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Resistance                                                             

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

R = V / 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 VS is applied across series connected resistances R1 and R2, the current IS which flows through the

series circuit is:

IS = VS / RS = VS / (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)

In general terms, for resistances R1, R2, R3, ... connected in series:

IS = VS / RS = VS / (R1 + R2 + R3 + ...)

Vn = ISRn = VSRn / RS = VSRn / (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 = IPR2 / (R1 + R2)

I2 = VP / 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

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

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, 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

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 + ...

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