Analysis of Single Phase AC Circuits: Voltage, Resistors, Inductors, and Capacitors, Lecture notes of Electronics

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Single Phase Alternating Current

Circuits Analysis

Alternating Current Circuits

• Electrical appliances in the house use alternating current

(AC) circuits.

• If an AC source applies an alternating voltage to a series

circuit containing resistor, inductor, and capacitor, what

are the amplitude and time characteristics of the

alternating current.

• An AC circuit consists of a combination of circuit

elements and a power source.

• The power source provides an alternating voltage, D v.

AC Voltage, cont.

• The angular frequency is - ƒ is the frequency of

the source.

• T is the period of the

source.

• The voltage is positive

during one half of the cycle

and negative during the

other half.

2 ƒ

π

ω π

T

Resistors in an AC Circuit

Consider a circuit consisting

of an AC source and a

resistor.

The AC source is symbolized

By;

Δ v

R

= D V

max

= V

max

sin w t

Δ v

R

is the instantaneous

voltage across the resistor.

Resistors in an AC Circuit, final

• The graph shows the current through

and the voltage across the resistor.

• The current and the voltage reach

their maximum values at the same

time.

• The current and the voltage are said

to be in phase.

• For a sinusoidal applied voltage, the

current in a resistor is always in phase

with the voltage across the resistor.

• The direction of the current has no

effect on the behavior of the resistor.

• Resistors behave essentially the same

way in both DC and AC circuits.

Phasor Diagram

• To simplify the analysis of AC

circuits, a graphical constructor

called a phasor diagram can be

used.

• A phasor is a vector whose length

is proportional to the maximum

value of the variable it represents.

• The vector rotates

counterclockwise at an angular

speed equal to the angular

frequency associated with the

variable.

• The projection of the phasor onto

the vertical axis represents the

instantaneous value of the quantity

it represents.

Inductors in an AC Circuit

Kirchhoff’s loop rule can

be applied and gives:

0 or

0

max

,

sin

L

v v

di

v L

dt

di

v L V ωt

dt

   

  

  

Current in an Inductor

The equation obtained from Kirchhoff's loop rule can

be solved for the current;

This shows that the instantaneous current i

L

in the

inductor and the instantaneous voltage Δ v

L

across the

inductor are out of phase by;

(p/2) rad = 90

o

max

sin

2

max

max max

max

cos

sin I

L

L

V V

i ωt dt ωt

L ωL

V π V

i ωt

ωL ωL

 

 

 

 

  

 

 



Phasor Diagram for an Inductor

The phasors are at 90

o

with respect to each other.

This represents the phase

difference between the

current and voltage.

Specifically, the current

lags behind the voltage by

o

Inductive Reactance

The factor ωL has the same units as resistance and is

related to current and voltage in the same way as

resistance.

Because ωL depends on the frequency, it reacts

differently, in terms of offering resistance to current,

for different frequencies.

The factor is the inductive reactance and is given

by:

X

L

= ωL

Voltage Across the Inductor

The instantaneous voltage across the inductor is;

max

max

sin

sin

L

L

di

v L

dt

V ωt

I X ωt

Numerical;

Numerical;

Capacitor in an AC Circuit

The circuit contains a

capacitor and an AC

source.

Kirchhoff’s loop rule

gives:

Δ v + Δ v

c

= 0 and so

Δ v = Δ v

C

= Δ V

max

sin ωt

Δ v

c

is the instantaneous

voltage across the

capacitor.