#### Transcript AC Circuits - GTU e

```AC CIRCUITS
Enrollment no.:
130010111001- Abhi P. Choksi
130010111051-Anuj Watal
130010111023- Esha N. Patel
Guidied by: M. K. Joshi, P.R.Modha
A.D.PATEL.INSTITUTE OF TECHNOLOGY
NEW V.V.NAGAR
AC Circuits
Capacitive Reactance
Phasor Diagrams
Inductive Reactance
RCL Circuits
Resonance
Ohm’s Law:
V
Vrms
I
I rms 
R
R
Vt  V0 sin 2ft 
Vt V0
I t   sin 2ft   I 0 sin 2ft 
R R
R is constant – does not depend on frequency
No phase difference between V and I
Capacitive Reactance
At the moment a capacitor is connected to a
voltage source:
I0
V
+
-
Current is at its maximum
Voltage across capacitor is zero
+
C
V=0
-
Capacitive Reactance
After a long time, the capacitor is charged:
V
+
-
+
C
V
-
Current is zero
Voltage is at its maximum (= supply voltage)
Capacitive Reactance
Now, we reverse the polarity of the applied voltage:
I
V
+
+
-
+
C
Current is at its maximum (but reversed)
Voltage hasn’t changed yet
V
-
Capacitive Reactance
Apply an AC voltage source:
I(t) = I0sin(2ft+/2)
V(t) = V0sin(2ft)
+
an AC current is present in the circuit
a 90° phase difference is found between the
voltage and the current
C
Capacitive Reactance
We want to find a relationship between the voltage
and the current that we can use like Ohm’s Law
for an AC circuit with a capacitive load:
Vrms  I rms  X C
We call XC the capacitive reactance, and calculate it
as:
1
XC 
2fC
units of capacitive reactance: ohms (W)
Capacitive Reactance
A particular example:
V0 = 50 V
f = 100 Hz
-
C = 750 mF
+
1
1
XC 

 2.12 W
-4
2fC 2  100 Hz  7.5  10 F
Capacitive Reactance
Power is zero each time either the voltage or current
is zero
Power is positive whenever V and I have the same
sign
Power is negative whenever V and I have opposite
signs
Power spends equal amounts of time being negative
and positive
Average power over time: zero
Capacitive Reactance
X 
C
1
2fC
The larger the capacitance, the smaller the capacitive
reactance
As frequency increases, reactance decreases
DC: capacitor is an “open circuit” and
XC  
high frequency: capacitor is a “short circuit”
andX  0
C
Phasor Diagrams
Consider a vector which rotates
counterclockwise withan
speed
 angular
2f
:
V
V0 sin(2ft)
This vector is called
a “phasor.” It is a
visualization tool.
V0
t=2ft
Phasor Diagrams
For a resistive load: the current is always
proportional to the voltage
V, I
V0 sin(2ft)
I0 sin(2ft)
I0 = V0 / R
V0
voltage phasor
t=2ft
current phasor
Phasor Diagrams
voltage by /2 (or 90°)
V, I
current phasor
V0 sin(2ft)
I0 sin(2ft + /2)
voltage phasor
t=2ft
Inductive Reactance
A coil or inductor also acts as a reactive load in
an AC circuit.
AC
Inductive Reactance
As the current increases through zero, its time
rate of change is a maximum – and so is the
induced EMF
I
EMF   L
t
AC
Inductive Reactance
As the current reaches its maximum value, its
rate of change decreases to zero – and so does
the induced EMF
I
EMF   L
t
AC
Inductive Reactance
The inductive reactance is the Ohm’s Law
constant of proportionality:
Vrms  I rms X L
AC
X L  2fL
units of inductive reactance:
ohms (W)
Inductive Reactance
The voltage-current relationship in an inductive
load in an AC circuit can be represented by a
phasor diagram:
V, I
V0 sin(2ft)
voltage phasor
t=2ft
current phasor
I0 sin(2ft - /2)
Inductive Reactance
X L  2fL
 Larger inductance: larger reactance (more induced EMF
to oppose the applied AC voltage)
 Higher frequency: larger impedance (higher frequency
means higher time rate of change of current, which means
more induced EMF to oppose the applied AC voltage)
RLC Circuit
Here is an AC circuit containing series-connected resistive,
R
C
AC
L
The voltages across the loads at any instant are different, but
a common current is present.
RCL Circuit
The current is in phase with voltage in
the resistor.
VL
I
The capacitor voltage trails the
it.
We want to calculate the entire
applied voltage from the generator.
VC
VR
RCL Circuit
We will add the voltage phasors as vectors
(which is what they are.)
We start out by adding the reactive
voltages (across the capacitor and the
inductor).
This is easy because those phasors are
opposite in direction. The resultant’s
magnitude is the difference of the two,
and its direction is that of the larger
one.
VR
VL - V C
I
RLC Circuit
Now we use Pythagoras’ Theorem to add
the VL – VC phasor to the VR phasor.
V = VR2 + (VL - VC)2
VR
VL - V C
I
RCL Circuit
The current phasor is unaffected by our
It now makes an angle f with the overall
applied voltage phasor.
V = VR2 + (VL - VC)2
f
I
RLC Circuit
We can make Ohm’s Law substitutions for the voltages:
VC  IX C
VL  IX L
VR  IR
V  VR  VL  VC   I R  IX L  IX C 
2
2
2
V  I R  I X L  X C 
2
2
2
2
V  I R  X L  X C 
2
2
2
2
RCL Circuit
Our result:
V I R
2
2


 X L  XC
suggests an Ohm’s Law relationship for the combined loads in
the series RCL circuit:
V  IZ  I R   X L  X C 
2
2
Z  R2  X L  X C 
2
Z is called the impedance of the RCL circuit.
SI units: ohms (W)
RCL Circuit -- Power
If the load is purely resistive, the
2
average power dissipated is P  I rms R
V = VR2 + (VL - VC)2
We can use the phasor diagram to
relate R to Z trigonometrically:
VR I rms R R

  cos f
V
I rms Z Z
f
VL - VC
R  Z cos f
“power factor”
P  I rms R  I rms Z cos f
2
2
I
VR
RLC Circuit -- Resonance
Series-connected inductor and capacitor:
Capacitor is initially charged.
+
discharge current I flows
through inductor. Induced EMF
opposes and limits discharge
current.
C
L
I
RCL Circuit -- Resonance
When capacitor is discharged,
current I slows and stops.
Decrease of magnetic flux in
inductor induces EMF that
opposes the decrease (and
continues the current I).
+
C
L
I
-
RCL Circuit -- Resonance
Induced current charges C with
the opposite polarity to its
original state. When the
capacitor is charged, the current I is stopped.
+
C
L
RCL Circuit -- Resonance
Now the capacitor begins to
discharge through the inductor
again – this time in the opposite
direction (new discharge current = -I).
+
Opposite induced EMF across
inductor again limits this new
discharge current.
The cycle continues.
C
-I
L
RLC Circuit -- Resonance
Energy is alternately stored in
the capacitor (in the form of
the electrical potential
energy of separated charges)
and in the inductor (in its
magnetic field). When the
magnetic field collapses, it
charges the capacitor; when
the capacitor discharges, it
builds the magnetic field in
the inductor.
C
L
RCL Circuit -- Resonance
This “LC oscillator” or “tuned tank circuit” oscillates at a natural
or resonant frequency of
f res 
1
2 LC
C
L
RLC Circuit -- Resonance
At the resonant frequency, how are the inductive and capacitive reactances
related?
f res 
XC 
1
2 LC
1

2f resC
1
LC

2C
C
2 LC
The reactances are equal to each other.
X L  2f res L  2L 
1
2 LC

L
L LC
LC


 XC
LC
C
LC
RCL Circuit -- Resonance
At the resonant frequency, when the inductive and capacitive
reactances are equal, what is the situation in the circuit?
VL
I
VC
VR
I
VR
RLC Circuit –SERIES Resonance
At the resonant frequency, when the inductive and capacitive
reactances are equal, what is the impedance of the circuit?
Z  R2  X L  X C   R2  R
2
At resonance, the circuit’s impedance is simply equal to its
resistance, and its voltage and current are in phase.
If the resistance is small, the current may be quite large.
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