1. The document discusses AC circuits and components including inductors, capacitors, resistors, and transformers. 2. Key concepts covered include inductive and capacitive reactance, impedance, phase relationships between voltage and current, and calculations of effective voltage and current. 3. Transformers can be used to step up or down voltages in an AC circuit by changing the ratio of turns in the primary and secondary coils. An ideal transformer does not lose energy.

2. Ronak S Sutariya
Ronak S Sutariya
Branch :
Branch :Computer
Computer
Sub:
Sub: Elements Of Electrical Engineering
Elements Of Electrical Engineering
Enrollment No:
Enrollment No: 151290107052
151290107052
Topic :
Topic : A.
A. C CIRCUITS
C CIRCUITS

3. θ
450
900
1350
1800
2700
3600
E
R = Emax
E = Emax sin θ
Rotating Vector Description
Rotating Vector Description
The coordinate of the emf at any instant is the
value of Emax sin θ. Observe for incremental
angles in steps of 450
. Same is true for i.
The coordinate of the emf at any instant is the
value of Emax sin θ. Observe for incremental
angles in steps of 450
. Same is true for i.
θ
450
900
1350
1800
2700
3600
E
Radius = Emax
E = Emax sin θ

4. Effective AC Current
Effective AC Current
i
imax
max
The average current
The average current
in a cycle is zero—
in a cycle is zero—
half + and half -.
half + and half -.
But energy is expended,
But energy is expended,
regardless of direction.
regardless of direction.
So the
So the “root-mean-
“root-mean-
square”
square” value is useful.
value is useful.
2
2 0.707
rms
I I
I = =
I = imax
The
The rms
rms value
value I
Irms
rms is
is
sometimes called the
sometimes called the
effective
effective current
current I
Ieff
eff:
:
The effective ac current:
ieff = 0.707 imax

5. AC Definitions
AC Definitions
One
One effective ampere
effective ampere is that ac current for
is that ac current for
which the power is the same as for one
which the power is the same as for one
ampere of dc current.
ampere of dc current.
One
One effective volt
effective volt is that ac voltage that
is that ac voltage that
gives an effective ampere through a
gives an effective ampere through a
resistance of one ohm.
resistance of one ohm.
Effective current: ieff = 0.707 imax
Effective current: ieff = 0.707 imax
Effective voltage: Veff = 0.707 Vmax
Effective voltage: Veff = 0.707 Vmax

6. Pure Resistance in AC Circuits
Pure Resistance in AC Circuits
A
a.c. Source
R
V
Voltage and current are in phase, and Ohm’s
Voltage and current are in phase, and Ohm’s
law applies for effective currents and voltages.
law applies for effective currents and voltages.
Voltage and current are in phase, and Ohm’s
Voltage and current are in phase, and Ohm’s
law applies for effective currents and voltages.
law applies for effective currents and voltages.
Ohm’s law: Veff = ieffR
Vmax
i
imax
max
Voltage
Current

7. AC and Inductors
AC and Inductors
Time, t
I
i
Current
Current
Rise
Rise
τ
0.63I
Inductor
The voltage
The voltage V
V peaks first, causing rapid rise in
peaks first, causing rapid rise in i
i
current which then peaks as the emf goes to zero.
current which then peaks as the emf goes to zero.
Voltage
Voltage leads
leads (
(peaks before
peaks before) the current by 90
) the current by 900
0
.
.
Voltage and current are out of phase
Voltage and current are out of phase.
.
Time, t
I i
Current
Current
Decay
Decay
τ
0.37I
Inductor

8. A Pure Inductor in AC Circuit
A Pure Inductor in AC Circuit
A
L
V
a.c.
Vmax
i
imax
max
Voltage
Current
The voltage peaks 90
The voltage peaks 900
0
before the current peaks.
before the current peaks.
One builds as the other falls and vice versa.
One builds as the other falls and vice versa.
The voltage peaks 90
The voltage peaks 900
0
before the current peaks.
before the current peaks.
One builds as the other falls and vice versa.
One builds as the other falls and vice versa.
The
The reactance
reactance may be defined as the
may be defined as the nonresistive
nonresistive
opposition
opposition to the flow of ac current.
to the flow of ac current.

9. Inductive Reactance
Inductive Reactance
A
L
V
a.c.
The
The back
back emf
emf induced
induced
by a changing current
by a changing current
provides opposition to
provides opposition to
current, called
current, called inductive
inductive
reactance X
reactance XL
L.
.
Such losses are
Such losses are temporary
temporary, however, since the
, however, since the
current
current changes direction
changes direction, periodically re-supplying
, periodically re-supplying
energy so that no net power is lost in one cycle.
energy so that no net power is lost in one cycle.
Inductive reactance
Inductive reactance X
XL
L is a function of both the
is a function of both the
inductance
inductance and the
and the frequency
frequency of the ac current.
of the ac current.

10. Calculating Inductive Reactance
Calculating Inductive Reactance
A
L
V
a.c.
Inductive Reactance:
2 Unit is the
L
X fL
π
= Ω
Ohm's law: L L
V iX
=
The
The voltage
voltage reading
reading V
V in the above circuit at the
in the above circuit at the
instant the
instant the ac
ac current is
current is i
i can be found from the
can be found from the
inductance
inductance in
in H
H and the
and the frequency
frequency in
in Hz
Hz.
.
(2 )
L
V i fL
π
= Ohm’s law: VL = ieffXL

11. AC and
AC and
Capacitance
Capacitance
Time, t
Qmax
q
Rise in
Rise in
Charge
Charge
Capacitor
τ
0.63 I
Time, t
I
i
Current
Current
Decay
Decay
Capacitor
τ
0.37 I
The voltage
The voltage V
V peaks ¼ of a cycle after the current
peaks ¼ of a cycle after the current
i
i reaches its maximum. The voltage
reaches its maximum. The voltage lags
lags the
the
current.
current. Current
Current i
i and V out of phase
and V out of phase.
.

12. A Pure Capacitor in AC
A Pure Capacitor in AC
Circuit
Circuit
Vmax
i
imax
max
Voltage
Current
A V
a.c.
C
The voltage peaks 90
The voltage peaks 900
0
after
after the current peaks.
the current peaks.
One builds as the other falls and vice versa.
One builds as the other falls and vice versa.
The voltage peaks 90
The voltage peaks 900
0
after
after the current peaks.
the current peaks.
One builds as the other falls and vice versa.
One builds as the other falls and vice versa.
The diminishing current
The diminishing current i
i builds charge on
builds charge on C
C
which increases the
which increases the back emf
back emf of
of V
VC
C.
.
The diminishing current
The diminishing current i
i builds charge on
builds charge on C
C
which increases the
which increases the back emf
back emf of
of V
VC
C.
.

13. Capacitive Reactance
Capacitive Reactance
No
No net power
net power is lost in a complete cycle, even
is lost in a complete cycle, even
though the capacitor does provide nonresistive
though the capacitor does provide nonresistive
opposition (
opposition (reactance
reactance) to the flow of ac current.
) to the flow of ac current.
Capacitive reactance
Capacitive reactance X
XC
C is affected by both the
is affected by both the
capacitance
capacitance and the
and the frequency
frequency of the ac current.
of the ac current.
A V
a.c.
C
Energy
Energy gains and
gains and
losses are also
losses are also
temporary
temporary for capacitors
for capacitors
due to the constantly
due to the constantly
changing ac current.
changing ac current.

14. Calculating Inductive Reactance
Calculating Inductive Reactance
Capacitive Reactance:
1
Unit is the
2
C
X
fC
π
= Ω
Ohm's law: VC C
iX
=
The
The voltage
voltage reading
reading V
V in the above circuit at the
in the above circuit at the
instant the
instant the ac
ac current is
current is i
i can be found from the
can be found from the
inductance
inductance in
in F
F and the
and the frequency
frequency in
in Hz
Hz.
.
2
L
i
V
fL
π
=
A V
a.c.
C
Ohm’s law: VC = ieffXC

15. Series LRC Circuits
Series LRC Circuits
L
VR VC
C
R
a.c.
VL
VT
A
Series ac circuit
Consider an
Consider an inductor
inductor L
L,
, a
a capacitor
capacitor C
C,
, and
and
a
a resistor
resistor R
R all connected in
all connected in series
series with
with an
an
ac source
ac source. The instantaneous current and
. The instantaneous current and
voltages can be measured with meters.
voltages can be measured with meters.
Consider an
Consider an inductor
inductor L
L,
, a
a capacitor
capacitor C
C,
, and
and
a
a resistor
resistor R
R all connected in
all connected in series
series with
with an
an
ac source
ac source. The instantaneous current and
. The instantaneous current and
voltages can be measured with meters.
voltages can be measured with meters.

16. Phase in a Series AC Circuit
Phase in a Series AC Circuit
The voltage
The voltage leads
leads current in an inductor and
current in an inductor and lags
lags
current in a capacitor.
current in a capacitor. In phase
In phase for resistance
for resistance R
R.
.
θ
450
900
1350
1800
2700
3600
V V = Vmax sin θ
VR
VC
VL
Rotating
Rotating phasor diagram
phasor diagram generates voltage
generates voltage
waves for each element
waves for each element R
R,
, L
L, and
, and C
C showing
showing
phase relations. Current
phase relations. Current i
i is always
is always in phase
in phase with
with
V
VR.
R.

17. Impedance in an AC Circuit
Impedance in an AC Circuit
φ
R
XL - XC
Z
Z
Impedance
Impedance 2 2
( )
T L C
V i R X X
= + −
Impedance
Impedance Z
Z is defined:
is defined:
2 2
( )
L C
Z R X X
= + −
Ohm’s law for ac current
Ohm’s law for ac current
and impedance:
and impedance:
or T
T
V
V iZ i
Z
= =
The impedance is the combined opposition to ac
current consisting of both resistance and reactance.
The impedance is the combined opposition to ac
current consisting of both resistance and reactance.

18. Power in an AC Circuit
Power in an AC Circuit
No power is consumed by inductance or
No power is consumed by inductance or
capacitance. Thus power is a function of the
capacitance. Thus power is a function of the
component of the impedance along resistance:
component of the impedance along resistance:
No power is consumed by inductance or
No power is consumed by inductance or
capacitance. Thus power is a function of the
capacitance. Thus power is a function of the
component of the impedance along resistance:
component of the impedance along resistance:
In terms of ac voltage:
In terms of ac voltage:
P = iV cos φ
P = iV cos φ
In terms of the resistance R:
In terms of the resistance R:
P = i2
R
P = i2
R
φ
R
XL - XC
Z
Z
Impedance
Impedance
P
P lost in
lost in R
R only
only
The fraction
The fraction Cos
Cos φ
φ is known as the
is known as the power factor.
power factor.

19. The Transformer
The Transformer
A
A transformer
transformer is a device that uses induction
is a device that uses induction
and ac current to step voltages up or down.
and ac current to step voltages up or down.
R
a.c.
Np Ns
Transformer
P P
N
t
∆Φ
= −
∆
E S S
N
t
∆Φ
= −
∆
E
Induced
emf’s are:
Induced
emf’s are:
An ac source of emf
An ac source of emf
E
Ep
p is connected to
is connected to
primary coil with
primary coil with N
Np
p
turns. Secondary has
turns. Secondary has
N
Ns
s turns and emf of
turns and emf of E
Es
s.
.
An ac source of emf
An ac source of emf
E
Ep
p is connected to
is connected to
primary coil with
primary coil with N
Np
p
turns. Secondary has
turns. Secondary has
N
Ns
s turns and emf of
turns and emf of E
Es
s.
.

20. Transformers (Continued):
Transformers (Continued):
R
a.c.
Np Ns
Transformer
P P
N
t
∆Φ
= −
∆
E
S S
N
t
∆Φ
= −
∆
E
Recognizing that
Recognizing that ∆φ
∆φ/
/∆
∆t
t is the same in each coil,
is the same in each coil,
we divide first relation by second and obtain:
we divide first relation by second and obtain:
The transformer
equation:
The transformer
equation:
P P
S S
N
N
=
E
E

21. Transformer Efficiency
Transformer Efficiency
There is no power gain in stepping up the voltage
There is no power gain in stepping up the voltage
since voltage is increased by reducing current. In
since voltage is increased by reducing current. In
an ideal transformer with no internal losses:
an ideal transformer with no internal losses:
or S
P
P P S S
s P
i
i i
i
= =
E
E E
E
An ideal
An ideal
transformer:
transformer:
R
a.c.
Np Ns
Ideal Transformer
The above equation assumes no internal energy
The above equation assumes no internal energy
losses due to heat or flux changes.
losses due to heat or flux changes. Actual
Actual
efficiencies
efficiencies are usually between
are usually between 90 and 100%.
90 and 100%.
The above equation assumes no internal energy
The above equation assumes no internal energy
losses due to heat or flux changes.
losses due to heat or flux changes. Actual
Actual
efficiencies
efficiencies are usually between
are usually between 90 and 100%.
90 and 100%.

22. Summary
Summary
Effective current: ieff = 0.707 imax
Effective current: ieff = 0.707 imax
Effective voltage: Veff = 0.707 Vmax
Effective voltage: Veff = 0.707 Vmax
Inductive Reactance:
2 Unit is the
L
X fL
π
= Ω
Ohm's law: L L
V iX
=
Capacitive Reactance:
1
Unit is the
2
C
X
fC
π
= Ω
Ohm's law: VC C
iX
=

23. Summary (Cont.)
Summary (Cont.)
2 2
( )
T R L C
V V V V
= + − tan L C
R
V V
V
φ
−
=
2 2
( )
L C
Z R X X
= + −
or T
T
V
V iZ i
Z
= =
tan L C
X X
R
φ
−
=
1
2
r
f
LC
π
=

24. Summary (Cont.)
Summary (Cont.)
In terms of ac voltage:
In terms of ac voltage:
P = iV cos φ
P = iV cos φ
In terms of the resistance R:
In terms of the resistance R:
P = i2
R
P = i2
R
Power in AC Circuits:
Power in AC Circuits:
P P
S S
N
N
=
E
E P P S S
i i
=
E E
Transformers:
Transformers:

25. THANK YOU
THANK YOU