Series/Parallel Calculations

Here are the formulas for calculation of the parameters of series and parallel circuits of lamps, speakers, or impedances.

The rows marked "SAME" have identical impedances for all of the lamps, speakers, or impedance items in the circuit.

The rows marked "DIFF" do not have identical impedances for all of the lamp or impedance items in the circuit.

Light Bulbs, Speakers, and Impedances


TYPE		Parameter	Number n	Resistance R	Current I	Voltage V	Power P
	R_i	Original (o) Values	n = 1	R_o = V_o / I_o	I_o = P_o / V_o	V_o = Rated V	P_o = Rated W

S E R I E S	D I F F	Totals	n = # Lamps	R_t = Σ(R_i, i=1,n)	I_t = V_t / R_t	V_t = V_s (Supply)	P_t = V_t × I_t
		Individual	n = # Lamps	R_i = V_t / I_t	I_i = I_t	V_i = I_t × R_i	P_i = V_i × I_i
		Individual	n = # Lamps	R_i = V_i ² / P_i	I_i = V_t / R_t	V_i = P_i / I_t	P_i = V_i ² / R_i

	S A M E	Totals (V_s = V_o)	n = # Lamps	R_t = R_o × n	I_t = I_o / n	V_t = V_o	P_t = P_o / n
		Individual (V_s = V_o)	n = # Lamps	R_i = R_o	I_i = I_o / n	V_i = V_o / n	P_i = P_o / n ²
		Total (V_s ≠ V_o)	n = # Lamps	R_t = R_o × n	I_t = V_s / R_t	V_t = V_s	P_t = V_s ² / R_t
		Individual (V_s ≠ V_o)	n = # Lamps	R_i = R_o	I_i = V_s / R_t	V_i = V_s / n	P_i = P_t / n ²

P A R A L L E L	D I F F	Totals	n = # Lamps	R_t = 1 / Σ(1 / R_i, i=1,n)	I_t = V_t / R_t	V_t = V_s (Supply)	P_t = V_t × I_t
		Individual	n = # Lamps	R_i = V_s / I_i	I_i = P_i / V_s	V_i = V_s	P_i = V_s × I_i
		Individual	n = # Lamps	R_i = V_s ² / P_i	I_i = V_s / R_i	V_i = V_t	P_i = V_s ² / R_i

	S A M E	Totals (V_s = V_o)	n = # Lamps	R_t = R_o / n	I_t = I_o × n	V_t = V_o	P_t = P_o × n
		Individual (V_s = V_o)	n = # Lamps	R_i = R_o	I_i = I_o	V_i = V_o	P_i = P_o
		Total (V_s < V_o)	n = # Lamps	R_t = R_o / n	I_t = V_s / R_t	V_t = V_s	P_t = V_s ² / R_t
		Individual (V_s ≠ V_o)	n = # Lamps	R_i = R_o	I_i = V_s / R_i	V_i = V_s	P_i = P_t / n

P A R − S E R	S A M E	Totals (V_s = V_o)	n ² = # Lamps n = # Lamps / Series n = # of Series	R_t = R_o	I_t = I_o	V_t = V_o	P_t = P_o
		Individual (V_s = V_o)	n ² = # Lamps n = # Lamps / Series n = # of Series	R_i = R_o	I_i = I_o / n	V_i = V_o / n	P_i = P_o / n ²
		Total (V_s < V_o)	n ² = # Lamps n = # Lamps / Series n = # of Series	R_t = R_o	I_t = V_s / R_o	V_t = V_s	P_t = V_s ² / R_o
		Individual (V_s ≠ V_o)	n ² = # Lamps n = # Lamps / Series n = # of Series	R_i = R_o	I_i = V_s / R_o / n	V_i = V_s / n	P_i = P_t / n ²

Adjustments to calculation for lamps with tungsten filaments

Because tungsten has a relatively high temperature coefficient, calculations for tungsten-filament lamps are different from the calculations above.

Tungsten has a linear response to power. Its resistance changes linearly from its cold resistance to full-power illumination.

Calculations for series and parallel circuits of batteries

Here are the formulas for calculation of the parameters of series and parallel circuits of batteries.

The rows marked "SAME" have identical characteristics for all of the batteries in the circuit.

Use of batteries with different characteristics causes problems with battery life.

Batteries


TYPE		Parameter	Number n	Current I_max	Current Iτ	Voltage V	Energy ε
Cells		Original (o) Values	n = 1	I_o = Rated I_max	Iτ_o = Rated A Hr	V_o = Rated V	ε_o = Rated W Hr

S E R I E S	S A M E	Totals	n = # Cells	I_t = I_o	Iτ_t = Iτ_o	V_t = V_o × n	ε_t = ε_o × n
S E R I E S	S A M E	Individual Cells	n = # Cells	I_i = I_o	Iτ_i = Iτ_o	V_i = V_o	ε_i = ε_o

P A R A L L E L	S A M E	Totals	n = # Cells	I_t = I_o × n	Iτ_t = Iτ_o × n	V_t = V_o	ε_t = ε_o × n
P A R A L L E L	S A M E	Individual Cells	n = # Cells	I_i = I_o	Iτ_i = Iτ_o	V_i = V_o	ε_i = ε_o

P A R − S E R	S A M E	Totals	n ² = # Cells n = # Cells / Series n = # of Series	I_t = I_o × n	Iτ_t = Iτ_o × n	V_t = V_o × n	ε_t = ε_o × n ²
P A R − S E R	S A M E	Individual Cells	n ² = # Cells n = # Cells / Series n = # of Series	R_i = R_o	I_i = I_o	V_i = V_o	ε_i = ε_o

CALCULATIONS FOR CIRCUITS SELECTING
BETWEEN SERIES OR PARALLEL OPERATION
OF VARIOUS ELECTRICAL DEVICES

Calculations for series and parallel circuits of loads

Adjustments to calculation for lamps with tungsten filaments

Calculations for series and parallel circuits of batteries

LINKS

CALCULATIONS FOR CIRCUITS SELECTING BETWEEN SERIES OR PARALLEL OPERATION OF VARIOUS ELECTRICAL DEVICES

Calculations for series and parallel circuits of loads

Adjustments to calculation for lamps with tungsten filaments

Calculations for series and parallel circuits of batteries

LINKS

CALCULATIONS FOR CIRCUITS SELECTING
BETWEEN SERIES OR PARALLEL OPERATION
OF VARIOUS ELECTRICAL DEVICES