Enrollment no.: 130010111001- Abhi P. Choksi 130010111051-Anuj Watal 130010111023- Esha N. Patel...
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Transcript of Enrollment no.: 130010111001- Abhi P. Choksi 130010111051-Anuj Watal 130010111023- Esha N. Patel...
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 TECHNOLOGYNEW V.V.NAGAR
AC CIRCUITS
AC Circuits
Capacitive ReactancePhasor DiagramsInductive ReactanceRCL CircuitsResonance
Resistive Loads in AC Circuits
Ohm’s Law:
R is constant – does not depend on frequencyNo phase difference between V and I
ftIftR
V
R
VI
ftVVR
VI
R
VI
tt
t
rmsrms
2sin2sin
2sin
00
0
Capacitive Reactance
At the moment a capacitor is connected to a voltage source:
Current is at its maximumVoltage across capacitor is zero
+
-V C
I 0
V = 0
+
-
Capacitive Reactance
After a long time, the capacitor is charged:
Current is zeroVoltage is at its maximum (= supply voltage)
V C V
+
-
+
-
Capacitive Reactance
Now, we reverse the polarity of the applied voltage:
Current is at its maximum (but reversed)Voltage hasn’t changed yet
V C V
+
-+
- +
-
I
Capacitive Reactance
Apply an AC voltage source:
an AC current is present in the circuita 90° phase difference is found between the
voltage and the current
V(t) = V0sin(2f t)
C+
-
I (t) = I 0sin(2f t+/ 2)
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:
We call XC the capacitive reactance, and calculate it as:
units of capacitive reactance: ohms ()
Crmsrms XIV
fCXC 2
1
Capacitive Reactance
A particular example:
2.12 F 107.5Hz 1002
1
2
14-fC
XC
V0 = 50 V
+
- C = 750 mFf = 100 Hz
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
The larger the capacitance, the smaller the capacitive reactance
As frequency increases, reactance decreasesDC: capacitor is an “open circuit” and
high frequency: capacitor is a “short circuit”
and
CX
0CX
fCXC 2
1
Phasor Diagrams
Consider a vector which rotates counterclockwise with an angular speed :
This vector is called
a “phasor.” It is a
visualization tool.
f 2
t=2f t
V0V0 sin(2f t)
V
Phasor Diagrams
For a resistive load: the current is always proportional to the voltage
t=2f t
V0V0 sin(2f t)
V, I
I 0 sin(2f t)
I 0 = V0 / R
voltage phasor
current phasor
Phasor Diagrams
For a capacitive load: the current “leads” the voltage by /2 (or 90°)
t=2f t
V0 sin(2f t)
V, I
I 0 sin(2f t + / 2)
voltage phasor
current phasor
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
AC
t
ILEMF
Inductive Reactance
As the current reaches its maximum value, its rate of change decreases to zero – and so does the induced EMF
AC
t
ILEMF
Inductive Reactance
The inductive reactance is the Ohm’s Law constant of proportionality:
units of inductive reactance:
ohms ()
AC
fLX
XIV
L
Lrmsrms
2
Inductive Reactance
The voltage-current relationship in an inductive load in an AC circuit can be represented by a phasor diagram:
t=2f t
V0 sin(2f t)
V, I
I 0 sin(2f t - / 2)
voltage phasor
current phasor
Inductive Reactance
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)
fLX L 2
RLC Circuit
Here is an AC circuit containing series-connected resistive, capacitive, and inductive loads:
The voltages across the loads at any instant are different, but a common current is present.
AC
R
C
L
RCL Circuit
The current is in phase with voltage in the resistor.
The capacitor voltage trails the current; the inductor voltage leads it.
We want to calculate the entire applied voltage from the generator.
VR
VL
VC
I
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 - VC I
RLC Circuit
Now we use Pythagoras’ Theorem to add the VL – VC phasor to the VR phasor.
VR
VL - VC I
V = VR2 + (VL - VC)
2
RCL Circuit
The current phasor is unaffected by our addition of the voltage phasors.
It now makes an angle with the overall applied voltage phasor.
I
V = VR2 + (VL - VC)
2
RLC Circuit
We can make Ohm’s Law substitutions for the voltages:
22
2222
22222
CL
CL
CLCLR
RLLCC
XXRIV
XXIRIV
IXIXRIVVVV
IRVIXVIXV
RCL Circuit
Our result:
suggests an Ohm’s Law relationship for the combined loads in the series RCL circuit:
Z is called the impedance of the RCL circuit.SI units: ohms ()
22CL XXRIV
22
22
CL
CL
XXRZ
XXRIIZV
RCL Circuit -- Power
If the load is purely resistive, the average power dissipated is
We can use the phasor diagram to relate R to Z trigonometrically:
VR
VL - VC I
V = VR2 + (VL - VC)
2
RIP rms2
cos
cos
cos
22 ZIRIP
ZR
Z
R
ZI
RI
V
V
rmsrms
rms
rmsR
“power factor”
RLC Circuit -- Resonance
Series-connected inductor and capacitor:
C L
+
-
Capacitor is initially charged.
When connection is made, discharge current I flows through inductor. I nduced EMF opposes and limits discharge current.
I
RCL Circuit -- Resonance
C L
+
-
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 ).
I
RCL Circuit -- Resonance
C L+
-
I nduced current charges C with the opposite polarity to its original state. When the capacitor is charged, the current I is stopped.
RCL Circuit -- Resonance
C L+
-
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.
-I
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
LCfres 2
1 C L
RLC Circuit -- Resonance
At the resonant frequency, how are the inductive and capacitive reactances related?
The reactances are equal to each other.
CresL
resC
res
XC
LC
LC
LCL
LC
L
LCLLfX
C
LC
LC
CCfX
LCf
2
122
2
21
2
1
2
1
RCL Circuit -- Resonance
At the resonant frequency, when the inductive and capacitive reactances are equal, what is the situation in the circuit?
VR
VL
VC
IVR
I
RLC Circuit –SERIES Resonance
At the resonant frequency, when the inductive and capacitive reactances are equal, what is the impedance of the circuit?
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.
RRXXRZ CL 222