Transcript Chapter 21: AC Circuits

My
Chapter 21
Lecture
Outline
1
Chapter 21: Alternating Currents
•Sinusoidal Voltages and Currents
•Capacitors, Resistors, and Inductors in AC Circuits
•Series RLC Circuits
•Resonance
2
§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 .
3
  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.
4
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).
5
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
6
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
7
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
8
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.
9
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 Wat t s
2
R
R  12 
10
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
11
§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
12
13
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.
14
For a resistor in an AC circuit,
V (t )  I (t ) R.
The voltage and current will be in phase with each other.
15
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
16
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.
17
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.
18
A similar result is found for inductors; no energy is dissipated
by an ideal inductor.
19
§21.5 Series RLC Circuits
20
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


21
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

VL  VC
IR  IX L  IX C 
2
2
 I R2  X L  X C 
2
VR
VC
X
 IZ
Z is called impedance.
22
The phase angle between the
current in the circuit and the
input voltage is:
y
VL
0
VL  VC

VR
VC
X
VL  VC X L  X C
tan 

VR
R
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.
23
Example (text problem 21.86): 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 2   X L  X C   25.0 
2
24
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
25
Example continued:
(e) The phase angle:
X L  X C 35  20
tan 

 0.75
R
20
  tan1 0.75  0.644 rads (Or 37°)
(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
26
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
27
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?).
28
§21.6 Resonance in RLC Circuits
A plot of I vs.
 for a series
RLC circuit
has a peak at
 = 0.
29
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).
XL  XC
0 L 
0 
1
0C
1
LC
This is the resonance
frequency for the circuit.
30
At resonance:
XL  XC
t an 
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.
31
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
32