Electrical Formulas for Series and Parallel Circuits

Contents

DC Circuit containing one resistor (Figure One)

Rules of Series Circuits

DC Series Circuit containing two resistors R1 and R2 (Figure One)

Rules of Parallel Circuits

DC Parallel Circuit containing two resistors R1 and R2 (Figure One)

AC Series Circuit containing R, C and L (Figure Two)

AC Parallel Circuit containing R and L (Figure Two)

AC Parallel Circuit containing R and C (Figure Two)

AC Parallel Circuit containing L and C (Figure Two)

AC Parallel Circuit containing R, L and C (Figure Two)

AC Power

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DC Circuit containing one resistor (Figure One)

E: electromotive force - Unit of measurement: volts

I: current - Unit of measurement: amperes

R: resistance - Unit of measurement: ohms

P: power - Unit of measurement: watts

A^2 means A squared.

A*(1/2) means square root of A

E = I * R

R = E/I

I = E/R

P = E*I = (I*R) * I = (I^2) * R

E = P/I = (I^2) * R/I = I * R

I = E/P

Rules of Series Circuits

Total voltage = sum of voltage drops

Total current = current through each component

Total Resistance = sum of resistances

DC Series Circuit containing two resistors R1 and R2 (Figure One)

R1: Resistor

R2: Resistor

Vr1: voltage drop across R1

Vr2: voltage drop across R2

Rt: total resistance

E = I * R1 + I * R2

I = same through each component

I = E/(R1 + R2)

Vr1 = I * R1

R1 = Vr1/I

I = Vr1/R

Vr2 = I * R2

R1 = Vr2/I

I = Vr2/R

E = Vr1 + Vr2

Rt = R1 + R2

P = E * I

Rules of Parallel Circuits

Voltage = same across each branch

Current = sum of currents through each branch

Resistance = less than least resistance

DC Parallel Circuit containing two resistors R1 and R2 (Figure One)

I = I*R1 + I*R2

E = same through each branch

E = Vr1 = Vr2

1/Rt = 1/R1 + 1/R2

Rt = R1 * R2/(R1 + R2)

P = E * I

Rules of Series Circuits

Total voltage = sum of voltage drops

Total current = current through each component

Total Resistance = sum of resistances

AC Series Circuit containing R, C and L (Figure Two)

L: Inductor - Unit of Measurement: Henries

C: Capacitor - Unit of Measurement: Farads

XL: Inductive Reactance - Unit of Measurement: Ohms

XC: Capacitive Reactance - Unit of Measurement: Ohms

Z: Impedance - Unit of measurement: Ohms

o: angle - Unit of Measurement: Degrees

X: algebraic sum of XC and XL

j: (-1)^(1/2)

XL = 2 * pi * F * L

XC = 1/(2 * pi * F * C)

Z = R + j*XL - j*XC

Z = R^2 + (XL - XC)^2

Z^2 = R^2 + X^2

Z = Z * (cos o) + j*Z * (sin o)

I = E/Z = same through each component

E = I * (R^2 + (XL - XC)^2 )^(1/2)

angle o = arctan (X/R)

Rules of Parallel Circuits

Voltage = same across each branch

Current = sum of currents through each branch

Resistance = less than least resistance

AC Parallel Circuit containing R and L (Figure Two)

1/Z = 1/R + 1/( j*XL)

I = sum of currents through each branch

I = (I * cos o) + j * (I * sin o)

E = same through each branch

E = I*Z

angle o = arctan (XL/R)

AC Parallel Circuit containing R and C (Figure Two)

Ir: current through R

Ixc: current through XC

1/Z = 1/R + 1/(1/( j*XC))

I = sum of currents through each branch

Ir = E/R

Ixc = E/(- j*XC)

I = Ir - j*Ixc

I = I * cos o - j * I * sin o

E = same through each branch

E = I*Z

angle o = arctan (-XC/R)

AC Parallel Circuit containing L and C (Figure Two)

IXL: current through XL

IXC: current through XC

1/Z = 1/(j*XL) + 1/(1/(j*XC))

I = sum of currents through each branch

IXL = E/(j*XL)

IXC = E/(j*XC)

I = j*IXL - j*IXC

E = same through each branch

E = I*Z

If XL > XC then

angle o = 90 degrees

If Xc > XL then

angle o = -90 degrees

AC Parallel Circuit containing R, L and C (Figure Two)

1/Z = 1/R + 1/(j*XL) + 1/(1/(j*XC))

E = same through each branch

E = I*Z

I equals sum of curents through each branch

I = IR + jIX

IX = I * j*(IXL - IXC)

If XL > XC then

0 < angle o <= 90

If XL < XC then

-90 <= angle o < 0

If XL = XC then

Angle o = 0 degrees

AC Power

AP = Apparent Power

RP = Reactive Power

TP = True Power

PF = Power Factor

TP = V * I * cos o = I^2 * R

RP = V * I * sine o = I^2 * X

AP = V * I = V * I * cos o + j * V * I * sine o

(AP)^2 = (TP)^2 + (RP)^2

PF = cos o = TP/AP

References:
I have a Bachelor of Science in Electrical Engineering.

Introductory Circuit Analysis 3rd Edition
Robert l Boylestad
ISBN 0-675-8559-4