Download Chapter 21: AC Circuits

Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Chapter 21: Alternating Currents
•Sinusoidal Voltages and Currents
•Capacitors, Resistors, and Inductors in AC Circuits
•Series RLC Circuits
•Resonance
•AC to DC Conversion
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
1
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§21.1 Sinusoidal Currents and
Voltage
A power supply can be set to give an EMF of form:
 (t )   0 sin t
This EMF is time dependent, has an amplitude 0, and
varies with angular frequency .
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
2
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
  2f
angular
frequency
in rads/sec
frequency in
cycles/sec or Hz
The current in a resistor is still given by Ohm’s Law:
I (t ) 
 (t )
R

0
R
sin t  I 0 sin t
The current has an amplitude of I0=0/R.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
3
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The instantaneous power dissipated in a resistor will be:
P  I (t )VR (t )
 I 0 sin t  0 sin t   I 0 0 sin 2 t
The power dissipated depends on t (where in the cycle the
current/voltage are).
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
4
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What is the average power dissipated by a resistor in one
cycle?
The average value sin2t over one cycle is 1/2.
1
The average power is Pav  I 0 0 .
2
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
5
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
I rms
I0

2
and  rms 
0
2
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
6
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
In terms of rms quantities, the power dissipated by a resistor
can be written as:
I0  0
1
Pav  I 0 0 
2
2 2
2
 I rms  rms  I rms
R
2
 rms
R
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
7
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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:
Pav  I rms  rms
100 Watts  I rms 120 V 
I rms  0.83 Amps
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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
8
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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?
Pav 
 2 rms
R

120 V 
1200 Watts 
2
R
R  12 
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
9
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) What is the rms current drawn by the hair dryer?
Pav  I rms  rms
1200 Watts  I rms 120 V 
I rms  10 Amps
(c) What is the maximum instantaneous power that the
resistance must withstand?
P  I 0 0 sin t  Pmax  I 0 0
2
1
Pav  I 0 0
2
Pmax = 2Pav = 2400 Watts
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
10
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§21.3-4 Capacitors, Resistors and
Inductors in AC circuits
For a capacitor:
Q(t )  CVC (t )
Q(t )
 VC (t ) 
 C

In the circuit: I (t ) 
t
 t 
Slope of the
plot V(t) vs. t
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
11
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
12
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 VCI. The equality is Vc = IXC where XC
is called capacitive reactance. (Think Ohm’s Law!)
1
XC 
C
Reactance has
units of ohms.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
13
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
For a resistor in an AC circuit, V (t )
 I (t ) R.
The voltage and current will be in phase with each other.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
14
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
For an inductor in an AC circuit:
 I (t ) 
VL  L

 t 
Slope of an
I(t) vs. t plot
Also, VL = IXL where the inductive reactance is:
X L  L
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
15
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
16
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
17
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
A similar result is found for inductors; no energy is dissipated
by an ideal inductor.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
18
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§21.5 Series RLC Circuits
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
19
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Applying Kirchhoff’s loop rule:
 (t )  VL (t )  VR (t )  VC (t )  0
 (t )   0 sin t   




 VL sin  t    VR sin t   VC sin  t  
2
2


Copyright © 2008 – The McGraw-Hill Companies s.r.l.
20
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
To find the amplitude (0) and phase () of the total voltage
we add VL, VR, and VC together by using phasors.
y
 0  V  VL  VC 
2
2
R

VL
0
IR   IX L  IX C 
2
2
 I R  X L  X C 
2
VR
X
2
 IZ
VC
Z is called impedance.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
21
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
y
The phase angle between the
current in the circuit and the input
voltage is:
VL
0
VL  VC X L  X C
tan  

VR
R

VR
VC
X
VR
R
cos  

0 Z
>0 when XL> XC and the voltage leads the current
(shown above).
<0 when XL< XC and the voltage lags the current.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
22
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 1.0103
rad/sec. Find…
(a) The reactances of the capacitor and the inductor.
X L  L  35.0 
1
XC 
 20.0 
C
(b) The impedance.
Z  R   X L  X C   25.0 
2
2
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
23
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(c) The rms current:
 rms  I rms Z
 rms 100.0 V
I rms 
Z

25.0 
 4.00 Amps
(d) The current amplitude:
I rms
I0

2
I 0  2 I rms  5.66 Amps
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
24
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(e) The phase angle:
X L  X C 35  20
tan  

 0.75
R
20
(Or 37°)
  tan 1 0.75  0.644 rads
(f) The rms voltages across each circuit element:
Vrms , R  I rms R  80.0 V
Vrms , L  I rms X L  140 V
Vrms ,C  I rms X C  80.0 V
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
25
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(g) Does the current lead or lag the voltage?
Since XL>XC,  is a positive angle. The voltage leads
the current.
(h) Draw a phasor diagram.
y
VL
rms

VC
VR
X
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
26
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The power dissipated by a resistor is:
Pav  I rms  rms, R  I rms  rms cos 
where cos is called the power factor (compare to slide 7;
Why is there a difference?).
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
27
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§21.6 Resonance in RLC Circuits
A plot of I vs.
 for a series
RLC circuit
has a peak at
 = 0.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
28
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The peak occurs at the resonant frequency for the circuit.
I

Z


R2  X L  X C 
2
The current will be a maximum when Z is a minimum. This
occurs when XL = XC (or when Z=R).
X L  XC
0 L 
1
0C
0 
1
LC
This is the resonance
frequency for the circuit.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
29
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
At resonance:
X L  XC
tan  
0
R
R
cos    1
R
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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
30
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
31
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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).
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
32
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
33
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
A high-pass filter. This will allow high frequency signals
to pass through while filtering out low frequency signals.
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
34
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
35