alternating current r-l-c series circuits
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- PRESENTED BY
JAY B.THAKAR
M.TECH (POWER SYSTEM)
11
ALTERNATING CURRENT
CIRCUITS
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OUTLINES
ResistorInductor
R L series circuit
CapacitorR C series circuit
R L C series circuit
Impedance
Phasor diagram
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WHAT ISRESISTOR?????
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RESISTOR
A resistor is a two-terminalelectronic component
which implements electrical resistance as a circuit
element.
When a voltage V is applied across the terminals of a
resistor, a current I will flow through the resistor in
direct proportion to that voltage.
The reciprocal of the constant of proportionality isknown as the resistance R, since, with a given voltage
V, a larger value of R further "resists" the flow of
current.
Resistors can be either fixed or variable in value.44
http://en.wikipedia.org/wiki/Terminal_(electronics)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Direct_proportionhttp://en.wikipedia.org/wiki/Resistancehttp://en.wikipedia.org/wiki/Resistancehttp://en.wikipedia.org/wiki/Direct_proportionhttp://en.wikipedia.org/wiki/Electrical_resistancehttp://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Terminal_(electronics) -
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(CONTD.)
Fixed resistors come in a variety of different shapes,
sizes and forms.
Resistors can be either fixed or variable in value
Fixed resistors come in a variety of different shapes,
sizes and forms
Axial lead resistors have the value of resistance printed
on them or as a colour code
Surface mount resistors have a numerical code indicatinga value
All resistors have a tolerance value.
Ohms Law for a resistor, R, in an AC circuit 55
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TYPES OF RESISTORS
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RESISTOR
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THE VALUE OF
RESISTORS???The values of the resistor are calculated from the
colour of the bands. The values of the colours are
shown in Table on next slide
88
Four band resistor colour code
1st band provides the first digitof the code2nd band provides the second digitof the code
3rd band is themultiplier
4th band indicates the
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CIRCUIT
Consider a circuitconsisting of an ACsource and a resistor
The graph shows the
current through and thevoltage across theresistor
The current and the
voltage reach theirmaximum values at thesame time
The current and the
voltage are said to be in 1010
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WHAT ISINDUCTOR?????
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INDUCTOR
An inductor or a is a passive electrical component that
can store energy in a magnetic field created by the
electric current passing through it.
An inductor's ability to store magnetic energy is
measured by its inductance, in units of henries. Typically
an inductor is a conducting wire shaped as a coil; the loops
help to create a strong magnetic field inside the coil dueto Ampere's Law.
Due to the time-varying magnetic field inside the coil, a
voltage is induced, according to
Faraday's law of electromagnetic induction, which by1212
http://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Passivity_(engineering)http://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Ampere's_Lawhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Ampere's_Lawhttp://en.wikipedia.org/wiki/Henry_(unit)http://en.wikipedia.org/wiki/Inductancehttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Electronic_componenthttp://en.wikipedia.org/wiki/Passivity_(engineering) -
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(CONTD)
Inductors are one of the basic components used in
electronics where current and voltage change with time,
due to the ability of inductors to delay and reshape
alternating currents.
Inductors called chokes are used as parts of filters inpower supplies or to block AC signals from passing
through a circuit.
The effective resistance of a coil in an AC circuit iscalled its inductive reactanceand is given by
XL = 2 L
When is in Hz and L is in H, XL will be in ohms 1313
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INDUCTORS
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BEHAVIOR OF AC CIRCUITOF R-L SERIES
CIRCUIT!!!!!
1515
I UIT DIA AM
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I UIT DIA AMPHASOR DIAGRAMS &
EQUATIONS
1616
WAVEFORMS
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WAVEFORMS
Consider an AC circuit
with a source and an
inductor
The current in the
circuit is impeded bythe back EMF of the
inductor
The voltage across theinductor always leads
the current by 90
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WHAT ISCAPACITOR?????
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CAPACITORS
A basic capacitor has two parallel plates separated
by an insulating material
A capacitor stores an electrical charge betweenthe two plates
The unit of capacitance is Farads (F)
Capacitance values are normally smaller, such as
F, nF or pF
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(CONTD...)
Basic capacitor construction
2020
Dielectric
material
Plate
1
Plate2
The dielectric
material is aninsulator
therefore no
current flows
through the
capacitor
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(CONTD...)
The function of capacitor is toStoring a charge between theplates
Electrons on the left plate
are attracted toward thepositive terminal of thevoltage source
This leaves an excess ofpositively charged holes
The electrons are pushedtoward the right plate
Excess electrons leave a
negative charge 2121
+ -
+_
+ _
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CAPACITOR
2222
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BEHAVIOR OF AC CIRCUITOF R-C SERIES
CIRCUIT!!!!!
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CIRCUITConsider a circuit containing a capacitor and an AC source
The current starts out at a large value and charges the plates of the
capacitor
There is initially no resistance to hinder the flow of the current
while the plates are not chargedAs the charge on the plates increases, the voltage across the plates
increases and the current flowing in the circuit decreases
The impeding effect of a capacitor on the current in an AC circuit is
called the capacitive reactanceand is given by
2424
C2
1
XC =
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PHASOR DIAGRAMS
& EQUATIONS
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CAPACITORS IN AN AC
CIRCUITThe current reversesdirection
The voltage across theplates decreases as theplates lose the chargethey had accumulated
The voltage across thecapacitor lags behindthe current by 90
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ELEMENTS, IMPEDANCE
AND PHASE ANGLES
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BEHAVIOR OF AC CIRCUITOF R-L-C SERIES
CIRCUIT!!!!!
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CIRCUIT
The resistor, inductor,
and capacitor can be
combined in a circuitThe current in the
circuit is the same at
any time and varies
sinusoidally with time
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RELATIONSHIPS IN AN RLC
CIRCUITThe instantaneous voltage
across the resistor is inphase with the current
The instantaneous voltageacross the inductor leadsthe current by 90
The instantaneous voltageacross the capacitor lagsthe current by 90
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PHASOR DIAGRAMS
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PHASOR DIAGRAMS
To account for the
different phases of
the voltage drops,vector techniques are
used
Represent the voltage
across each element asa rotating vector,
called a phasor
The diagram is called a 3131
PHASOR DIAGRAMS
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PHASOR DIAGRAMS
3232
Vp
Ip
t
Vp
Ip
t
Vp
Ip
t
Resistor Capacitor Inductor
A phasor is an arrow whose length represents the
amplitude of an AC voltage or current.
The phasor rotates counterclockwise about the origin
with the angular frequency of the AC quantity.
Phasor diagrams are useful in solving complex ACcircuits.
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-DIAGRAMS
3333
Vp
Ip
t
Vp
Ip
t
Vp
Ip
t
Resistor Capacitor Inductor
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RS
From these equations, we can draw thephasor diagram to the right.
Assume:
imR
im
L
im
C
m
LC
R
Given: = m tsin
i i tm= sin( )
Qi
tm=
cos( )
di
dti tm= cos( )
V Ri Ri tR m= = sin( )
V Ldi
dtLi tL m= = cos( )
V
Q
C C i tC m= =
1
cos( )
l This picture corresponds to a snapshot att=0.The projections of these phasors alongthe vertical axis are the actual values of thevoltages at the given time.
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Impedance of an AC Circuit
3535
R
L
C~
The impedance, Z, of a circuitrelates peakcurrent to peak voltage:
I VZ
pp= (Units:
OHMS)
(This is the AC equivalent of
Ohms law.)
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Impedance of an RLC Circuit
R
L
C~E
As in DC circuits, we can use the loopmethod:
E - VR - VC - VL = 0I is same through all
components.
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Impedance of an RLC Circuit
3737
R
L
C~E
As in DC circuits, we can use the loopmethod:
E - VR - VC - VL = 0I is same through all
components.BUT: Voltages have differentPHASES
they add as PHASORS.
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Impedance of an RLC Circuit
3838
Solve for thecurrent:
Ip = Vp
R2 + (Xc XL )2= Vp
Z
R
L
C~
d f C Ci i
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Impedance of an RLC Circuit
3939
Solve for thecurrent:
Impedance:
Ip = Vp
R2 + (Xc XL )2= Vp
Z
Z= R2 +1
CL
2
R
L
C~
I d f RLC Ci i
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The circuit hits resonance when 1/C-L=0: r=1/When this happens the capacitor and inductorcancel each otherand the circuit behaves purely resistively: IP=VP/R.
Impedance of an RLC Circuit
4040
Ip =Vp
Z
Z= R2 +1
C L
2
The currents magnitudedepends onthe driving frequency. When Zis a
minimum, the current is amaximum.This happens at a resonancefrequency:
LC
The current diesawayat both low and
high
IP
0
1 0
2
1 0
3
1 0
4
1 0
5
R = 1 0 0
R = 1 0
r
L=1mHC=10F
PHASOR DIAGRAM FOR
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PHASOR DIAGRAM FORRLC SERIES CIRCUIT
The voltage across theresistor is on the +x axissince it is in phase withthe current
The voltage across theinductor is on the +ysince it leads the currentby 90
The voltage across thecapacitor is on the yaxis since it lags behind
the current by 90 4141
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(CONTD)
The phasors are addedas vectors to account
for the phase
differences in the
voltages
VL and VC are on
the same line and so
the net y component isVL - VC
4242
asors or a er es
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asors or a er esCircuit
4343
I
pVRp
(VCp-VLp)
VP
VCp
VLp
asors or a er es
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asors or a er esCircuit
4444
By Pythagoras theorem:(VP )2 = [ (VRp )2 + (VCp- VLp)2 ]
I
pVRp
(VCp-VLp)
VP
VCp
VLp
asors or a er es
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asors or a er esCircuit
4545
By Pythagoras theorem:(VP )2 = [ (VRp )2 + (VCp- VLp)2 ]
= -
I
pVRp
(VCp-VLp)
VP
VCp
VLp
Phase in an RLC Circuit
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Phase in an RLC Circuit
4646
I
pVRp
(VCp-VLp)
V
P
VCp
VLp
We can also find the
phase:
tan = (VCp -VLp)/ VRp
or;tan = (XC-XL)/R.
or
tan = (1/C -
L) / R
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Power in an AC
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Power in an ACCircuit
V(t) = VP sin(t)I(t) = IP sin
(t)
P(t) = IV = IP VP sin
2(t)Note this oscillates
twice as fast.
V
t
2I
2
P
=0
(This is for apurely resistivecircuit.)
4848
Power in an AC
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The power is P=IV. Since both I and V vary
in time, sodoes the power: P is a function of time.
Power in an ACCircuit
Use, V = VP sin (t) and I = IP sin ( t+ ) :
P(t) = IpVpsin(t) sin ( t+ )
This wiggles in time, usually very fast. Whatwe usually
care about is the time average of this:P
TP t dt
T=
10
( ) (T=1/f)
4949
Power in an AC
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Power in an ACCircuit
Now: sin( ) sin( )cos cos( ) sin t t t+ = +
5050
Power in an AC
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Power in an ACCircuit
P t I V t t I V t t t
P P
P P
( ) sin( ) sin( ) sin ( )cos sin( )cos( ) sin
= += +
2
Now: sin( ) sin( )cos cos( ) sin t t t+ = +
5151
Power in an AC
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Power in an ACCircuit
P t I V t t I V t t t
P P
P P
( ) sin( ) sin( ) sin ( )cos sin( )cos( ) sin
= += +
2
sin ( )
sin( ) cos( )
2 1
2
0
t
t t
=
=
Use:
and:So P I V P P=1
2cos
Now: sin( ) sin( )cos cos( ) sin t t t+ = +
5252
Power in an AC
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Power in an ACCircuit
P t I V t t I V t t t
P P
P P
( ) sin( ) sin( ) sin ( )cos sin( )cos( ) sin
= += +
2
sin ( )
sin( ) cos( )
2 1
2
0
t
t t
=
=
Use:
and:So P I V P P=1
2cos
Now:
which we usually
write as
P I V rms rms= cos
sin( ) sin( )cos cos( ) sin t t t+ = +
5353
Power in an AC
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Power in an ACCircuit
P I V rms rms= cos( goes from -900 to 900, so the averagepower is positive)
cos() is called thepower factor.
For a purely resistive circuit the power factoris 1.
When R=0, cos()=0 (energy is traded butnot dissipated).Usually the power factor depends on
frequency. 5454
Power in an AC
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Power in an ACCircuit
P I V rms rms= cosWhat if f is not zero?I
V PHere I and V are 900
out of phase. (f=
900)
(It is purely reactive)
The time averageof
P is zero.
t
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POWERThe phase f between the current and the driving emf depends on the
relative magnitudes of the inductive and capacitive reactances.
R
XL
XC
tan = X XR
L CiZ
mm= X LL
XC
C 1
XL > XC > 0current
R
XL
XC
XL < XC < 0current
XL = XC = 0current
R
XL
XC
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