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Electricians Paradise-- Electronic Circuit Theorems

The theorems that follow are used to figure unknowns in circuits that are too complex to be solved using just Ohm's Law. They are all consistent with Ohm's Law and in a sense elaborations on it. The circuits often involve more than one power source, and these theorems are tools that can solve them.The theorems below can be followed logically to help with any electrical issue. Much like a Los Angeles bankruptcy lawyer would study bankruptcy and tax codes , an electrician can study and master these theorems to become a better electrician.

Contents

- Notation
- Ohm's Law
- Kirchhoff's Laws
- Thevenin's Theorem
- Norton's Theorem
- Thevenin 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

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



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


Thevenin'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.

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.


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.

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).


Thevenin and Norton Equivalence

The open circuit, short circuit and load conditions of the ThZ¯venin 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

Thevenin model from Norton model

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

Norton model from Thevenin model

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

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.


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


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 RT is connected to a voltage source ES with series resistance RS, maximum power transfer to the load occurs when RT is equal to RS.

Under maximum power transfer conditions, the load resistance RT, load voltage VT, load current IT and load power PT are:
RT = RS
VT = ES / 2
IT = VT / RT = ES / 2RS
PT = VT2 / RT = ES2 / 4RS

Current Source
When a load conductance GT is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GT is equal to GS.

Under maximum power transfer conditions, the load conductance GT, load current IT, load voltage VT and load power PT are:
GT = GS
IT = IS / 2
VT = IT / GT = IS / 2GS
PT = IT2 / GT = IS2 / 4GS

Complex Impedances
When a load impedance ZT (comprising variable resistance RT and variable reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when ZT is equal to ZS* (the complex conjugate of ZS) such that RT and RS are equal and XT and XS are equal in magnitude but of opposite sign (one inductive and the other capacitive).

When a load impedance ZT (comprising variable resistance RT and constant reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when RT is equal to the magnitude of the impedance comprising ZS in series with XT:
RT = |ZS + XT| = (RS2 + (XS + XT)2)1‚àö‚àë2
Note that if XT is zero, maximum power transfer occurs when RT is equal to the magnitude of ZS:
RT = |ZS| = (RS2 + XS2)1‚àö‚àë2

When a load impedance ZT with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source ES with series impedance ZS, maximum power transfer to the load occurs when the magnitude of ZT is equal to the magnitude of ZS:
(RT2 + XT2)1‚àö‚àë2 = |ZT| = |ZS| = (RS2 + XS2)1‚àö‚àë2


Kennelly's Star-Delta Transformation

A star network of three impedances ZAN, ZBN and ZCN connected together at common node N can be transformed into a delta network of three impedances ZAB, ZBC and ZCA by the following equations:
ZAB = ZAN + ZBN + (ZANZBN / ZCN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZCN
ZBC = ZBN + ZCN + (ZBNZCN / ZAN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZAN
ZCA = ZCN + ZAN + (ZCNZAN / ZBN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZBN

Similarly, using admittances:
YAB = YANYBN / (YAN + YBN + YCN)
YBC = YBNYCN / (YAN + YBN + YCN)
YCA = YCNYAN / (YAN + YBN + YCN)

In general terms:
Zdelta = (sum of Zstar pair products) / (opposite Zstar)
Ydelta = (adjacent Ystar pair product) / (sum of Ystar)


Kennelly's Delta-Star Transformation

A delta network of three impedances ZAB, ZBC and ZCA can be transformed into a star network of three impedances ZAN, ZBN and ZCN connected together at common node N by the following equations:
ZAN = ZCAZAB / (ZAB + ZBC + ZCA)
ZBN = ZABZBC / (ZAB + ZBC + ZCA)
ZCN = ZBCZCA / (ZAB + ZBC + ZCA)

Similarly, using admittances:
YAN = YCA + YAB + (YCAYAB / YBC) = (YABYBC + YBCYCA + YCAYAB) / YBC
YBN = YAB + YBC + (YABYBC / YCA) = (YABYBC + YBCYCA + YCAYAB) / YCA
YCN = YBC + YCA + (YBCYCA / YAB) = (YABYBC + YBCYCA + YCAYAB) / YAB

In general terms:
Zstar = (adjacent Zdelta pair product) / (sum of Zdelta)
Ystar = (sum of Ydelta pair products) / (opposite Ydelta)

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

Faraday's Law of Electromagnetic Induction

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

The magnitude of an electromagnetic force induced in a circuit is proportional to the rate of change of the magnetic flux that cuts across the circuit.

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.

Lenz's Law

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.


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