Alternating Current

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Alternating Current Circuits
Lectures 25, 26, 27
Ph 2B lectures by George M. Fuller, UCSD
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George M. Fuller 427 SERF gfuller_at_ucsd.edu 822-1
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TH 700 PM - 800 PM WLH 2204
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no quiz
Ch 29,30,31
Ch 31,32
March 5 Ch 32,33
problem assignment TBA
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Sinusoidal currents and voltages are
characterized by three quantities amplitude,
frequency, and phase (or, more correctly, phase
constant).
amplitude
peak-to-peak value is twice the amplitude
root mean square or rms values
Ph 2B lectures by George M. Fuller, UCSD
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relation between angular frequency w (radians per
second s-1) and frequency f in cycles per
second (Hz)
angular frequency
phase constant
wt
sin(wtf)
one wavelength 2p radians
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some mathematics . . .
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some more math . . .
So the average of the square of a circular
function over a cycle (one period) is
This is why we define the root-mean-square the
way we do it is the square root of the average
of the square of the quantity over one cycle
(period).
Ph 2B lectures by George M. Fuller, UCSD
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even more math . . .
Why bother with any of this? It is easier to work
with exponentials than sins and coss.
Ph 2B lectures by George M. Fuller, UCSD
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We can write a complex number in several ways . .
.
For example, we could write it as a real
part plus an imaginary part
The complex conjugate of this number is
obtained by changing the sign of the imaginary
part
The magnitude of this number is
Another way to write this complex number is then
The complex conjugate is
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Behavior of Circuit Elements in AC Circuits
current and voltage are in phase for a resistor
resistors
capacitors
we say that the current leads the voltage by p /2
Note that we can write a relation between the
peak current and peak voltage
The reactance acts like a resistance, but it
is frequency dependent. At very high frequency it
disappears. At low freq. it causes the capacitor
to behave like an open circuit.
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Another look at the capacitor in an AC circuit .
. .
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Behavior of Circuit Elements in AC Circuits
continued . . .
inductors
The loop law for this circuit gives
L
Let us employ the relation between inductor EMF
and the rate of change of current and re-write
the loop law
We want a relationship between voltage and
current, so integrate this expression
the voltage across the inductor leads the current
by 90 degrees
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from this we can conclude that the relation
between the peak current and the peak voltage is
given by
Note that the inductive reactance increases with
increasing frequency and increasing inductance.
This makes physical sense since the back EMF from
the inductive circuit element increases with
increased L and/or increased rate of change of
current.
At very high frequencies an inductor behaves like
an open circuit, and at very low frequencies is
behaves like a short circuit.
Ph 2B lectures by George M. Fuller, UCSD
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Another look at an inductor in an AC circuit . . .
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(complex) impedances of circuit elements
voltage lags current by 900
voltage leads current by 900
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Note impedances and reactances and resistances
are measured in Ohms. Why?
We add impedances in series or in parallel in the
same way in which we add resistances in these
situations.
Note that impedance is a complex number in
general and we can write it as the sum of a real
and imaginary part
Physically, this means that we can represent any
circuit segment with an equivalent impedance
which, in turn, we can represent as a resistance
in series with a pure imaginary impedance ( the
real coefficient of which we have called the
reactance).
R
Z
iX
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Consider an RLC circuit as shown, in which a
resistance, inductor, and capacitor are connected
in series with a generator which can deliver a
time-varying driving voltage.
The loop law for this circuit gives
We can re-write this using the definition of
current and back EMF from the inductance
Does this equation look familiar? It is of the
same form as the equation for a driven, or
forced, oscillator (e.g., a mass on a spring),
including a damping force
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First, a general solution for our loop equation
We will represent the driving voltage and the
charge (e.g., on the capacitor) as the real part
of complex numbers
Where the resonant frequency (determined by the
capacitance and the inductance) and the damping
factor are defined, respectively, as
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The complex amplitudes of the charge and the
driving voltage are related by the equation
we found on the previous slide
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The response of the magnitude (squared) of the
charge in terms of the voltage is an example of a
Lorentzian curve
Ph 2B lectures by George M. Fuller, UCSD
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Ph 2B lectures by George M. Fuller, UCSD
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Example Consider a series RLC circuit with R
10 Ohms, L 100 mH, and C 1.0 mF.
What is the rms current that flows if the rms
driving voltage is 100 V and
driven at one half the resonant angular frequency
for this circuit?
Watch out! The elements are in series and have
the same current flowing through them. However,
we cannot simply use the scalar version of Ohms
law because the voltage responses across the
capacitor and inductor are out of phase by 180
degrees.
We must add the impedances (with their phases) as
vectors or phasors or complex numbers. The
magnitude of the resultant total impedance will
be
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To evaluate this we need to know the angular
frequency, which we are told is half of the
resonant frequency. The resonant angular
frequency we can calculate from
We are told that the driving freq. is 1/2 of this
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Special Case Consider an LC circuit as shown, in
which an inductor and a capacitor are connected
in series.
The loop law for this circuit gives
We can re-write this using the definition of
current and back EMF from the inductance
Does this equation look familiar? It is of the
same form as the equation for the displacement of
a mass on a spring
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For our LC circuit, we found that the charge on
the capacitor is given by
where the resonant frequency is
Equations like this have a simple solution
because of the differentiation properties of
sines and cosines
If initially (t 0) the charge on the capacitor
is its maximum value q(0) and the current is
I(0)0, then the constants in the general
solution above are fixed to be
giving time-dependent charge current as
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If initially (t 0) the charge on the capacitor
is its maximum value q(0) and the current is
I(0)0, then the solution of this equation is
So, you see that the charge decreases with
time as the current through the inductor
increases. After a quarter period there is no
charge on the capacitor and the current is
maximal, implying that the magnetic field
through the inductor is maximal.
The current keeps flowing (because of the
inductor), charging up the capacitor to the
opposite sign of charge (at 1/2 period). Then
this drives a current through the inductor in the
other direction, etc., etc., etc.. Note that the
energy sloshes back and forth between being
contained in the electric field of the capacitor
and the magnetic field of the inductor.
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Special Case Consider an RLC circuit as shown,
in which an inductor and a capacitor and a
resistor are connected in series. No driving
voltage in this case so . . .
plug all this into the loop eqn.
solve this quadratic equation to get
simplify to get
Now, for small damping, R/2L ltlt 1/(LC)1/2 , we
have
Take the positive value
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We take the real part of this to get the charge
on the capacitor
which could be, for example,
Remember, this solution obtains in the small
damping limit
critical damping
over-damping
Since this circuit has a resistor it exhibits
damping energy is lost to heat dissipated

by the resistor.
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Power in AC circuits
The power in an AC circuit, like for any circuit,
is the product of the current and the voltage.
Watch out here, though, as the current and
voltage are time dependent quantities and so may
or may not be in phase.
We know that current and voltage are related
through a complex impedance that can be written
as the sum of a resistance and a (pure imaginary)
reactance term.
If we let the current be the real part of
The instantaneous voltage is the real part of
Ph 2B lectures by George M. Fuller, UCSD
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The average power (rate of energy loss) in this
circuit is the integral of the instantaneous
power (product of current and voltage) over one
period divided by the period, or
0
There is no energy loss from the reactive part
of the circuit, that is, from the inductors and
capacitors, but only from the resistance.
Ph 2B lectures by George M. Fuller, UCSD
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Through any circuit or circuit element where the
current and voltage differ in phase constant we
can write, for example,
The so-called power factor is
Ph 2B lectures by George M. Fuller, UCSD
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