TheoremTheorem

It states that a linear two-terminal circuit (Fig a) can be It states that a linear two terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTH in series with a resistor RTH

VTH is the open-circuit voltage at the terminals RTH is the input or equivalent resistance at the terminals when

the independent sources are turned off Voltage sources are replaced by short circuits Voltage sources are replaced by short circuits Current sources are replaced by open circuits

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Thevenin’sThevenin’s TheoremTheorem

Example:

Thevenin sThevenin s TheoremTheorem

Example:

Rth=8Ω

Vth=20V

RthVth

Rth

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Thevenin’sThevenin’s TheoremTheorem

Example:

Thevenin sThevenin s TheoremTheorem

Example: Measuring ETH :

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Thevenin’sThevenin’s TheoremTheorem

Example:

Thevenin sThevenin s TheoremTheorem

Example: Measuring RTH :

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Norton’s TheoremNorton’s Theorem

It states that a linear two-terminal circuit can be

Norton s TheoremNorton s Theorem

It states that a linear two terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN

IN is the short circuit current through the terminals R is the input or equivalent resistance at the terminals when RN is the input or equivalent resistance at the terminals when

the independent sources are turned off Voltage sources are replaced by short circuits Current sources are replaced by open circuits

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Norton’s TheoremNorton’s Theorem

Experimental Procedure

Norton s TheoremNorton s Theorem

Experimental Procedure IN :

Norton current can be determined by placing an ammeter across the output terminals of the network

RN : Since R = R the experimental procedures described for Since RN = RTH , the experimental procedures described for

the Thevenin equivalent circuit can be applied here also

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Norton’s TheoremNorton’s Theorem

Example: Be solved by Norton’s theorem

Norton s TheoremNorton s Theorem

Example: Be solved by Norton s theorem

(a) Original circuit. (b) Short circuit across terminals A and B.

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Norton’s TheoremNorton’s Theorem

The Norton Equivalent Circuit

Norton s TheoremNorton s Theorem

e o o qu a e C cu Replace R2 with a short and determine IN Determine RN = RTH The current source provides 12 A total flow, regardless of

what is connected across it When shorted all of the current will flow in the short When shorted, all of the current will flow in the short

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(c) The short-circuit current IN is 36/3 = 12 A. (d) Open terminals A and B but short-circuit V to find RAB is 2 Ω, the same as RTH

Norton’s TheoremNorton’s TheoremNorton s TheoremNorton s Theorem

212 6 A4

NL N

N L

RI IR R

(e) Norton equivalent circuit. (f) RL reconnected to terminals A and B to find that IL is 6A.

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L

Source TransformationsSource Transformations

The Norton equivalent circuit can be obtained directly

Source TransformationsSource Transformations

The Norton equivalent circuit can be obtained directly from the Thevenin equivalent circuit using a simple source conversion

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