# Electrical terms POWER QUALITY GLOSSARY

## Library - Electrical Circuit Theorems

 Notation The 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

#### Th�venin'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.

#### Th�venin and Norton Equivalence

The open circuit, short circuit and load conditions of the Th�venin model are:
Voc = E
Isc = E / Z

The open circuit, short circuit and load conditions of the Norton model are:
Voc = I / Y
Isc = I

Th�venin model from Norton model

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

Norton model from Th�venin model

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

When performing network reduction for a Th�venin 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.

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.

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)
Note that if XT is zero, maximum power transfer occurs when RT is equal to the magnitude of ZS:
RT = |ZS| = (RS2 + XS2)

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) = |ZT| = |ZS| = (RS2 + XS2)

#### 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

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)

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)

#### Resistance

The resistance R of a circuit is equal to the applied direct voltage E divided by the resulting steady current I:
R = E / I

#### Resistances in Series

When resistances R1, R2, R3, ... are connected in series, the total resistance RS is:
RS = R1 + R2 + R3 + ...

#### Voltage Division by Series Resistances

When a total voltage ES is applied across series connected resistances R1 and R2, the current IS which flows through the series circuit is:
IS = ES / RS = ES / (R1 + R2)

The voltages V1 and V2 which appear across the respective resistances R1 and R2 are:
V1 = ISR1 = ESR1 / RS = ESR1 / (R1 + R2)
V2 = ISR2 = ESR2 / RS = ESR2 / (R1 + R2)

In general terms, for resistances R1, R2, R3, ... connected in series:
IS = ES / RS = ES / (R1 + R2 + R3 + ...)
Vn = ISRn = ESRn / RS = ESRn / (R1 + R2 + R3 + ...)
Note that the highest voltage drop appears across the highest resistance.

#### Resistances in Parallel

When resistances R1, R2, R3, ... are connected in parallel, the total resistance RP is:
1 / RP = 1 / R1 + 1 / R2 + 1 / R3 + ...

Alternatively, when conductances G1, G2, G3, ... are connected in parallel, the total conductance GP is:
GP = G1 + G2 + G3 + ...
where Gn = 1 / Rn

For two resistances R1 and R2 connected in parallel, the total resistance RP is:
RP = R1R2 / (R1 + R2)
RP = product / sum

The resistance R2 to be connected in parallel with resistance R1 to give a total resistance RP is:
R2 = R1RP / (R1 - RP)
R2 = product / difference

#### Current Division by Parallel Resistances

When a total current IP is passed through parallel connected resistances R1 and R2, the voltage VP which appears across the parallel circuit is:
VP = IPRP = IPR1R2 / (R1 + R2)

The currents I1 and I2 which pass through the respective resistances R1 and R2 are:
I1 = VP / R1 = IPRP / R1 = IPR2 / (R1 + R2)
I2 = VP / R2 = IPRP / R2 = IPR1 / (R1 + R2)

In general terms, for resistances R1, R2, R3, ... (with conductances G1, G2, G3, ...) connected in parallel:
VP = IPRP = IP / GP = IP / (G1 + G2 + G3 + ...)
In = VP / Rn = VPGn = IPGn / GP = IPGn / (G1 + G2 + G3 + ...)
where Gn = 1 / Rn
Note that the highest current passes through the highest conductance (with the lowest resistance).

#### Capacitance

When a voltage is applied to a circuit containing capacitance, current flows to accumulate charge in the capacitance:
Q = idt = CV

Alternatively, by differentiation with respect to time:
dq/dt = i = C dv/dt
Note that the rate of change of voltage has a polarity which opposes the flow of current.

The capacitance C of a circuit is equal to the charge divided by the voltage:
C = Q / V = idt / V

Alternatively, the capacitance C of a circuit is equal to the charging current divided by the rate of change of voltage:
C = i / dv/dt = dq/dt / dv/dt = dq/dv

#### Capacitances in Series

When capacitances C1, C2, C3, ... are connected in series, the total capacitance CS is:
1 / CS = 1 / C1 + 1 / C2 + 1 / C3 + ...

For two capacitances C1 and C2 connected in series, the total capacitance CS is:
CS = C1C2 / (C1 + C2)
CS = product / sum

#### Voltage Division by Series Capacitances

When a total voltage ES is applied to series connected capacitances C1 and C2, the charge QS which accumulates in the series circuit is:
QS = iSdt = ESCS = ESC1C2 / (C1 + C2)

The voltages V1 and V2 which appear across the respective capacitances C1 and C2 are:
V1 = iSdt / C1 = ESCS / C1 = ESC2 / (C1 + C2)
V2 = iSdt / C2 = ESCS / C2 = ESC1 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in series:
QS = iSdt = ESCS = ES / (1 / CS) = ES / (1 / C1 + 1 / C2 + 1 / C3 + ...)
Vn = iSdt / Cn = ESCS / Cn = ES / Cn(1 / CS) = ES / Cn(1 / C1 + 1 / C2 + 1 / C3 + ...)
Note that the highest voltage appears across the lowest capacitance.

#### Capacitances in Parallel

When capacitances C1, C2, C3, ... are connected in parallel, the total capacitance CP is:
CP = C1 + C2 + C3 + ...

#### Charge Division by Parallel Capacitances

When a voltage EP is applied to parallel connected capacitances C1 and C2, the charge QP which accumulates in the parallel circuit is:
QP = iPdt = EPCP = EP(C1 + C2)

The charges Q1 and Q2 which accumulate in the respective capacitances C1 and C2 are:
Q1 = i1dt = EPC1 = QPC1 / CP = QPC1 / (C1 + C2)
Q2 = i2dt = EPC2 = QPC2 / CP = QPC2 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in parallel:
QP = iPdt = EPCP = EP(C1 + C2 + C3 + ...)
Qn = indt = EPCn = QPCn / CP = QPCn / (C1 + C2 + C3 + ...)
Note that the highest charge accumulates in the highest capacitance.

#### Inductance

When the current changes in a circuit containing inductance, the magnetic linkage changes and induces a voltage in the inductance:
dy/dt = e = L di/dt
Note that the induced voltage has a polarity which opposes the rate of change of current.

Alternatively, by integration with respect to time:
Y = edt = LI

The inductance L of a circuit is equal to the induced voltage divided by the rate of change of current:
L = e / di/dt = dy/dt / di/dt = dy/di

Alternatively, the inductance L of a circuit is equal to the magnetic linkage divided by the current:
L = Y / I

Note that the magnetic linkage Y is equal to the product of the number of turns N and the magnetic flux F:
Y = NF = LI

#### Mutual Inductance

The mutual inductance M of two coupled inductances L1 and L2 is equal to the mutually induced voltage in one inductance divided by the rate of change of current in the other inductance:
M = E2m / (di1/dt)
M = E1m / (di2/dt)

If the self induced voltages of the inductances L1 and L2 are respectively E1s and E2s for the same rates of change of the current that produced the mutually induced voltages E1m and E2m, then:
M = (E2m / E1s)L1
M = (E1m / E2s)L2
Combining these two equations:
M = (E1mE2m / E1sE2s) (L1L2) = kM(L1L2)
where kM is the mutual coupling coefficient of the two inductances L1 and L2.

If the coupling between the two inductances L1 and L2 is perfect, then the mutual inductance M is:
M = (L1L2)

#### Inductances in Series

When uncoupled inductances L1, L2, L3, ... are connected in series, the total inductance LS is:
LS = L1 + L2 + L3 + ...

When two coupled inductances L1 and L2 with mutual inductance M are connected in series, the total inductance LS is:
LS = L1 + L2 � 2M
The plus or minus sign indicates that the coupling is either additive or subtractive, depending on the connection polarity.

#### Inductances in Parallel

When uncoupled inductances L1, L2, L3, ... are connected in parallel, the total inductance LP is:
1 / LP = 1 / L1 + 1 / L2 + 1 / L3 + ...

#### Time Constants

Capacitance and resistance
The time constant of a capacitance C and a resistance R is equal to CR, and represents the time to change the voltage on the capacitance from zero to E at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

Similarly, the time constant CR represents the time to change the charge on the capacitance from zero to CE at a constant charging current E / R (which produces a rate of change of voltage E / CR across the capacitance).

If a voltage E is applied to a series circuit comprising a discharged capacitance C and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:
i = (E / R)e - t / CR
vR = iR = Ee - t / CR
vC = E - vR = E(1 - e - t / CR)
qC = CvC = CE(1 - e - t / CR)

If a capacitance C charged to voltage V is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vC across the capacitance and the charge qC on the capacitance are:
i = (V / R)e - t / CR
vR = iR = Ve - t / CR
vC = vR = Ve - t / CR
qC = CvC = CVe - t / CR

Inductance and resistance
The time constant of an inductance L and a resistance R is equal to L / R, and represents the time to change the current in the inductance from zero to E / R at a constant rate of change of current E / L (which produces an induced voltage E across the inductance).

If a voltage E is applied to a series circuit comprising an inductance L and a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:
i = (E / R)(1 - e - tR / L)
vR = iR = E(1 - e - tR / L)
vL = E - vR = Ee - tR / L
yL = Li = (LE / R)(1 - e - tR / L)

If an inductance L carrying a current I is discharged through a resistance R, then after time t the current i, the voltage vR across the resistance, the voltage vL across the inductance and the magnetic linkage yL in the inductance are:
i = Ie - tR / L
vR = iR = IRe - tR / L
vL = vR = IRe - tR / L
yL = Li = LIe - tR / L

Rise Time and Fall Time
The rise time (or fall time) of a change is defined as the transition time between the 10% and 90% levels of the total change, so for an exponential rise (or fall) of time constant T, the rise time (or fall time) t10-90 is:
t10-90 = (ln0.9 - ln0.1)T 2.2T

The half time of a change is defined as the transition time between the initial and 50% levels of the total change, so for an exponential change of time constant T, the half time t50 is :
t50 = (ln1.0 - ln0.5)T 0.69T

Note that for an exponential change of time constant T:
- over time interval T, a rise changes by a factor 1 - e -1 ( 0.63) of the remaining change,
- over time interval T, a fall changes by a factor e -1 ( 0.37) of the remaining change,
- after time interval 3T, less than 5% of the total change remains,
- after time interval 5T, less than 1% of the total change remains.

#### Power

The power P dissipated by a resistance R carrying a current I with a voltage drop V is:
P = V2 / R = VI = I2R

Similarly, the power P dissipated by a conductance G carrying a current I with a voltage drop V is:
P = V2G = VI = I2 / G

The power P transferred by a capacitance C holding a changing voltage V with charge Q is:
P = VI = CV(dv/dt) = Q(dv/dt) = Q(dq/dt) / C

The power P transferred by an inductance L carrying a changing current I with magnetic linkage Y is:
P = VI = LI(di/dt) = Y(di/dt) = Y(dy/dt) / L

#### Energy

The energy W consumed over time t due to power P dissipated in a resistance R carrying a current I with a voltage drop V is:
W = Pt = V2t / R = VIt = I2tR

Similarly, the energy W consumed over time t due to power P dissipated in a conductance G carrying a current I with a voltage drop V is:
W = Pt = V2tG = VIt = I2t / G

The energy W stored in a capacitance C holding voltage V with charge Q is:
W = CV2 / 2 = QV / 2 = Q2 / 2C

The energy W stored in an inductance L carrying current I with magnetic linkage Y is:
W = LI2 / 2 = YI / 2 = Y2 / 2L

#### Batteries

If a battery of open-circuit voltage EB has a loaded voltage VL when supplying load current IL, the battery internal resistance RB is:
RB = (EB - VL) / IL

VL = ILRL = EB - ILRB = EBRL / (RB + RL)
IL = VL / RL = (EB - VL) / RB = EB / (RB + RL)

The battery short-circuit current Isc is:
Isc = EB / RB = EBIL / (EB - VL)

#### Voltmeter Multiplier

The resistance RS to be connected in series with a voltmeter of full scale voltage VV and full scale current drain IV to increase the full scale voltage to V is:
RS = (V - VV) / IV

The power P dissipated by the resistance RS with voltage drop (V - VV) carrying current IV is:
P = (V - VV)2 / RS = (V - VV)IV = IV2RS

#### Ammeter Shunt

The resistance RP to be connected in parallel with an ammeter of full scale current IA and full scale voltage drop VA to increase the full scale current to I is:
RP = VA / (I - IA)

The power P dissipated by the resistance RP with voltage drop VA carrying current (I - IA) is:
P = VA2 / RP = VA(I - IA) = (I - IA)2RP

#### Wheatstone Bridge

The Wheatstone Bridge consists of two resistive potential dividers connected to a common voltage source. If one potential divider has resistances R1 and R2 in series and the other potential divider has resistances R3 and R4 in series, with R1 and R3 connected to one side of the voltage source and R2 and R4 connected to the other side of the voltage source, then at the balance point where the two resistively divided voltages are equal:
R1 / R2 = R3 / R4

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