1) The document is a lecture on steady state analysis of R, L, and C circuits by I V VEERANJANEYULU of the Electrical & Electronics Engineering department at Aditya Engineering College. 2) It introduces phasor diagrams and explains that resistors have voltage and current phasors in phase, inductors have voltage leading current by 90 degrees, and capacitors have current leading voltage by 90 degrees. 3) It then provides the steady state analysis of RL and RC series circuits using phasor diagrams and expressions for impedance, conductance, susceptance, and admittance.

1. ADITYA ENGINEERING COLLEGE (A)
Steady state analysis of R, L
and C circuits
By
I V VEERANJANEYULU
Dept of Electrical & Electronics Engineering
Aditya Engineering College(A)
Surampalem.

2. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Concept of Phase angle and Phase difference
A phasor diagram can be used to represent a sine wave in terms of its magnitude and angular position.
The length of the arrow represents the magnitude of the sine wave and angle θ represents the angular
position of the sine wave.


1 0
30
1 4∠1350
3∠2250

3. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU


1 0
360
1 

or 135
1
0
225
1 

or
)
30
sin(
10
)
(
)
45
sin(
10
)
(
sin
7
)
(
0
0





t
t
v
t
t
v
t
t
v
C
B
A



0
0
0
45
10
30
5
0
7







C
B
A
V
V
V

4. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Draw the phasor diagram to represent the two sine waves shown in fig.

5. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Phasor Relationships for Circuit Elements
Resistor
If the current through a resistor R is i(t) = Im cos(ωt+Φ),
then voltage across it is given by Ohm’s law as
v(t) = i(t)R = RIm cos(ωt+Φ)
The phasor form of this voltage is V = RIm ∠ Φ
But the phasor representation of the current is I = Im ∠ Φ.
Hence, V =R I Phasor diagram for the resistor.
In case of resistor, Voltage wave form follows current wave form. Voltage phasor and current phasor are
in phase.
The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in
ohms (Ω).
R
I
V
Z 






m
m
I
RI

6. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Inductor
reactance
inductive
the
is
L
where
L
90
L
I
90
LI
I
V
Z
Impedance
LI
90
LI
V
and
I
I
notation
phasor
The
)
90
cos(
LI
)
90
cos(
LI
).
sin(
LI
dt
t)
(
d
L
t)
(
is
inductor
the
across
voltage
The
).
cos(
I
t)
(
is
L
inductor
hrough
current t
the
Assume
m
m
m
m
m
m
m
m
m



















































L
L
X
jX
j
j
t
t
t
i
v
t
i
In case of inductor, current waveform lags voltage waveform by 900.
Voltage phasor and current phasor are out of phase by 900,
voltage phasor leads current phasor by 900 or current phasor lags voltage phasor by 900

7. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Capacitor
reactance
capacitive
the
is
C
1
where
C
90
C
1
90
CV
V
I
V
Z
Impedance
CV
90
CV
I
and
V
V
notation
phasor
The
)
90
cos(
CV
)
90
cos(
CV
).
sin(
CV
dt
t)
(
d
C
t)
(
is
capacitor
rough
Current th
The
).
cos(
V
t)
(
is
C
capacitor
across
voltage
the
Assume
m
m
m
m
m
m
m
m
m






















































C
C
X
jX
j
j
t
t
t
v
i
t
v
In case of capacitor current waveform leads voltage waveform by 900.
Voltage phasor and current phasor are out of phase by 900,
voltage phasor lags current phasor by 900 or current phasor leads voltage phasor by 900.

8. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
The admittance Y is the reciprocal of
impedance, measured in siemens (S)
Frequency (ω) is 0 for dc. Frequency (ω) is ∞ at high
frequencies
Z = jωL = 0
Z = jωL = ∞




C
j
Z

0



C
j
Z


9. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Steady state analysis of RL series circuit
inductor
the
across
voltage
t)
(
resistor,
the
across
voltage
t)
(
V
is
t)
(
of
notation
phasor
the
circuit.
the
across
voltage
is
t)
(
0
I
I
is
t)
(
of
notation
phasor
the
sin
I
t)
(
Let
circuit
series
RL
Consider
m
m





L
R
v
v
v
v
i
t
i 

10. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU















sin
and
cos
R
R
tan
and
where
R
Z
R
L
R
I
L)I
R
(
I
V
Z
Impedance
L)I
R
(
LI
RI
V
V
V
circuit
the
to
KVL
Applying
LI
90
LI
V
is
t)
(
of
notation
phasor
the
)
90
sin(
LI
cos
LI
dt
t)
(
d
L
t)
(
RI
0
RI
V
is
t)
(
of
notation
phasor
the
sin
RI
t)
(
R
t)
(
1
-
2
2
m
m
m
m
m
L
R
m
m
L
m
m
m
m
R
m
Z
X
Z
X
X
R
Z
Z
jX
jX
j
j
j
j
j
v
t
t
i
v
v
t
i
v
L
L
L
L
L
L
L
R
R




































angle
an
by
V
lags
I
angle
an
by
I
leads
v

11. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Siemens
in
measured
are
e
susceptanc
and
e
conductanc
nce)
B(Suscepta
R
nce)
G(conducta
where
R
R
R
1
Z
1
Y
Admittance
2
2
2
2
2
2
2
2
2
2
L
L
L
L
L
L
L
L
L
X
R
X
X
R
jB
G
X
R
X
j
X
R
X
R
jX
jX



















12. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Steady state analysis of RC series circuit
capacitor
the
across
voltage
t)
(
resistor,
the
across
voltage
t)
(
V
is
t)
(
of
notation
phasor
the
circuit.
the
across
voltage
is
t)
(
0
I
I
is
t)
(
of
notation
phasor
the
sin
I
t)
(
Let
circuit
series
RC
Consider
m
m





C
R
v
v
v
v
i
t
i 

13. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU















sin
and
cos
R
R
tan
and
where
R
Z
R
R
I
)I
R
(
I
V
Z
Impedance
)I
R
(
I
RI
V
V
V
circuit
the
to
KVL
Applying
I
90
I
1
V
is
t)
(
of
notation
phasor
the
)
90
sin(
I
1
cos
I
1
t)dt
(
1
t)
(
RI
0
RI
V
is
t)
(
of
notation
phasor
the
sin
RI
t)
(
R
t)
(
1
-
2
2
m
m
m
m
m
C
R
m
m
C
m
m
m
m
R
m
Z
X
Z
X
X
R
Z
Z
jX
jX
C
j
C
j
C
j
C
j
C
j
C
v
t
C
t
C
i
C
v
v
t
i
v
C
C
C
C
C
C
C
R
R










































angle
an
by
V
leads
I
angle
an
by
I
lag
v

14. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Siemens
in
measured
are
e
susceptanc
and
e
conductanc
nce)
B(Suscepta
R
nce)
G(conducta
where
R
R
R
1
Z
1
Y
Admittance
2
2
2
2
2
2
2
2
2
2
C
C
C
C
C
C
C
C
C
X
R
X
X
R
jB
G
X
R
X
j
X
R
X
R
jX
jX

















15. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
The impedance may be expressed in rectangular form as Z = R ± jX
where Re[Z] is the resistance R and Im[Z ]is the reactance X.
The reactance X may be positive or negative.
The impedance is inductive when X is positive or capacitive when X is negative.
Thus, impedance Z = R + jX is said to be inductive or lagging since current lags voltage,
while impedance Z = R - jX is capacitive or leading because current leads voltage.
The impedance, resistance, and reactance are all measured in ohms.
Admittance, conductance, and susceptance are all expressed in the unit of Siemens (or mhos).

16. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
Steady state analysis of RLC series circuit
capacitor
the
across
voltage
t)
(
inductor
the
across
voltage
t)
(
resistor,
the
across
voltage
t)
(
V
is
t)
(
of
notation
phasor
the
circuit.
the
across
voltage
is
t)
(
0
I
I
is
t)
(
of
notation
phasor
the
sin
I
t)
(
Let
circuit
series
RLC
Consider
m
m






C
L
R
v
v
v
v
v
i
t
i 

17. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
m
m
C
m
m
m
m
L
m
m
m
m
R
m
I
90
I
1
V
is
t)
(
of
notation
phasor
the
)
90
sin(
I
1
cos
I
1
t)dt
(
1
t)
(
LI
90
LI
V
is
t)
(
of
notation
phasor
the
)
90
sin(
LI
cos
LI
dt
t)
(
d
L
t)
(
RI
0
RI
V
is
t)
(
of
notation
phasor
the
sin
RI
t)
(
R
t)
(
C
j
C
v
t
C
t
C
i
C
v
j
v
t
t
i
v
v
t
i
v
C
C
L
L
R
R





































18. Aditya Engineering College (A)
Basic Electrical Circuits Tuesday, December 12, 2023
I V VEERANJANEYULU
R
tan
R
tan
and
,
X
where
R
Z
)
(
R
-
L
R
I
)I
-
L
R
(
I
V
Z
Impedance
)I
-
L
R
(
I
LI
RI
V
V
V
V
circuit
the
to
KVL
Applying
1
-
1
-
2
2
m
m
m
m
m
m
C
L
R
C
L
C
L
C
L
X
X
X
X
R
Z
X
X
Z
jX
X
X
j
C
j
j
C
j
j
C
j
j
C
j
j





































age.
with volt
inphase
current
and
circuit
resistive
as
behaves
circuit
the
and
zero
is
X
then
,
If
voltage.
leads
current
and
circuit
capacitive
as
behaves
circuit
the
and
negative
is
X
then
,
If
voltage.
lags
current
and
circuit
inductive
as
behaves
circuit
the
and
positive
is
X
then
,
If
L
C
L
C
C
L
X
X
X
X
X
X


