Electronic Circuit Formulas

Contents

- Notation
- Resistance
- Resistances in Series
- Voltage Division by Series Resistances
- Resistances in Parallel
- Current Division by Parallel Resistances
- Capacitance
- Capacitances in Series
- Capacitances in Parallel
- Inductance
- Inductances in Series
- Inductances in Parallel
- Time Constants
- Power
- Energy
- Disclaimer

Notation

The library uses the symbol font for some of the notation and formulas. If the symbols for the letters 'alpha beta delta' do not appear here [a b d] then the symbol font needs to be installed before all notation and formulas will be displayed correctly.

C
E
G
I
i
L
M 		capacitance
voltage source
conductance
current
instantaneous current
inductance
mutual inductance 		(Farads)
(Volts)
(Siemens)
(Amps)
(Amps)
(Henrys)
(Henrys) 		P
Q
R
t
V
v
W
 power
charge
resistance
time
voltage drop
instantaneous voltage
energy
 (Watts)
(Coulombs)
(Ohms)
(seconds)
(Volts)
(Volts)
(Joules)

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 / (R1 + R2)
V2 = ISR2 = ESR2 / (R1 + R2)

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 G = 1 / R

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 = IPR2 / (R1 + R2)
I2 = VP / R2 = IPR1 / (R1 + R2)

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

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 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 + ...

Capacitances in Parallel

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

Inductance

When the current changes in a circuit containing inductance, a voltage is induced in the circuit:
E = - L di/dt
The negative sign indicates that the induced voltage opposes the rate of change of current.

The inductance L of a circuit is equal to the induced voltage divided by the rate of change of current:
L = - E / (di/dt)

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 = - E1 / (di2/dt)
M = - E2 / (di1/dt)
If the coupling between the two inductances is perfect, then:
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).

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 and the voltage vC across the capacitance are:
i = (E / R)e - t / CR
vR = iR = Ee - t / CR
vC = E - vR = E(1 - e - 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 a pure inductance L and a resistance R, then after time t the current i, the voltage vR across the resistance and the voltage vL across the inductance are:
i = (E / R)(1 - e - tR / L)
vR = iR = E(1 - e - tR / L)
vL = E - vR = Ee - tR / L

Power

When a voltage E causes a current I to flow through a resistance R, the power P dissipated is:
P = E2 / R = EI = I2R

Energy

The energy W consumed due to power P over time t is:
W = Pt

The energy W consumed by a resistance R carrying a current I due to a voltage E is:
W = E2t / R = EIt = I2Rt

The energy W stored in a capacitance C holding a voltage V is:
W = ½CV2

The energy W stored in an inductance L carrying a current I is:
W = ½LI2

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