Ohm's Law

Voltage (E) in Volts E = I × R E = √(P × R) E = P / I

Resistance (R) in Ohms R = E2 / P R = P / I2 R = E / I

Current (I) in Amps I = E / R I = P / E I = √(P / R)

Power (P) in Watts P = I × E P = E2/R P = I2×R

Resistor Network: Series
When resistances R₁, R₂, R₃, ... are connected in series, the total resistance R_S is:
R_S = R₁ + R₂ + R₃ + ...

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

Alternatively, when conductances G₁, G₂, G₃, ... are connected in parallel, the total conductance G_P is:
G_P = G₁ + G₂ + G₃ + ...
where G_n = 1 / R_n

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

The resistance R₂ to be connected in parallel with resistance R₁ to give a total resistance R_P is:
R₂ = R₁R_P / (R₁ - R_P)
R₂ = product / difference

Resistor Network: Voltage Divider
When a total voltage E_S is applied across series connected resistances R₁ and R₂, the current I_S which flows through the series circuit is:
I_S = E_S / R_S = E_S / (R₁ + R₂)

The voltages V₁ and V₂ which appear across the respective resistances R₁ and R₂ are:
V₁ = I_SR₁ = E_SR₁ / (R₁ + R₂)
V₂ = I_SR₂ = E_SR₂ / (R₁ + R₂)

In general terms, for resistances R₁, R₂, R₃, ... connected in series:
I_S = E_S / R_S = E_S / (R₁ + R₂ + R₃ + ...)
V_n = I_SR_n = E_SR_n / R_S = E_SR_n / (R₁ + R₂ + R₃ + ...)
Note that the highest voltage drop appears across the highest resistance.

Resistor Network: Current Divider
When a total current I_P is passed through parallel connected resistances R₁ and R₂, the voltage V_P which appears across the parallel circuit is:
V_P = I_PR_P = I_PR₁R₂ / (R₁ + R₂)

The currents I₁ and I₂ which pass through the respective resistances R₁ and R₂ are:
I₁ = V_P / R₁ = I_PR₂ / (R₁ + R₂)
I₂ = V_P / R₂ = I_PR₁ / (R₁ + R₂)

In general terms, for resistances R₁, R₂, R₃, ... (with conductances G₁, G₂, G₃, ...) connected in parallel:
V_P = I_PR_P = I_P / G_P = I_P / (G₁ + G₂ + G₃ + ...)
I_n = V_P / R_n = V_PG_n = I_PG_n / G_P = I_PG_n / (G₁ + G₂ + G₃ + ...)
where G_n = 1 / R_n
Note that the highest current passes through the highest conductance (with the lowest resistance).

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

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

Capacitor Network: Parallel
When capacitances C₁, C₂, C₃, ... are connected in parallel, the total capacitance C_P is:
C_P = C₁ + C₂ + C₃ + ...

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.

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 = E_2m / (di₁/dt)
M = E_1m / (di₂/dt)

If the self induced voltages of the inductances L₁ and L₂ are respectively E_1s and E_2s for the same rates of change of the current that produced the mutually induced voltages E_1m and E_2m, then:
M = (E_2m / E_1s)L₁
M = (E_1m / E_2s)L₂
Combining these two equations:
M = (E_1mE_2m / E_1sE_2s)^1/2 (L₁L₂)^1/2 = k_M(L₁L₂)^1/2
where k_M 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)^1/2

Inductor Network: Series
When uncoupled inductances L₁, L₂, L₃, ... are connected in series, the total inductance L_S is:
L_S = L₁ + L₂ + L₃ + ...

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

Inductor Network: Parallel
When uncoupled inductances L₁, L₂, L₃, ... are connected in parallel, the total inductance L_P is:
1 / L_P = 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 v_R across the resistance, the voltage v_C across the capacitance and the charge q_C on the capacitance are:
i = (E / R)e ^{- t / CR}
v_R = iR = Ee ^{- t / CR}
v_C = E - v_R = E(1 - e ^{- t / CR})
q_C = Cv_C = 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 v_R across the resistance, the voltage v_C across the capacitance and the charge q_C on the capacitance are:
i = (V / R)e ^{- t / CR}
v_R = iR = Ve ^{- t / CR}
v_C = v_R = Ve ^{- t / CR}
q_C = Cv_C = 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 v_R across the resistance, the voltage v_L across the inductance and the magnetic linkage y_L in the inductance are:
i = (E / R)(1 - e ^{- tR / L})
v_R = iR = E(1 - e ^{- tR / L})
v_L = E - v_R = Ee ^{- tR / L}
y_L = 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 v_R across the resistance, the voltage v_L across the inductance and the magnetic linkage y_L in the inductance are:
i = Ie ^{- tR / L}
v_R = iR = IRe ^{- tR / L}
v_L = v_R = IRe ^{- tR / L}
y_L = 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) t_10-90 is:
t_10-90 = (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 E_B has a loaded voltage V_L when supplying load current I_L, the battery internal resistance R_B is:
R_B = (E_B - V_L) / I_L

The load voltage V_L and load current I_L for a load resistance R_L are:
V_L = I_LR_L = E_B - I_LR_B = E_BR_L / (R_B + R_L)
I_L = V_L / R_L = (E_B - V_L) / R_B = E_B / (R_B + R_L)

The battery short-circuit current I_sc is:
I_sc = E_B / R_B = E_BI_L / (E_B - V_L)

Voltmeter Multiplier
The resistance R_S to be connected in series with a voltmeter of full scale voltage V_V and full scale current drain I_V to increase the full scale voltage to V is:
R_S = (V - V_V) / I_V

The power P dissipated by the resistance R_S with voltage drop (V - V_V) carrying current I_V is:
P = (V - V_V)² / R_S = (V - V_V)I_V = I_V²R_S

Ammeter Shunt
The resistance R_P to be connected in parallel with an ammeter of full scale current I_A and full scale voltage drop V_A to increase the full scale current to I is:
R_P = V_A / (I - I_A)

The power P dissipated by the resistance R_P with voltage drop V_A carrying current (I - I_A) is:
P = V_A² / R_P = V_A(I - I_A) = (I - I_A)²R_P