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Electrical Circuit Theorems

Source: BOWest Pty Ltd.

Contents

- Notation
- Ohm's Law
- Kirchhoff's Laws
- ThŽvenin's Theorem
- Norton's Theorem
- ThŽvenin and Norton Equivalence
- Superposition Theorem
- Reciprocity Theorem
- Compensation Theorem
- Millman's Theorem
- Joule's Law
- Maximum Power Transfer Theorem
- Star-Delta Transformation
- Delta-Star Transformation

Notation

E G I R P

voltage source conductance current resistance power

[volts, V] [siemens, S] [amps, A] [ohms, W] [watts]

V X Y Z

voltage drop reactance admittance impedance

[volts, V] [ohms, W] [siemens, S] [ohms, W]

Ohm's Law

Similarly, when a voltage E is applied across an impedance Z, the resulting current I through the impedance is equal to the voltage E divided by the impedance Z.

Current = Voltage / Impedance

I = E / Z

Similarly, when a current I is passed through an impedance Z, the resulting voltage drop V across the impedance is equal to the current I multiplied by the impedance Z.

Voltage = Current * Impedance

V = IZ

Alternatively, using admittance Y which is the reciprocal of impedance Z:

Voltage = Current / Admittance

V = I / Y

Kirchhoff's Laws

Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
SI = 0

Kirchhoff's Voltage Law
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit:
SE = SIZ

Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
SE - SIZ = 0

ThŽvenin's Theorem

Any linear voltage network which may be viewed from two terminals can be replaced by a voltage-source equivalent circuit comprising a single voltage source E and a single series impedance Z. The voltage E is the open-circuit voltage between the two terminals and the impedance Z is the impedance of the network viewed from the terminals with all voltage sources replaced by their internal impedances.

Norton's Theorem

Any linear current network which may be viewed from two terminals can be replaced by a current-source equivalent circuit comprising a single current source I and a single shunt admittance Y. The current I is the short-circuit current between the two terminals and the admittance Y is the admittance of the network viewed from the terminals with all current sources replaced by their internal admittances.

ThŽvenin and Norton Equivalence

The open circuit, short circuit and load conditions of the Norton model are:
V_oc = I / Y
I_sc = I
V_load = I / (Y + Y_load)
I_load = I - V_loadY

ThŽvenin model from Norton model

Voltage = Current / Admittance Impedance = 1 / Admittance

E = I / Y Z = Y -1

Norton model from ThŽvenin model

Current = Voltage / Impedance Admittance = 1 / Impedance

I = E / Z Y = Z -1

When performing network reduction for a ThŽvenin or Norton model, note that:
- nodes with zero voltage difference may be short-circuited with no effect on the network current distribution,
- branches carrying zero current may be open-circuited with no effect on the network voltage distribution.

Superposition Theorem

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

Reciprocity Theorem

If a voltage source E acting in one branch of a network causes a current I to flow in another branch of the network, then the same voltage source E acting in the second branch would cause an identical current I to flow in the first branch.

Compensation Theorem

If the impedance Z of a branch in a network in which a current I flows is changed by a finite amount dZ, then the change in the currents in all other branches of the network may be calculated by inserting a voltage source of -IdZ into that branch with all other voltage sources replaced by their internal impedances.

Millman's Theorem (Parallel Generator Theorem)

If any number of admittances Y₁, Y₂, Y₃, ... meet at a common point P, and the voltages from another point N to the free ends of these admittances are E₁, E₂, E₃, ... then the voltage between points P and N is:
V_PN = (E₁Y₁ + E₂Y₂ + E₃Y₃ + ...) / (Y₁ + Y₂ + Y₃ + ...)
V_PN = SEY / SY

The short-circuit currents available between points P and N due to each of the voltages E₁, E₂, E₃, ... acting through the respective admitances Y₁, Y₂, Y₃, ... are E₁Y₁, E₂Y₂, E₃Y₃, ... so the voltage between points P and N may be expressed as:
V_PN = SI_sc / SY

Joule's Law

When a current I is passed through a resistance R, the resulting power P dissipated in the resistance is equal to the square of the current I multiplied by the resistance R:
P = I²R

By substitution using Ohm's Law for the corresponding voltage drop V (= IR) across the resistance:
P = V² / R = VI = I²R

Maximum Power Transfer Theorem

Note that power is zero for an open-circuit (zero current) and for a short-circuit (zero voltage).

Voltage Source
When a load resistance R_T is connected to a voltage source E_S with series resistance R_S, maximum power transfer to the load occurs when R_T is equal to R_S.

Under maximum power transfer conditions, the load resistance R_T, load voltage V_T, load current I_T and load power P_T are:
R_T = R_S
V_T = E_S / 2
I_T = V_T / R_T = E_S / 2R_S
P_T = V_T² / R_T = E_S² / 4R_S

Current Source
When a load conductance G_T is connected to a current source I_S with shunt conductance G_S, maximum power transfer to the load occurs when G_T is equal to G_S.

Under maximum power transfer conditions, the load conductance G_T, load current I_T, load voltage V_T and load power P_T are:
G_T = G_S
I_T = I_S / 2
V_T = I_T / G_T = I_S / 2G_S
P_T = I_T² / G_T = I_S² / 4G_S

Complex Impedances
When a load impedance Z_T (comprising variable resistance R_T and variable reactance X_T) is connected to an alternating voltage source E_S with series impedance Z_S (comprising resistance R_S and reactance X_S), maximum power transfer to the load occurs when Z_T is equal to Z_S^* (the complex conjugate of Z_S) such that R_T and R_S are equal and X_T and X_S are equal in magnitude but of opposite sign (one inductive and the other capacitive).

When a load impedance Z_T (comprising variable resistance R_T and constant reactance X_T) is connected to an alternating voltage source E_S with series impedance Z_S (comprising resistance R_S and reactance X_S), maximum power transfer to the load occurs when R_T is equal to the magnitude of the impedance comprising Z_S in series with X_T:
R_T = |Z_S + X_T| = (R_S² + (X_S + X_T)²)^*
Note that if X_T is zero, maximum power transfer occurs when R_T is equal to the magnitude of Z_S:
R_T = |Z_S| = (R_S² + X_S²)^*

When a load impedance Z_T with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source E_S with series impedance Z_S, maximum power transfer to the load occurs when the magnitude of Z_T is equal to the magnitude of Z_S:
(R_T² + X_T²)^* = |Z_T| = |Z_S| = (R_S² + X_S²)^*

Kennelly's Star-Delta Transformation

Similarly, using admittances:
Y_AB = Y_ANY_BN / (Y_AN + Y_BN + Y_CN)
Y_BC = Y_BNY_CN / (Y_AN + Y_BN + Y_CN)
Y_CA = Y_CNY_AN / (Y_AN + Y_BN + Y_CN)

In general terms:
Z_delta = (sum of Z_star pair products) / (opposite Z_star)
Y_delta = (adjacent Y_star pair product) / (sum of Y_star)

Kennelly's Delta-Star Transformation

Similarly, using admittances:
Y_AN = Y_CA + Y_AB + (Y_CAY_AB / Y_BC) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_BC
Y_BN = Y_AB + Y_BC + (Y_ABY_BC / Y_CA) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_CA
Y_CN = Y_BC + Y_CA + (Y_BCY_CA / Y_AB) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_AB

In general terms:
Z_star = (adjacent Z_delta pair product) / (sum of Z_delta)
Y_star = (sum of Y_delta pair products) / (opposite Y_delta)