Chapter 31
Alternating Current
In this chapter we will learn how resistors, inductors, and capacitors behave in
circuits with sinusoidally vary voltages and currents. We will define the relationship
between the voltage and the current in AC circuits, as well as the concept of
resonance in AC circuits.
1
Phasors and Alternating Currents
To supply an alternating current to a circuit, a source of alternating emf or voltage
is required. The sinusoidal voltage might be described by a function such as
v = V cos ωt
where V is the peak voltage, and ω = 2πf , and f is the frequency in Hz.
Figure 1:
Sinusoidal Alternating Current
i(t) = I cos(ωt)
1
1.1
Phasor Diagrams
Figure 2:
2
1.2
Rectified Alternating Current
Figure 3:
Rectified Average Current
Irav =
2
I = 0.6371 I
π
3
1.3
Root-Mean-Square (rms) Values
Figure 4:
What do we mean when we talk about the root-mean-square current? Why is this
important?
i2 = I 2 sin2 ωt
i2 is never less than zero as a function of time. The time-average of i2 is:
< i2 >T
1
=
T
Z
0
T
1
I 2 sin2 (ωt) dt = I 2
2
p
I
< i2 >T = √
(root-mean-square current)
2
where I is the maximum current in the ac circuit
Irms =
4
(1)
The same sequence of calculations occur for defining the root-mean-square voltage.
V
Vrms = √
(root-mean-square voltage)
2
where V is the maximum voltage in the ac circuit.
Example:
5
(2)
2
Resistance and Reactance
Figure 5:
vR = iR = (IR) cos ωt
vR = VR cos ωt
The current i and v are in-phase.
6
2.1
Inductor in an ac Circuit
Figure 6:
vL = L
di
= − ILω sin ωt = − I(ωL) sin ωt = −I XL sin ωt
dt
XL = ωL
(inductive reactance)
(3)
VL = I ωL
VL = I X L
(Ohm’s Law)
7
(4)
2.2
Capacitor in an ac Circuit
Figure 7:
i =
q =
vC =
I
sin ωt
ωC
dq
= I cos ωt
dt
I
sin ωt
ω
⇒
⇒
XC =
VC = I X C
vC C =
1
ωC
I
sin ωt
ω
(capacitive reactance)
(Ohm’s Law)
8
(5)
2.3
Comparing ac Circuit Elements
Figure 8:
9
3
The L-R-C Series Circuit
Figure 9:
V =
VR = IR
VL = I XL
q
)2
VR2
+ (VL − VC
= I
p
10
VC = I X C
R2 + (XL − XC )2 = I Z
Z =
p
R2 + (XL − XC )2
Figure 10:
11
(the impedance)
(6)
3.1
The Meaning of Impedance and Phase Angle
Z =
p
R2 + (XL − XC )2
tan φ =
tan φ =
4
(the impedance)
VL − V C
XL − X C
=
VR
R
ωL − 1/ωC
R
(the phase angle)
(7)
Power in Alternating-Current Circuits
p = vi
(the instantaneous power)
Figure 11:
4.1
Power in a General ac Circuit
Pav =
1
V I cos φ = Vrms Irms cos φ
2
where φ is the phase angle and cos φ is called the power-factor.
12
(8)
Figure 12:
5
Resonance in Alternating-Current Circuits
The resonance angular frequency occurs when the impedance Z is at its minimum,
thus causing the current I to be at its maximum. This occurs when XL = XC , or
when:
ωo = √
1
LC
(Resonance Angular Frequency)
13
(9)
Figure 13:
14
Figure 14:
6
Transformers
Figure 15:
15