LECTURE 33
AC CIRCUITS (RC & L & LC)
Instructor: Kazumi Tolich
Lecture 33
2
!
Reading chapter 24-3 to 24-4.
!  Resistors
and capacitors in AC circuits (RC circuit)
!  Inductors in AC circuits
!  Resistors and inductors in AC circuits (RL circuit)
RC circuit voltages
3
!
!
Consider an RC circuit with a generator with a maximum
voltage of Vmax, a resistor and a capacitor.
The voltage across the resistor and the voltage across the
capacitor are not in phase; they do not peak at the same time.
Therefore
Vmax ≠ Vmax, R + Vmax, C
where
Vmax, R = I max R
Vmax, C = I max X C =
I max
ωC
Phasor diagram for an RC circuit
4
!
!
!
!
Current phasor with a magnitude Imax.
Resistor-voltage phasor with a magnitude
ImaxR. The resistor-voltage phasor is in
phase with the current phasor.
Capacitor-voltage phasor with a
magnitude ImaxXC. The capacitor-voltage
phasor lags current phasor by 90°.
The total voltage phasor is the vector
sum of the resistor-voltage and
capacitor-voltage phasors.
Impedance for an RC circuit
5
!
The magnitude of the total voltage is
Vmax =
!
(I
) (
2
R + I max X C
max
)
2
= I max R 2 + X C2
This has the exact same form as Ohm’s
law (V = IR) if we define the
impedance, Z (in Ω) for an RC circuit:
⎛ 1 ⎞
Z = R + X = R +⎜
⎝ ω C ⎟⎠
2
!
2
C
2
2
The maximum current in an RC circuit is
I max =
Vmax
Z
Example: 1
6
!
An ac generator with a
frequency of f = 105 Hz and
an rms voltage of
Vrms = 22.5 V is connected in
series with a resistor with a
resistance of R = 10.0 kΩ,
and a capacitor with a
capacitance of C = 0.250 µF.
What is the rms current in this
circuit?
Phase angle
7
There is a phase angle ϕ between
the total voltage and the current.
!  The phase angle can be found by
!
Power factor for an RC circuit
8
!
The average power delivered to the circuit is
⎛ Vrms ⎞
Pav = I R = I rms ⎜
R = I rmsVrms cos φ
⎟
⎝ Z ⎠
2
rms
!
Therefore cos ϕ is called the power factor.
Purely resistive
Purely capacitive
Example: 2
9
a)
b)
Sketch the phasor diagram for an
ac circuit with a resistor with a
resistance of R = 105 Ω in series
with a capacitor with a
capacitance of C = 32.2 µF. The
frequency of the generator is
f = 60.0 Hz.
If the rms voltage of the
generator is Vrms = 120 V, what is
the average power consumed by
the circuit?
Inductors in AC circuits
10
!
The relationship among the rms current Irms in the inductor,
its inductance L, the angular frequency ω, and the rms
voltage across the inductor Vrms is given by
I rms =
!
Vrms
ωL
In analogy with ohm’s law applied to resistance
(Vrms = IrmsR), we write:
Vrms = I rmsω L ≡ I rms X L
Inductive reactance
11
!
Inductive reactance is defined by
X L ≡ ω L = 2π f L
and has a unit of Ohms, Ω.
!  Rms current and rms voltage, and max current and
max voltage are related by
I rms =
Vrms
XL
I max =
Vmax
XL
Behaviors at high or low frequencies
!
!
!
!
The inductive reactance depends on the
frequency.
X L ≡ ω L = 2π fL
In the low frequency limit, XL is small, an
inductor acts like a short circuit.
In the high frequency limit, XL is large, a
inductor acts like a open circuit.
The behavior as a function of frequency
is opposite from that of capacitors.
Reactance, or resistance
12
I and V in an ac inductor circuit
13
!
The voltage across an inductor leads the current by 90° or π/2.
!
The current and voltage are
( )
I = I max sin ω t
!
(
V = Vmax sin ω t + 90!
)
The phase difference ϕ between the current and the voltage is
-90°, giving the power factor of cos ϕ = 0.
Power
14
!
!
!
!
!
The instantaneous power for any
circuit is P = IV.
P > 0 in 0 < ωt < π/2: the inductor
delivers energy to the generator.
P < 0 in π/2 < ωt < π: the inductor
draws energy from the generator.
The average power as a function of
time is zero.
An inductor in an ac circuit consumes
zero net energy.
Clicker question: 1
15
Example: 3
16
!
An inductor with an
inductance of L = 0.22 µH
is connected to an ac
generator with an rms
voltage of Vrms = 12 V.
For what range of
frequencies will the rms
current in the circuit less
than 1.0 mA?
RL circuit voltages
17
!
!
Consider an RL circuit with a generator oscillating at ω with a
maximum voltage of Vmax, a resistor with a resistance R and an
inductor with an inductance L.
The voltage across the resistor and the voltage across the
inductor are not in phase; they do not peak at the same time.
Therefore
Vmax ≠ Vmax, R + Vmax, L
where
Vmax, R = I max R
Vmax, C = I max X L = I maxω L
Phasor diagram for an RL circuit
18
!
!
!
!
Current phasor with a magnitude Imax.
Resistor-voltage phasor with a magnitude
ImaxR. The resistor-voltage phasor is in
phase with the current phasor.
Inductor-voltage phasor with a
magnitude ImaxXL. The inductor-voltage
phasor leads current phasor by 90°.
The total voltage phasor is the vector
sum of the resistor-voltage and inductorvoltage phasors.
Impedance for an RL circuit
19
!
The magnitude of the total voltage is
Vmax =
!
(I
) (
2
R + I max X L
max
)
2
= I max R 2 + X L2
This has the exact same form as Ohm’s
law (V = IR) if we define the
impedance, Z (in Ω) for an RL circuit:
( )
Z = R + X = R + ωL
2
!
2
L
2
2
The maximum current in an RL circuit is
I max =
Vmax
Z
Power factor for an RL circuit
20
!
The power factor for an RL circuit is:
Irms comparisons
21
!
Rms currents in a resistor-only, an RC, and an RL
circuits as a function of angular frequency:
Ferrite beads
22
A ferrite bead contains an inductor made of a special
material called a ferrite with a large resistance.
!  Electronic devices might be near other devices that
radiate damaging high-frequency signals.
!  Such high-frequency noise signals can be reduced by
ferrite beads by dissipating the unwanted signals as
heat.
!