Contents

Notation

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.

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.

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

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

In 1847, G. R. Kirchhoff extended the use of Ohm's law by developing a simple concept concerning the voltages contained in a series circuit loop. Kirchhoff's voltage law states:

"The algebraic sum of the voltage drops in any closed path in a circuit and the electromotive forces in that path is equal to zero."

To state Kirchhoff's law another way, the voltage drops and voltage sources in a circuit are equal at any given moment in time. If the voltage sources are assumed to have one sign (positive or negative) at that instant and the voltage drops are assumed to have the opposite sign, the result of adding the voltage sources and voltage drops will be zero.

NOTE: The terms electromotive force and emf are used when explaining Kirchhoff's law because Kirchhoff's law is used in alternating current circuits (covered in Module 2). In applying Kirchhoff's law to direct current circuits, the terms electromotive force and emf apply to voltage sources such as batteries or power supplies.

Through the use of Kirchhoff's law, circuit problems can be solved which would be difficult, and often impossible, with knowledge of Ohm's law alone. When Kirchhoff's law is properly applied, an equation can be set up for a closed loop and the unknown circuit values can be calculated.

This information courtesy of Integrated Publishing

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.

Thevenin's Theorem says you can simplify any linear circuit, regardless of complexity, to an equivalent circuit with a single voltage source and series resistance connected to a load. As in the Superposition Theorem, it must be linear. In other words, passive components such as resistors, inductors and capacitors are okay. Non-linear components such as semiconductors, do not fall under this 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.

Norton's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin's Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots).

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

Thevenin model from Norton model

Norton model from Thevenin model

When performing network reduction for a Thevenin 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.

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.

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

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

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)^{1‚àö‚àë2}
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)^{1‚àö‚àë2}

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)^{1‚àö‚àë2} = |Z_T| = |Z_S| = (R_S² + X_S²)^{1‚àö‚àë2}

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)

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)

The above information provided by BOWest Pty Ltd Electrical & Project Engineering

Faraday's law of electromagnetic induction deals with the relationship between changing magnetic flux and induced electromotive force. It states:

The amount of induced voltage is determined by:

1. The amount of magnetic flux

The greater the number of magnetic field lines cutting across the conductor, the greater the induced voltage.

2. The rate at which the magnetic field lines cut across the conductor

The faster the field lines cut across a conductor, or the conductor cuts across the field lines, the greater the induced voltage.

The Russian physicist Heinrich Lenz discovered in 1833 the directional relationships among the forces, voltages, and currents of electromagnetic induction. Lenz's law says:

An induced electromotive force generates a current that induces a counter magnetic field that opposes the magnetic field generating the current.

Thus, when an external magnetic field approaches a conductor, the current that is produced in the conductor will induce a magnetic field in opposition to the approaching external magnetic field. But when the external magnetic field moves away from the conductor, the induced magnetic field in the conductor reverses direction and opposes the change in the direction of the external magnetic field.