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# Theorems And Laws • Notation
• Ohm’s Law
• Kirchhoff’s Laws
• Thévenin’s Theorem
• Norton’s Theorem
• Thévenin 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 .The symbol font is used for some notation and formulae. If the Greek symbols for alpha beta delta .do not appear here [ a b d ] the symbol font needs to be installed for correct display of notation and .formulae. 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

When an applied voltage E causes a current I to flow through an impedance Z, the value of the impedance Z is equal to the voltage E divided by the current I

.

 Impedance = Voltage / Current Z = E / I

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

Kirchhoff’s Current Law At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node: SIin = SIout 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

#### Thévenin’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.

#### Thévenin and Norton Equivalence

The open circuit, short circuit and load conditions of the Thévenin model are: Voc = E Isc = E / Z Vload = E – IloadZ Iload = E / (Z + Zload) The open circuit, short circuit and load conditions of the Norton model are: Voc = I / Y Isc = I Vload = I / (Y + Yload) Iload = I – VloadY Thévenin model from Norton model

 Voltage = Current / Admittance Impedance = 1 / Admittance E = I / Y Z = Y -1

Norton model from Thévenin model

 Current = Voltage / Impedance Admittance = 1 / Impedance I = E / Z Y = Z -1

When performing network reduction for a Thévenin 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: VPN = (E1Y1 + E2Y2 + E3Y3 + …) / (Y1 + Y2 + Y3 + …) VPN = 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: VPN = SIsc / 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