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Ohm's low

 

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

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).