Chapter 22: Alternating current
What will we learn in this chapter?
Contents:
Phasors and alternating currents
Resistance and reactance
Series R–L–C circuit
Power in ac-circuits
Series resonance
Parallel resonance
vintage radio
In more general terms:
How do R, L and C behave in ac currents/voltages?
Technological relevance of R–L–C circuits.
Notation: angular frequencies are always in s-1, frequencies in Hz.
Phasors and alternating currents
Reminder:
Alternators (ac generators) produce
a sinusoidally-alternating current.
L–C circuits can produce alternating
currents of variable frequency.
Nomenclature:
We use the term ac source for any device that supplies a
harmonically-varying potential difference v or current i:
v = V cos ωt
i = I cos ωt
V [I] is the maximum potential difference or voltage amplitude
[current amplitude], v [i] is the instantaneous potential difference
[current], and ω = 2πf is the angular frequency
f = cycles/second
ac sources contd.
Because the current
and voltage in the
circuit oscillate, the
direction of the current
in the emf sources
changes continuously
with time.
For time-dependent
quantities we use
lower-case letters.
Representing time-dependent currents
Phasors:
Sinusoidally-varying voltages/currents are best represented with
vector diagrams (see harmonic motion in physics 201).
Elements of a phasor:
Length = maximal current I.
The projection onto the
horizontal axis corresponds
to the instantaneous value
of the current:
i = I cos ωt
The rotation frequency is
ω = 2πf
At t = 0 i = I, hence cosine.
phasor diagram
Phasors contd.
Why the phasors?
Phasors are not physical quantities. They are merely a convenient
geometric construct.
Adding harmonic currents/voltages with phase differences
becomes a matter of vector addition.
The truth behind it...
Any harmonic motion can be represented
as a complex number in the plane:
x = Re + iIm = reiφ = r(cos φ + i sin φ)
�
√
i = −1
r = Re2 + Im2
Doing the calculus with complex numbers makes life much
simpler. But we are not allowed to in this class…
Characterizing currents/voltages
Two options:
Root-mean-square of the current Irms (1: graph I–t diagram,
2: square, 3: average, 4: take root).
Maximum of the current Imax.
I = 3A
Note:
Irms is always positive.
Calculation:
i2 = I 2 cos2 ωt
I2
= (1 + cos 2ωt)
2
Recall:
[cos x]ave = 0
I2
We obtain:
[i2 ]ave =
2
Characterizing currents/voltages contd.
rms value of harmonic currents/voltages: The rms current/voltage for a
harmonic current/voltage with amplitudes I (resp. V) is
I
Irms = √
2
V
Vrms = √
2
Notes:
Currents and voltages in power-distribution systems are always
described in terms of their rms values. Examples:
A 120V power supply has a peak voltage of ~170V.
A 0.5 A lightbulb has an amplitude of 0.71A.
Resistor in an ac circuit
Setup:
Resitor R in a sinusoidal current
i = I cos ωt .
Use Ohm’s law:
vR = iR = IR cos ωt
vR = VR cos ωt with VR = IR .
Because the voltage and the
current are both proportional to
cos ωt , the voltage is in step or
in phase with the current.
Phasors rotate together.
Inductor in an ac circuit
Setup: inductor L in an ac circuit.
i = I cos ωt
Recall: E = −L∆i/∆t
vL = L∆i/∆t = L∂i/∂t
The inductor is proportional
to the rate of change of the
current. This corresponds to
the derivative:
vL = −IωL sin ωt
in phase means
crests overlap
Inductor in an ac circuit contd.
Current and voltage are now out of phase by a quarter cycle.
Because the voltage peak occurs a quarter cycle before the current
peak, the current leads the voltage by 90º.
Using trigonometry we obtain: vL = IωL cos(ωt + π/2)
The phasor diagram shows the
situation.
Note:
In general, we describe the voltage
with respect to the current, i.e., if
i = I cos ωt
then
v = V cos(ωt + φ)
where φ is the phase angle.
Inductor in an ac circuit contd.
Voltage amplitude for an inductor: vL = IωL cos(ωt + π/2) = XL i
VL = IωL
Inductive reactance: We define the inductive reactance XL of an
inductor as the product of inductance and frequency, i.e.,
XL = ωL
VL = IXL
Note:
Because V = IXL like for a resistor, the unit of the reactance is also
Ohm (Ω).
Since the reactance behaves like a resistance and is directly
proportional to the frequency ω , it is used to filter high
frequencies out (also called a choke): XL big when ω big.
Capacitors in a ac circuit
Setup: Capacitor C in a harmonic
! ! current i = I cos ωt.
Goal: Compute the voltage between a and b.
The rate of change of the voltage is given by
∆vC
1 ∆q
i
∂vC
i
=
=
=
∆t
C ∆t
C
∂t
C
Performing an integral
vC =
I
sin ωt
ωC
Thus the voltage is again sinusoidal, and the current is largest
when the capacitor is charging or discharging.
The voltage and current are out of phase by a quarter cycle, the
voltage lags the current by 90º.
Capacitors in a ac circuit contd.
Graphic representation
of the voltage lag.
The voltage runs behind
the current.
Corresponding phasor
diagram.
Capacitors in a ac circuit contd.
Voltage lag:
vC =
I
cos(ωt − π/2)
ωC
The maximum voltage VC is given by
I
VC =
ωC
Capacitive reactance: We define the capacitive reactance XC of a
capacitor as the inverse product of capacitance and frequency, i.e.,
1
VC = IXC
XC =
ωC
Note:
The capacitive reactance is inversely proportional to the
frequency, i.e., low frequencies and dc currents are filtered.
Summary
Note:
The figure shows how resistance
and reactances vary with frequency.
When the frequency is zero (dc),
there is no current trough a
capacitor and there is no inductive
effect
XC |ω→0 → ∞
XL |ω→0 → 0
Application: Speakers
Question: !
! ! ! !
! ! ! !
There is one wire connecting speakers. How are low
and high frequencies filtered out to woofer and
tweeter?
The capacitor in the tweeter blocks
the low frequencies
The inductor in the woofer blocks
the high frequencies.
The series R–L–C circuit
Many electronic systems consist of R, L and C
in series with i = I cos ωt.
Analysis:
Following Kirchhoff’s rules, the total voltage
is the same for all components.
The phasor representing this total voltage
is the vector sum of the voltage phasors of all components.
Because the elements are connected in series, the current at any
instant can be represented by a single phasor I, representing all
elements of the circuit.
Nomenclature:
vn, n = R, L, C is used for instantaneous values e.g., vC = vcd.
Vn, n = R, L, C is used for the maximum values.
The series R–L–C circuit: Phasors
Resistor (in phase):
VR = IR
Inductor (leads by π/2):
VL = IXL
Capacitor (lags by π/2):
VC = IXC
One can now compute
the source voltage
phasor V as the vector
sum of VR, VL, VC.
After some algebra…
The series R–L–C circuit contd.
After performing the vector sum we obtain:
�
V = I R2 + (XL − XC )2
The quantity XL - XC is called the reactance X of the circuit.
The impedance of the circuit is defined as:
�
�
2
2
Z = R + (XL − XC ) = R2 + X 2
It follows:
V = IZ
Impedance of a series R–L–C circuit: The impedance is always a ratio of
voltage to current. The SI unit is the Ohm (Ω):
�
Z = R2 + X 2
�
= R2 + (XL − XC )2
�
= R2 + [ωL − (1/ωC)]2
The series R–L–C circuit contd.
Note:
�
The equation Z = R2 + [ωL − (1/ωC)]2 only gives the
impedance of a series R–L–C circuit.Via V = IZ the impedance of
any circuit can be computed.
The angle φ between current I and voltage V is given by
�
�
�
�
ωL − 1/ωC
XL − XC
φ = arctan
= arctan
R
R
If XL > XC the source voltage leads the current by an angle
between 0 and 90º and v = V cos(ωt + φ).
If XL < XC the vector V lies on the opposite
side of the current vector I and the
voltage lags the current. In this case
the angle lies between –90 and 0º.
If R/C/L are missing, drop them in
the equations.
Maximum vs rms
So far, we have only dealt with maximal values of current and voltage
(their amplitudes).
General rule:
√
For any harmonic quantity the rms value is always 1/ 2
times the amplitude.
Example:
Vrms = Irms Z
Note:
In all previous discussions we have described the steady state of
the system.
Transient effects have not been considered.
Power in ac circuits (resistance only)
Motivation: !
! ! ! !
! ! ! !
Assumption:
Because ac circuits are generally used for power
transfer and generation, we need to understand how
the power behaves in a harmonic environment.
i = I cos ωt
Resistance-only circuit (pure resistor):
i and v are in phase.
In general, P = VI.
Because i and v are in phase, p > 0.
Average power:
1
V I
P = V I = √ √ = Vrms Irms
only for
2
2 2
resistors!
Because Vrms = Irms R
2
Vrms
2
P = Irms R =
= Vrms Irms
R
Power in ac circuits (inductor/capacitor)
the average power is zero!
Because for inductors/capacitors the current and voltage are out of
phase, the product oscillates around zero.
It follows that the net energy transfer over one cycle is zero!
Power in ac circuits (general case)
In general:
i = I cos ωt
v = V cos(ωt + φ)
Power:
p = vi = [V cos(ωt + φ)][I cos ωt]
using trigonometry relations
p = V I cos φ cos2 ωt − V I sin φ cos ωt sin ωt
average over time and recall that [cos ωt]ave = 0
[cos2 ωt]ave = 1/2
to obtain
1
P = V I cos φ = Irms Vrms cos φ
2
When v and i are in phase, P > 0, when the phase is 90º, P = 0.
Power in ac circuits (general case) contd.
For an R-L-C circuit V cos φ is the voltage
amplitude for the resistor, hence
P = Irms Vrms cos φ
represents the power dissipated in the
resistor.
The power dissipated in the capacitor
and the inductor is zero.
The factor cos φ is called the power factor of the circuit:
For a resistance cos φ = 1
For a capacitor/inductance cos φ = 0
For a R-L-C circuit cos φ = R/Z.
Note: a large lead/lag angle is undesirable since for a given potential a
large current is needed to supply a given amount of power.
Series resonance
In a R-L-C circuit, the impedance depends on the frequency as
follows:
�
Z = R2 + X 2
�
= R2 + (XL − XC )2
�
= R2 + [ωL − (1/ωC)]2
For XC = XL the impedance Z
as a function of the frequency
has a minimum which only
depends on R.
We can now tune the frequency
of a potential source trough this
minimum and see what happens…
Series resonance contd.
Connect a potential source with fixed
V and variable frequency to the system.
current peaks at the
frequency where Z is least
As the frequency changes, the current
amplitude I has a maximum when Z
has a minimum.
This is called a resonance. The
angular frequency ω0 at which this
occurs is called the resonance
frequency. This is the case when
1
ω0 = √
XC = XL
LC
At the resonance, the voltage across the L–C part of the circuit is
zero (180º phase).
The resonance frequency can be tunes with L and C (radio…).
Series resonance contd.
At the resonance, the impedance is minimal and the current I is
maximal.
ω0 = 1000 rad/s
Example response curve:
V = 100V
R = 500Ω
L = 2.0H
C = 0.5µF
The height of the
peak is determined
by the value of R.
Note: in the old days the shape of the resonance (response) curve
was crucial for radio tuning.
Parallel resonance
A different kind of resonance occurs when R, L
and C are connected in parallel.
The system also shows resonant behavior, but
the roles of the voltage and current are
reversed.
In this case, the potential difference is the same
for all elements
v = V cos ωt
but the current is different.
The current iR is in phase with the source and
has a peak value IR = V /R .
The current iL lags by π/2 and has a peak value
IL = V /XL = V /ωL.
The current iC leads by π/2 and has a peak
value IC = V /XC = V ωC .
Parallel resonance contd.
At any frequency the inductor
current and the capacitor current
phase differs by π and cancel.
Again, when XL = XC we
have a resonance with
1
ω0 = √
LC
This is the same as before,
except that here the total
current reaches a minimum at
the resonance.
�
�
�2
For the parallel case
1
1
1
=
+ ωC −
Z
R2
ωL
hence Z has a maximum when Z = R.