 ## Library - Electrical Circuit Formulae

 Contents

 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. C E e G I i k L M N P capacitance voltage source instantaneous E conductance current instantaneous I coefficient inductance mutual inductance number of turns power [farads, F] [volts, V] [volts, V] [siemens, S] [amps, A] [amps, A] [number] [henrys, H] [henrys, H] [number] [watts, W] Q q R T t V v W F Y y charge instantaneous Q resistance time constant instantaneous time voltage drop instantaneous V energy magnetic flux magnetic linkage instantaneous Y [coulombs, C] [coulombs, C] [ohms, W] [seconds, s] [seconds, s] [volts, V] [volts, V] [joules, J] [webers, Wb] [webers, Wb] [webers, Wb]

#### 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 = òidt = 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 = òidt / 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 = òiSdt = ESCS = ESC1C2 / (C1 + C2)

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

In general terms, for capacitances C1, C2, C3, ... connected in series:
QS = òiSdt = ESCS = ES / (1 / CS) = ES / (1 / C1 + 1 / C2 + 1 / C3 + ...)
Vn = òiSdt / 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 = òiPdt = EPCP = EP(C1 + C2)

The charges Q1 and Q2 which accumulate in the respective capacitances C1 and C2 are:
Q1 = òi1dt = EPC1 = QPC1 / CP = QPC1 / (C1 + C2)
Q2 = òi2dt = EPC2 = QPC2 / CP = QPC2 / (C1 + C2)

In general terms, for capacitances C1, C2, C3, ... connected in parallel:
QP = òiPdt = EPCP = EP(C1 + C2 + C3 + ...)
Qn = òindt = 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 = òedt = 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)½ (L1L2)½ = kM(L1L2)½
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)½

#### 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

If the value of resistance R4 is unknown and the values of resistances R3, R2 and R1 at the balance point are known, then:
R4 = R3R2 / R1

 Updated 09 June 2008 Copyright ©1998-2008 BOWest Pty Ltd All content subject to disclaimer and copyright notices Home | Staff | Library | Links | Feedback | Email