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Electrical Circuit Theorems

Contents

- Notation- Ohm's Law- Kirchhoff's Laws- Thévenin's Theorem- Norton's Theorem- Thévenin and Norton Equivalence- Superposition Theorem- eciprocit! Theorem- Compensation Theorem- "illman's Theorem- #oule's Law

- "a$imum %ower Transfer Theorem- Star-&elta Transformation- &elta-Star Transformation

Notation

The library uses the symbol font for some of the notation and formulae. If the symbols for the

letters 'alpha beta delta' do not appear here [α β δ] then the symbol font needs to be installed

before all notation and formulae will be displayed correctly.

E

(%

voltage sourceconductance

currentresistancepower

[volts, V][siemens, S]

[amps, ][ohms, Ω]

[watts]

)*

+,

voltage dropreactance

admittanceimpedance

[volts, V]

[ohms, Ω]

[siemens, S][ohms, Ω]

Ohm's Law

!hen an applied voltage E causes a current ( to flow through an impedance ,, the value ofthe impedance , is e"ual to the voltage E divided by the current (.Impedance # Voltage $ %urrent , E . (

Similarly, when a voltage E is applied across an impedance ,, the resulting current ( through

the impedance is e"ual to the voltage E divided by the impedance ,.

%urrent # Voltage $ Impedance ( E . ,

Similarly, when a current ( is passed through an impedance ,, the resulting voltage drop ) across the impedance is e"ual to the current ( multiplied by the impedance ,.

Voltage # %urrent & Impedance ) (,

lternatively, using admittance + which is the reciprocal of impedance ,

Voltage # %urrent $ dmittance ) ( . +

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Kirchhoff's Laws

Kirchhoff's Current Law t any instant the sum of all the currents flowing into any circuit node is e"ual to the sum of allthe currents flowing out of that node

(in (out

Similarly, at any instant the algebraic sum of all the currents at any circuit node is (ero

( /

Kirchhoff's Voltage Law t any instant the sum of all the voltage sources in any closed circuit is e"ual to the sum of allthe voltage drops in that circuit

E (,

Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is (ero

E - (, /

Thévenin's Theorem

ny linear voltage networ) which may be viewed from two terminals can be replaced by avoltage*source e"uivalent circuit comprising a single voltage source E and a single seriesimpedance ,. The voltage E is the open*circuit voltage between the two terminals and theimpedance , is the impedance of the networ) viewed from the terminals with all voltagesources replaced by their internal impedances.

Norton's Theorem

ny linear current networ) which may be viewed from two terminals can be replaced by acurrent*source e"uivalent circuit comprising a single current source ( and a single shuntadmittance +. The current ( is the short*circuit current between the two terminals and theadmittance + is the admittance of the networ) viewed from the terminals with all currentsources replaced by their internal admittances.

Thévenin and Norton Equivalence

The open circuit, short circuit and load conditions of the Th+venin model are)oc E(sc E . ,)load E - (load,(load E . 0, 1 ,load2

The open circuit, short circuit and load conditions of the orton model are)oc ( . +(sc ()load ( . 0+ 1 +load2(load ( - )load +

Thévenin model from Norton model

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Voltage # %urrent $ dmittanceImpedance # - $ dmittance

E ( . +, + -3

Norton model from Thévenin model

%urrent # Voltage $ Impedance

dmittance # - $ Impedance

( E . ,

+ , -3

!hen performing networ) reduction for a Th+venin or orton model, note that* nodes with (ero voltage difference may be short*circuited with no effect on the networ)current distribution,* branches carrying (ero current may be open*circuited with no effect on the networ) voltagedistribution.

Superposition Theorem

In a linear networ) with multiple voltage sources, the current in any branch is the sum of thecurrents which would flow in that branch due to each voltage source acting alone with allother voltage sources replaced by their internal impedances.

eciprocit! Theorem

If a voltage source E acting in one branch of a networ) causes a current ( to flow in another branch of the networ), then the same voltage source E acting in the second branch wouldcause an identical current ( to flow in the first branch.

Compensation Theorem

If the impedance , of a branch in a networ) in which a current ( flows is changed by a finite

amount δ,, then the change in the currents in all other branches of the networ) may be

calculated by inserting a voltage source of -(δ, into that branch with all other voltage sources

replaced by their internal impedances.

"illman's Theorem 0%arallel enerator Theorem2

If any number of admittances +3, +4, +5, ... meet at a common point , and the voltages fromanother point to the free ends of these admittances are E3, E4, E5, ... then the voltagebetween points and is)%N 0E3 +3 1 E4 +4 1 E5 +5 1 6662 . 0+3 1 +4 1 +5 1 6662

)%N E+ . +

The short*circuit currents available between points and due to each of the voltages E3, E4,E5, ... acting through the respective admitances +3, +4, +5, ... are E3 +3, E4 +4, E5 +5, ... so thevoltage between points and may be e/pressed as

)%N (sc . +

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#oule's Law

!hen a current ( is passed through a resistance , the resulting power % dissipated in theresistance is e"ual to the s"uare of the current ( multiplied by the resistance % (4

0y substitution using 1hm's 2aw for the corresponding voltage drop ) 0 (2 across theresistance% )4 . )( (4

"a$imum %ower Transfer Theorem

!hen the impedance of a load connected to a power source is varied from open*circuit toshort*circuit, the power absorbed by the load has a ma/imum value at a load impedancewhich is dependent on the impedance of the power source.

ote that power is (ero for an open*circuit 3(ero current4 and for a short*circuit 3(ero voltage4.

Voltage Source!hen a load resistance T is connected to a voltage source ES with series resistance S,ma/imum power transfer to the load occurs when T is e"ual to S.

5nder ma/imum power transfer conditions, the load resistance T, load voltage )T, loadcurrent (T and load power %T areT S

)T ES . 4(T )T . T ES . 4S

%T )T4 . T ES

4 . 7S

Current Source!hen a load conductance T is connected to a current source (S with shunt conductance S,ma/imum power transfer to the load occurs when T is e"ual to S.

5nder ma/imum power transfer conditions, the load conductance T, load current (T, loadvoltage )T and load power %T areT S

(T (S . 4)T (T . T (S . 4S

%T (T4 . T (S

4 . 7S

Complex Impedances!hen a load impedance ,T 3comprising variable resistance T and variable reactance *T4 isconnected to an alternating voltage source ES with series impedance ,S 3comprisingresistance S and reactance *S4, ma/imum power transfer to the load occurs when ,T ise"ual to ,S

8 3the comple/ con6ugate of ,S4 such that T and S are e"ual and *T and *S aree"ual in magnitude but of opposite sign 3one inductive and the other capacitive4.

!hen a load impedance ,T 3comprising variable resistance T and constant reactance *T4 isconnected to an alternating voltage source ES with series impedance ,S 3comprisingresistance S and reactance *S4, ma/imum power transfer to the load occurs when T ise"ual to the magnitude of the impedance comprising ,S in series with *TT 9,S 1 *T9 0S

4 1 0*S 1 *T242:

ote that if *T is (ero, ma/imum power transfer occurs when T is e"ual to the magnitude of

,ST 9,S9 0S

4 1 *S42:

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!hen a load impedance ,T with variable magnitude and constant phase angle 3constantpower factor4 is connected to an alternating voltage source ES with series impedance ,S,ma/imum power transfer to the load occurs when the magnitude of ,T is e"ual to themagnitude of ,S0T

4 1 *T42: 9,T9 9,S9 0S

4 1 *S42:

Kennell!'s Star-&elta Transformation

star networ) of three impedances ,;N, ,<N and ,CN connected together at common node can be transformed into a delta networ) of three impedances ,;<, ,<C and ,C; by the followinge"uations,;< ,;N 1 ,<N 1 0,;N,<N . ,CN2 0,;N,<N 1 ,<N,CN 1 ,CN,;N2 . ,CN

,<C ,<N 1 ,CN 1 0,<N,CN . ,;N2 0,;N,<N 1 ,<N,CN 1 ,CN,;N2 . ,;N

,C; ,CN 1 ,;N 1 0,CN,;N . ,<N2 0,;N,<N 1 ,<N,CN 1 ,CN,;N2 . ,<N

Similarly, using admittances

+;< +;N +<N . 0+;N 1 +<N 1 +CN2 +<C +<N +CN . 0+;N 1 +<N 1 +CN2 +C; +CN +;N . 0+;N 1 +<N 1 +CN2

In general terms,delta # 3sum of ,star pair products4 $ 3opposite ,star 4 +delta # 3ad6acent +star pair product4 $ 3sum of +star 4

Kennell!'s &elta-Star Transformation

delta networ) of three impedances ,;<, ,<C and ,C; can be transformed into a star networ)of three impedances ,;N, ,<N and ,CN connected together at common node by the followinge"uations,;N ,C;,;< . 0,;< 1 ,<C 1 ,C;2,<N ,;<,<C . 0,;< 1 ,<C 1 ,C;2,CN ,<C,C; . 0,;< 1 ,<C 1 ,C;2

Similarly, using admittances +;N +C; 1 +;< 1 0+C; +;< . +<C2 0+;< +<C 1 +<C +C; 1 +C; +;<2 . +<C

+<N +;< 1 +<C 1 0+;< +<C . +C;2 0+;< +<C 1 +<C +C; 1 +C; +;<2 . +C;

+CN +<C 1 +C; 1 0+<C +C; . +;<2 0+;< +<C 1 +<C +C; 1 +C; +;<2 . +;<

In general terms,star # 3ad6acent ,delta pair product4 $ 3sum of ,delta4 +star # 3sum of +delta pair products4 $ 3opposite +delta4