# 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:

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

_{in}= SI_{out}**SI = 0**

**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:**

*Kirchhoff’s Voltage Law***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: **V _{oc} = E**

**I**

_{sc}= E / Z**V**

_{load}= E – I_{load}Z**I**The open circuit, short circuit and load conditions of the Norton model are:

_{load}= E / (Z + Z_{load})**V**

_{oc}= I / Y**I**

_{sc}= I**V**

_{load}= I / (Y + Y_{load})**I**

_{load}= I – V_{load}Y

*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 **Y _{1}**,

**Y**,

_{2}**Y**, … meet at a common point P, and the voltages from another point N to the free ends of these admittances are

_{3}**E**,

_{1}**E**,

_{2}**E**, … then the voltage between points P and N is:

_{3}**V**

_{PN}= (E_{1}Y_{1}+ E_{2}Y_{2}+ E_{3}Y_{3}+ …) / (Y_{1}+ Y_{2}+ Y_{3}+ …)**V**

_{PN}= SEY / SYThe short-circuit currents available between points P and N due to each of the voltages **E _{1}**,

**E**,

_{2}**E**, … acting through the respective admitances

_{3}**Y**,

_{1}**Y**,

_{2}**Y**, … are

_{3}**E**,

_{1}Y_{1}**E**,

_{2}Y_{2}**E**, … so the voltage between points P and N may be expressed as:

_{3}Y_{3}**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 = I ^{2}R**

By substitution using Ohm’s Law for the corresponding voltage drop **V (= IR)** across the resistance: **P = V ^{2} / R = VI = I^{2}R**

#### Maximum Power Transfer Theorem

When the impedance of a load connected to a power source is varied from open-circuit to short-circuit, the power absorbed by the load has a maximum value at a load impedance which is dependent on the impedance of the power source. 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**is connected to a voltage source

_{T}**E**with series resistance

_{S}**R**, maximum power transfer to the load occurs when

_{S}**R**is equal to

_{T}**R**. Under maximum power transfer conditions, the load resistance

_{S}**R**, load voltage

_{T}**V**, load current

_{T}**I**and load power

_{T}**P**are:

_{T}**R**

_{T}= R_{S}**V**

_{T}= E_{S}/ 2**I**

_{T}= V_{T}/ R_{T}= E_{S}/ 2R_{S}**P**

_{T}= V_{T}^{2}/ R_{T}= E_{S}^{2}/ 4R_{S}**When a load conductance**

*Current Source***G**is connected to a current source

_{T}**I**with shunt conductance

_{S}**G**, maximum power transfer to the load occurs when

_{S}**G**is equal to

_{T}**G**. Under maximum power transfer conditions, the load conductance

_{S}**G**, load current

_{T}**I**, load voltage

_{T}**V**and load power

_{T}**P**are:

_{T}**G**

_{T}= G_{S}**I**

_{T}= I_{S}/ 2**V**

_{T}= I_{T}/ G_{T}= I_{S}/ 2G_{S}**P**

_{T}= I_{T}^{2}/ G_{T}= I_{S}^{2}/ 4G_{S}**When a load impedance**

*Complex Impedances***Z**(comprising variable resistance

_{T}**R**and variable reactance

_{T}**X**) is connected to an alternating voltage source

_{T}**E**with series impedance

_{S}**Z**(comprising resistance

_{S}**R**and reactance

_{S}**X**), maximum power transfer to the load occurs when

_{S}**Z**is equal to

_{T}**Z**(the complex conjugate of

_{S}^{*}**Z**) such that

_{S}**R**and

_{T}**R**are equal and

_{S}**X**and

_{T}**X**are equal in magnitude but of opposite sign (one inductive and the other capacitive). When a load impedance

_{S}**Z**(comprising variable resistance

_{T}**R**and constant reactance

_{T}**X**) is connected to an alternating voltage source

_{T}**E**with series impedance

_{S}**Z**(comprising resistance

_{S}**R**and reactance

_{S}**X**), maximum power transfer to the load occurs when

_{S}**R**is equal to the magnitude of the impedance comprising

_{T}**Z**in series with

_{S}**X**:

_{T}**R**Note that if

_{T}= |Z_{S}+ X_{T}| = (R_{S}^{2}+ (X_{S}+ X_{T})^{2})^{½}**X**is zero, maximum power transfer occurs when

_{T}**R**is equal to the magnitude of

_{T}**Z**:

_{S}**R**When a load impedance

_{T}= |Z_{S}| = (R_{S}^{2}+ X_{S}^{2})^{½}**Z**with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source

_{T}**E**with series impedance

_{S}**Z**, maximum power transfer to the load occurs when the magnitude of

_{S}**Z**is equal to the magnitude of

_{T}**Z**:

_{S}**(R**

_{T}^{2}+ X_{T}^{2})^{½}= |Z_{T}| = |Z_{S}| = (R_{S}^{2}+ X_{S}^{2})^{½}#### Kennelly’s Star-Delta Transformation

A star network of three impedances **Z _{AN}**,

**Z**and

_{BN}**Z**connected together at common node N can be transformed into a delta network of three impedances

_{CN}**Z**,

_{AB}**Z**and

_{BC}**Z**by the following equations:

_{CA}**Z**

_{AB}= Z_{AN}+ Z_{BN}+ (Z_{AN}Z_{BN}/ Z_{CN}) = (Z_{AN}Z_{BN}+ Z_{BN}Z_{CN}+ Z_{CN}Z_{AN}) / Z_{CN}**Z**

_{BC}= Z_{BN}+ Z_{CN}+ (Z_{BN}Z_{CN}/ Z_{AN}) = (Z_{AN}Z_{BN}+ Z_{BN}Z_{CN}+ Z_{CN}Z_{AN}) / Z_{AN}**Z**Similarly, using admittances:

_{CA}= Z_{CN}+ Z_{AN}+ (Z_{CN}Z_{AN}/ Z_{BN}) = (Z_{AN}Z_{BN}+ Z_{BN}Z_{CN}+ Z_{CN}Z_{AN}) / Z_{BN}**Y**

_{AB}= Y_{AN}Y_{BN}/ (Y_{AN}+ Y_{BN}+ Y_{CN})**Y**

_{BC}= Y_{BN}Y_{CN}/ (Y_{AN}+ Y_{BN}+ Y_{CN})**Y**In general terms:

_{CA}= Y_{CN}Y_{AN}/ (Y_{AN}+ Y_{BN}+ Y_{CN})**Z**= (sum of

_{delta}**Z**pair products) / (opposite

_{star}**Z**)

_{star}**Y**= (adjacent

_{delta}**Y**pair product) / (sum of

_{star}**Y**

_{star})

#### Kennelly’s Delta-Star Transformation

A delta network of three impedances **Z _{AB}**,

**Z**and

_{BC}**Z**can be transformed into a star network of three impedances

_{CA}**Z**,

_{AN}**Z**and

_{BN}**Z**connected together at common node N by the following equations:

_{CN}**Z**

_{AN}= Z_{CA}Z_{AB}/ (Z_{AB}+ Z_{BC}+ Z_{CA})**Z**

_{BN}= Z_{AB}Z_{BC}/ (Z_{AB}+ Z_{BC}+ Z_{CA})**Z**Similarly, using admittances:

_{CN}= Z_{BC}Z_{CA}/ (Z_{AB}+ Z_{BC}+ Z_{CA})**Y**

_{AN}= Y_{CA}+ Y_{AB}+ (Y_{CA}Y_{AB}/ Y_{BC}) = (Y_{AB}Y_{BC}+ Y_{BC}Y_{CA}+ Y_{CA}Y_{AB}) / Y_{BC}**Y**

_{BN}= Y_{AB}+ Y_{BC}+ (Y_{AB}Y_{BC}/ Y_{CA}) = (Y_{AB}Y_{BC}+ Y_{BC}Y_{CA}+ Y_{CA}Y_{AB}) / Y_{CA}**Y**In general terms:

_{CN}= Y_{BC}+ Y_{CA}+ (Y_{BC}Y_{CA}/ Y_{AB}) = (Y_{AB}Y_{BC}+ Y_{BC}Y_{CA}+ Y_{CA}Y_{AB}) / Y_{AB}**Z**= (adjacent

_{star}**Z**pair product) / (sum of

_{delta}**Z**)

_{delta}**Y**= (sum of

_{star}**Y**pair products) / (opposite

_{delta}**Y**)

_{delta}