This document provides an overview of alternating current (AC) circuits, including concepts such as root mean square (RMS) values, single element circuits (resistors, capacitors, and inductors), and the use of phasors for analysis. It explains the differences between AC and direct current (DC), outlines the behavior of AC circuits with respect to voltage and current phases, and introduces key formulas related to impedance and reactance. The goal is to equip readers with basic techniques for analyzing AC circuits effectively.

1. Alternating Current Circuit
Bhavya Yadav

2. Outline
• Introduction
• Root Mean Square (RMS) Values
• A.C Circuits Containing Single
Elements
• RLC Series Circuit

3. Today, a “grid” of AC electrical
distribution systems spans the Nigeria
and other countries. Any device that
plugs into an electric outlet uses an AC
circuit. In this chapter, you will learn
some of the basic techniques for
analyzing AC circuits.
Chapter Goal: To understand and
apply basic techniques of AC circuit
analysis.
Introduction

4. Introduction
• What is alternating current?
• In alternating current (AC), the
flow of electric charge periodically
reverses direction, whereas
in direct current (DC, also dc), the
flow of electric charge is only in
one direction.
• Alternating current circuit
• An alternating voltage or current is
current or voltage which is an
oscillating function of time. The
general expression for the
alternating voltage is

5. Root Mean Square (rms) Values
• An alternating voltage or current is current or
voltage which is an oscillating function of
time.
where Vo is the amplitude of the voltage in volts,
f is the frequency in hertz,
ft
V
t
V
v o
o 
 2
sin
sin =
=
f

 2
= f
T
1
=
Meters in a.c. circuits read the effective, or
root mean square (rms) values of the current
and voltage. The rms values are
2
o
rms
V
V =
2
o
rms
I
I =
where Io (or Im) and Vo (or Vm) is the
maximum value of current and voltage
Meters in AC. circuits read the effective, or
root mean square (rms) values of the current
and voltage

6. In physics and engineering, a phasor (a portmanteau of phase vector), is a
complex number representing a sinusoidal function whose amplitude (A),
angular frequency (ω), and initial phase (θ) are time-invariant.
It is related to a more general concept called analytic
representation, which decomposes a sinusoid into the
product of a complex constant and a factor depending
on time and frequency.
The complex constant, which depends on amplitude and
phase, is known as a phasor, or complex amplitude.
A Phasor Diagram in these module will be use to
understand the vectorial sinusoidal varying voltages /
currents through the projection of a vector, with
length equal to the amplitude, onto a horizontal axis.

7. A.C Circuits Containing Single Elements
• An a.c circuit can contain either resistors (R),
capacitors (C) or inductors (L).
The output voltage and current for
this arrangement is
ft
V
v o
R 
2
sin
=
ft
I
ft
R
V
i o
o

 2
sin
2
sin =
=
Resistor in ac
The Equations 1 & 2 show that the
current and the voltage are in phase

8. In order to visualize the phase relationships between the current and voltage in ac
circuits,
we define phasors – vectors whose length is the maximum voltage or current, and
which rotate around an origin with the angular speed of the oscillating current.
Resistor in AC Circuit

9. Capacitor in AC
The output voltage for this arrangement is
t
V
C
q
v o
C 
sin
=
=
where q is the charge accumulated on the
capacitor, vC is the voltage across the
capacitor
t
CV
q o 
sin
=
t
V
C
dt
dq
i o 
 cos
=
=






+
=
2
sin


 t
V
C
i o
using trigonometry
identity






+
=
2
sin
cos









+
=
2
sin

t
I
i o
where we have written o
o V
C
I 
=
C
o
o
X
C
I
V
=
=

1
which
implies
The quantity is known as
the capacitive reactance,
measured in ohms.

C
X C
1
=

10. Capacitor in AC
t
V
C
q
v o
C 
sin
=
= 





+
=
2
sin

t
I
i o
Equations and
Physically, this means that the current reaches
its peaks a quarter of a cycle before the
voltage reaches its peaks.
shows that the current in the circuit leads the
voltage across the capacity by
π
2
, or one
quarter
1
4
cycle to reach its maximum value.

11. Inductor in AC
As the current flows through the inductor, there is
an induced EMF. in the inductor to oppose the
change of flux through it.
The e.m.f of the source must be equal to the
induced e.m.f. Thus, we have
t
V
v
dt
di
L o
L 
sin
=
=
tdt
L
V
di o

sin
=
t
L
V
tdt
L
V
i o
t
o


 cos
sin
0
−
=
=  





−
=
2
sin



t
L
V
i o






−
−
=
2
sin
cos


 t
t
using






−
=
2
sin

t
I
i o
L
V
I o
o

=
L
o
o
X
L
I
V
=
= 

12. Inductor in AC
The quantity is known as the
inductive reactance, measured in ohms.
Equations and
Physically, this means that the voltage reaches
its peaks a quarter of a cycle before the current
reaches its peaks.
L
X L 
=
t
V
v o
L 
sin
= 





−
=
2
sin

t
I
i o
that the voltage leads the current in the inductor
by
π
2
, as illustrated

13. Thus, from Pythagoras theorem, the applied voltage V
is given by
𝑉2 = 𝑉
𝑅
2
+ 𝑉
𝐿
– 𝑉
𝑐
2
𝑉2 = 𝐼
2
𝑅
2
+ 𝐼𝑋𝐿
– 𝐼𝑋𝑐
2
= 𝐼
2
[ 𝑅
2
+ 𝑋𝐿
– 𝑋𝑐
2
]
𝐼 =
𝑉
𝑅2 + 𝑋𝐿 − 𝑋𝑐
2
=
𝑉
𝑍
Where
𝑍 = 𝑅2 + 𝑋𝐿 = 𝑋𝐶
2
Z is the impedance of the circuit. It has the units of
ohms and it varies with frequency f.
LRC Series Circuit
If 𝐼 lags on 𝑉 by an angle θ
𝑡𝑎𝑛𝜃 =
𝑉𝐿 − 𝑉
𝑐
𝑉𝑅
=
𝐼𝑋𝐿 − 𝐼𝑋𝑐
𝐼𝑅
→
𝑡𝑎𝑛𝜃 =
𝑋𝐿 − 𝑋𝑐
𝑅

14. 14
( ) ( )
( ) ( )
( )2
2
2
2
2
2
C
L
o
C
o
L
o
o
C
L
R
o
X
X
R
I
X
I
X
I
R
I
V
V
V
V
−
+
=
−
+
=
−
+
=
( )





 −
=

−
=
−
=
−
=
−
+
=
=
−
R
X
X
R
X
X
R
I
X
I
X
I
X
X
R
I
V
Z
o
o
o
C
L
o
o
C
L
1
C
L
C
L
R
C
L
2
2
tan
V
V
V
tan
:
diagram
phasor
the
from
angle
phase
the
determine
also
can
We
e)
resistenac
to
analogous
quantity
(a
Z
impedance"
"
an
define
can
We


Phasor Diagram