A Tutorial on Series and Parallel AC Circuits

This article covers the principles of series and parallel AC Circuits. The article is divided into the following sections:

Inductive Reactance
Capacitive Reactance
Transcendental Functions
Complex Math vs Polar Notation
The Inductor
The Capacitor
The Series RL Circuit
The Series RC Circuit
The Parallel RL Circuit
The Parallel RC Circuit

INDUCTIVE REACTANCE

The inductive reactance equation is

XL = 2*pi*f*L

where

pi = 3.14

f = frequency in cycles per second

L = inductance in henries

XL = inductive reactance in ohms

CAPACITIVE REACTANCE

Xc = 1/( 2*pi*f*C)

where

where

pi = 3.14

f = frequency in cycles per second

C = capacitance in farads

XC = capacitive reactance in ohms.

TRANSCENDENTAL EQUATIONS

One of the first formulas we come across when solving AC circuit problems is an equation describing a sine wave:

v = V * sin (wt)

where

v = instantaneous voltage

V = peak voltage

where

wt = 2 * pi * f * t

and is measured in radians.

pi = 3.14

f = frequency in cycles per second

t = time in seconds

Remember that one radian is the distance around the the portion of the circle defined by an angle of 57.3 degrees

If the frequency = 1 cycle per second and the time = 1 second then

2 * pi * f * t = 6.28 radians.

Since the frequency is 1 cycle per second and the time is one second

6.28 radians = one complete cycle or 360 degrees.

Therefore

2 * pi = 360 degrees.

pi = 180 degrees

and

pi/2 = 90 degrees

Lets examine the difference between

v = V * sin (2 * pi * f * t) Equation one

and

v = V * sin (2 * pi * f * t + pi/2) Equation two

When t = 0, equation one becomes

v = V * sin (0 radians) = 0

and equation two becomes

v = V * sin (0 radians + (pi/2) radians

v = V * sin (90 degrees)

sin (90) = 1

Therefore

v = V * 1

Likewise

i = I * sin (wt)

where

I = peak current

and

i = instantaneous current.

And likewise

z = Z * sin (wt)

where

Z = peak current

and

z = instantaneous current.

COMPLEX MATH VS POLAR NOTATION

Complex math is of the form a + jb where

a is the resistance and b is the reactance.

Polar Notation is of the form

V/a

where

V is a polar vector

and the a is the angle between the vector and the horizontal axis.

To convert polar notation to a complex expression we use the equation

V/angle = V * cos (angle) + j * V * sin (angle)

where

V * cos (angle) is the voltage drop ac cross the resistive component of the complex expression.

Using the Cartesian Coordinate System we have a visual aid for polar notation and complex expressions. The magnitude of the voltage drop ac cross the resistance is on the horizontal axis.

and

V * sin (angle) is the magnitude of the voltage drop ac cross the reactive component of the complex expression. The magnitude of the voltage drop ac cross the reactive component is on the vertical axis. The positive half of the axis represents the voltage drop due to inductive reactance. The negative half of the axis represents the voltage drop due to capacitive reactance.

The same holds true for current and impedance.

When confronted with a problem that requires the use of complex numbers or polar coordinates,

my suggestion is to use whatever is more comfortable for solving a particular problem.

For multiplication and division, polar coordinates are easier to use.

5/40 * 10/50 = 5*10/40+50

(10/50)/(5/40) = (10/5)/50-40

For addition and subtraction, complex numbers are easier to use

(5 + j6) + (3 + j8) = (5 + 3) + j(6 + 8)

(5 + j6) - (3 + j8) = (5 - 3) + j(6 - 8)

THE INDUCTOR

In a circuit containing an inductor, the voltage leads the current by 90 degrees.

Here's how to remember this:

The formula for the instantaneous voltage across the inductor is

v = L*(di/dt)

where v is measured in volts,

L in measured in henries

di/dt is the change in current over a very small finite period of time.

When one first applies power to a simple circuit consisting of one inductor and a power source, the current tries to rise dramatically causing a voltage to appear across the inductor. Hence the voltage peaks out before the current. The voltage "leads" the current in an inductor.

Hence if

i = I* sin (wt)

then

v = V* sin (wt + (pi/2))

Hence, if we have a single inductor in a circuit, the voltage will lead the current by 90 degrees. When the current is changing at it's maximum rate, the voltage will be at it's peak.

Unlike in DC circuits, if a sine wave is applied to an inductor, the sine wave will not be distorted. The sine wave representing the voltage across the inductor will be shifted 90 degrees. Hence, at the instant power is provided, the sine wave representing the voltage will be at it peak and the sine wave representing the current will be at zero.

THE CAPACITOR

In a circuit containing a capacitor, the voltage lags the current by 90 degrees

Hence if we have a single capacitor in a circuit, the voltage will lag the current by 90 degrees.

Here is how to remember this:

The formula for the instantaneous current through a capacitor is

i = C * (dv/dt)

where

C is the capacitance in farads

and

dv/dt is the change in voltage over a small finite period of time.

When power is applied to a circuit containing one capacitor, the voltage rises steeply causing the current to peak.

Hence the current peaks out before the voltage does and hence we say that the current leads the voltage.

If

v = V * sin (wt)

then

i = I * sin (wt + (pi/2))

Unlike in DC circuits, if a sine wave is applied to an capacitor, the sine wave will not be distorted. The sine wave representing the current through the capacitor will be shifted 90 degrees. Hence, at the instant power is provided, the sine wave representing the current will be at it peak and the sine wave representing the voltage will be at zero.

AC SERIES CIRCUITS

Let's review the basic laws of a series circuit.

The total voltage is the algebraic sum of the voltage drops and gains across each component. The voltage gain would occur across the power supply.

The total current is the same throughout a series circuit.

The total resistance is the sum of the individual resistances.

THE SERIES RL CIRCUIT

The complex expression for voltage in an AC series circuit containing an inductor and a resistor is

V = V * cos O + j* V * sin O

or

V = Vr + jVx

The solution for this expression is

V^2 = Vr^2 + Vx^2

where ^2 means squared.

The angle is equal to arc tan (Vx)/(Vr)

In a series circuit containing an inductor and a resistor, the current through the resistor will be the same amplitude as the current through the inductor. Therefore if the current in a series circuit is 2/45 degrees, then the current vector through the resistor will be equal (in magnitude and direction) to the current vector through the inductor. This is something to be aware of because it is all too easy to make a mistake.

Therefore the vector calculation

It/arc tan (Vx/Vr) = Ir + jIx will find the total series current vector.

The current through the resistor is equal to the current through the inductor

Ir/arc tan (Vx/Vr) = Ix/arc tan (Vx/Vr)

Another words, the current through the inductor will be equal in phase and magnitude with the current through the resistor because the current is the same in a series circuit.

Using complex expressions

Ir + jIx = (Vr + jVx)/(R + jX)

where Z = R + jXL

The total voltage in a series circuit is the sum of the voltage drops across each component.

The voltage through the inductor will be out of phase with the voltage across the resistor.

The source voltage in polar form will be

V @ arc tan (Vx/Vr)

which is expressed as V/arc tan (Vx/Vr)

The complex expression is

V/arc tan (Vx/Vr) = Vr + jVx

The voltage across the resistor is

Vr = V * cos (arc tan (XL/R))

The voltage across the inductor is

Vx = V * sin (arc tan (XL/R))

The total voltage in the series circuit will be

V/arc tan (XL/R) = V * cos (arc tan (XL/R)) + j * V * sin (arc tan (XL/R))

THE SERIES RC CIRCUIT

In a series circuit containing a resistor and a capacitor, the total impedance is given by the equation

Z/arc tan(-Xc/R) = R - j*Xc

and the total voltage is given by the equation

V/arc tan(-Xc/R) = V * cos a - j * V * sin a

where a = arc tan(-Xc/R)THE PARALLEL CIRCUIT

Let's review the basic laws of a parallel circuit.

The voltage is the same across each branch of a parallel circuit.

The total current is the sum of the currents through each branch of a parallel circuit.

The total resistance is less than the least resistance.

THE PARALLEL RL CIRCUIT

In a parallel circuit the voltage amplitude and phase are the same throughout the circuit.

The voltage vector across inductor is the same as the voltage vector across resistor. That voltage vector is defined by the following complex equation:

V/arc tan (Vx)/(Vr) = Vr + jVx

This vector defines the voltage across the inductor and the voltage across the resistor.

Another words, the voltage across the inductor is defined by equation

Vx/arc tan (Vx)/(Vr) = Vr + jVx

The voltage across the resistor is defined by the equation

Vr/arc tan (Vx)/(Vr) = Vr + jVx

where

Vx/arc tan (Vx)/(Vr) = Vr/arc tan (Vx)/(Vr)

The angle is determined by

arc tan ((x)/(Vr))

The vector current has a different amplitude and phase in each branch of the parallel circuit.

The current through the resistor is ninety degrees out of phase with the current through the inductor.

The total current delivered by the power source is:

I/arc tan (Ix)/(Ir) = I * cos a + j * I * sin a

where a = arc tan (Ix)/(Ir)

I * cos a = the current through the resistor

and

I * sin a = the current through the inductor.

The total impedance Zt in a parallel RL circuit can be figured out by the formula

Zt = Z1*Z2/(Z1 + Z2)

Zt = R * jXL / (R + jXL)

I find it easier to use the equation

1/Zt = (Z1 + Z2)/(Z1*Z2)

1/Zt = (R + jXL) / (R * jXL)

THE PARALLEL RC CIRCUIT

In a parallel circuit containing a resistor and a capacitor, the total impedance is given by the equation

1/Z = 1/R + 1/jXc

and the total current is given by the equation

I/arc tan(-Xc/R) = I * cos a - j * I * sin a

where

I * cos a

is the current through the resistor

and

I * sin a

is the current through the capacitor.

note: a = arc tan(-Xc/R)

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

Introductory Circuit Analysis Third Edition
ISBN 0-675-8559-4