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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Chapter 21: Alternating Currents
•Sinusoidal Voltages and Currents
•Capacitors, Resistors, and Inductors in AC Circuits
•Series RLC Circuits
•Resonance
•AC to DC Conversion
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§21.1 Sinusoidal Currents and Voltage
A power supply can be set to give an EMF of form:
tt sin)( 0
This EMF is time dependent, has an amplitude 0, and varies with angular frequency .
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
f 2
angular frequency in rads/sec
frequency in cycles/sec or Hz
The current in a resistor is still given by Ohm’s Law:
tItRR
ttI
sinsin)(
)( 00
The current has an amplitude of I0=0/R.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The instantaneous power dissipated in a resistor will be:
tIttI
tVtIP R
20000 sinsinsin
)()(
The power dissipated depends on t (where in the cycle the current/voltage are).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
What is the average power dissipated by a resistor in one cycle?
The average value sin2t over one cycle is 1/2.
.2
100IPav The average power is
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
What are the averages of V(t) and I(t) over one cycle?
The “problem” here is that the average value of sin t over one complete cycle is zero! This is not a useful way to characterize the quantities V(t) and I(t).
To fix this problem we use the root mean square (rms) as the characteristic value over one cycle.
2 and
20
rms0
rms
I
I
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
In terms of rms quantities, the power dissipated by a resistor can be written as:
R
222
1
2rms2
rmsrmsrms
0000av
RII
IIP
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 21.4): A circuit breaker trips when the rms current exceeds 20.0 A. How many 100.0 W light bulbs can run on this circuit without tripping the breaker? (The voltage is 120 V rms.)
Each light bulb draws a current given by:
Amps 83.0
V 120 Watts100
rms
rms
rmsrmsav
I
I
IP
If 20 amps is the maximum current, and 0.83 amps is the current drawn per light bulb, then you can run 24 light bulbs without tripping the breaker.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 21.10): A hair dryer has a power rating of 1200 W at 120 V rms. Assume the hair dryer is the only resistance in the circuit.
(a) What is the resistance of the heating element?
12
V 120 Watts1200
2
rms2
av
RR
RP
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(b) What is the rms current drawn by the hair dryer?
Amps 10
V 120 Watts1200
rms
rms
rmsrmsav
I
I
IP
(c) What is the maximum instantaneous power that the resistance must withstand?
00max2
00 sin IPtIP 002
1 IPav
Pmax = 2Pav = 2400 Watts
Example continued:
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§21.3-4 Capacitors, Resistors and Inductors in AC circuits
)()( tCVtQ CFor a capacitor:
In the circuit:
t
tVC
t
tQtI C )()()(
Slope of the plot V(t) vs. t
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The current in the circuit and the voltage drop across the capacitor are 1/4 cycle out of phase. Here the current leads the voltage by 1/4 cycle.
Here it is true that VCI. The equality is Vc = IXC where XC is called capacitive reactance. (Think Ohm’s Law!)
CXC
1 Reactance has
units of ohms.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
For a resistor in an AC circuit,
.)()( RtItV
The voltage and current will be in phase with each other.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
For an inductor in an AC circuit:
t
tILVL
)(
Also, VL = IXL where the inductive reactance is:
LX L
Slope of an I(t) vs. t plot
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The current in the circuit and the voltage drop across the inductor are 1/4 cycle out of phase. Here the current lags the voltage by 1/4 cycle.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Plot of I(t), V(t), and P(t) for a capacitor.
The average power over one cycle is zero. An ideal capacitor dissipates no energy.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
A similar result is found for inductors; no energy is dissipated by an ideal inductor.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§21.5 Series RLC Circuits
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
2sinsin
2sin
sin)( 0
tVtVtV
tt
CRL
Applying Kirchhoff’s loop rule:
0)()()()( tVtVtVt CRL
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
To find the amplitude (0) and phase () of the total voltage we add VL, VR, and VC together by using phasors.
IZ
XXRI
IXIXIR
VVV
CL
CL
CLR
22
22
220
Z is called impedance.
X
y
VR
VL
VC
0
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The phase angle between the current in the circuit and the input voltage is:
Z
RV
R
XX
V
VV
R
CL
R
CL
0
cos
tan
>0 when XL> XC and the voltage leads the current (shown above).
<0 when XL< XC and the voltage lags the current.
X
y
VR
VL
VC
0
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 21.79): In an RLC circuit these three elements are connected in series: a resistor of 20.0 , a 35.0 mH inductor, and a 50.0 F capacitor. The AC source has an rms voltage of 100.0 V and an angular frequency of 1.0103 rad/sec. Find…
(a) The reactances of the capacitor and the inductor.
0.201
0.35
CX
LX
C
L
(b) The impedance.
0.2522CL XXRZ
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(c) The rms current:
Amps 00.4 25.0
V 0.100
Zrms
rms
rmsrms
I
ZI
(d) The current amplitude:
Amps 66.52
2
rms0
0rms
II
II
Example continued:
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(e) The phase angle:
rads 644.075.0tan
75.020
2035tan
1
R
XX CL
(f) The rms voltages across each circuit element:
V 0.80
V 140
V 0.80
rms,rms
rms,rms
rms,rms
CC
LL
R
XIV
XIV
RIV
(Or 37°)
Example continued:
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(g) Does the current lead or lag the voltage?
(h) Draw a phasor diagram.
Since XL>XC, is a positive angle. The voltage leads the current.
y
XVR
VL
VC
rms
Example continued:
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The power dissipated by a resistor is:
cosrmsrmsrms,rmsav IIP R
where cos is called the power factor (compare to slide 7; Why is there a difference?).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§21.6 Resonance in RLC Circuits
A plot of I vs. for a series RLC circuit has a peak at = 0.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The peak occurs at the resonant frequency for the circuit.
22CL XXRZ
I
The current will be a maximum when Z is a minimum. This occurs when XL = XC (or when Z=R).
LC
CL
XX CL
1
1
0
00
This is the resonance frequency for the circuit.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
At resonance:
1cos
0tan
R
RR
XX CL
The phase angle is 0; the voltage and the current are in phase. The current in the circuit is a maximum as is the power dissipated by the resistor.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§21.7 Converting AC to DC; Filters
A diode is a circuit element that allows current to pass through in one direction, but not the other.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The plot shows the voltage drop across the resistor. During half a cycle, it is zero.
Putting a capacitor in the circuit “smoothes” out VR, producing a nearly constant voltage drop (a DC voltage).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
A capacitor may be used as a filter.
Low-pass filter. When XC << R ( is large) the output voltage will be small compared to the input voltage.
When XC >> R ( is small), the output voltage will be comparable to the input voltage.
This circuit will allow low frequency signals to pass through while filtering out high frequency signals.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
A high-pass filter. This will allow high frequency signals to pass through while filtering out low frequency signals.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Summary
•Difference Between Instantaneous, Average, and rms Values
• Power Dissipation by R, L, and C
•Reactance for R, L, and C
•Impedance and Phase Angle
•Resonance in an RLC Circuit
•Diodes
•High- and Low-Pass Filters