AC Circuits & Phasors
We want to understand RLC circuits driven with a
sinusoidal emf.
First: engineers characterize the amplitude of a sinusoidal
emf with the “root mean square” (rms) value.
(This is because the power dissipated in a resistor is I2R
or V2/R… if we want to use an I or a V from a sine wave
emf, we have to use the rms I or V. )
Which phasor shows
a current that is
positive and
increasing?
Which phasor shows a current that is negative with an
increasing magnitude (i.e. getting more negative?)
X = “reactance”; the effective combination of Xs
in a circuit is called “impedance.”
At high frequencies, the reactance of what circuit
element decreases?
A] resistor
B] capacitor
C] inductor
D] none decrease
E] all decrease
Low reactance is like low resistance…
small voltage drop for a large current.
An AC voltage source drives a
sinusoidal current through two
resistors.
The amplitude of the sinusoidal voltage
across the top resistor is 4 V.
The amplitude of the sinusoidal voltage
across the bottom resistor is 3 V.
What is the amplitude of the sinusoidal voltage provided by the source?
A] 0 V
D] 7 V
B] 1 V
E] 12 V
C] 5 V
An AC voltage source drives a
sinusoidal current through a resistor
and an inductor in series.
The amplitude of the sinusoidal voltage
across the top resistor is 4 V.
The amplitude of the sinusoidal voltage
across the bottom inductor is 3 V.
What is the amplitude of the sinusoidal voltage provided by the source?
A] 0 V
D] 7 V
B] 1 V
E] 12 V
C] 5 V
Phasors
The x axis projection is the instantaneous value.
The length is the amplitude.
V=iX says the LENGTH of the voltage phasor is
proportional to the LENGTH of the current phasor.
The proportionality constant is the reactance, X.
Circuit elements in series have the same current phasor.
The voltage phasors add (like vectors) to give the total voltage.
Circuit elements in parallel have the same voltage phasor.
The current phasors add (like vectors) to give the total current.
An AC voltage source drives a
sinusoidal current through a capacitor
and a resistor in series.
The amplitude of the sinusoidal voltage
across the top capacitor is 4 V.
The amplitude of the sinusoidal voltage
across the bottom resistor is 3 V.
What is the amplitude of the sinusoidal voltage provided by the source?
A] 0 V
D] 7 V
B] 1 V
E] 12 V
C] 5 V
An AC voltage source drives a
sinusoidal current through a capacitor
and an inductor in series.
The amplitude of the sinusoidal voltage
across the top capacitor is 4 V.
The amplitude of the sinusoidal voltage
across the bottom inductor is 3 V.
What is the amplitude of the sinusoidal voltage provided by the source?
A] 0 V
D] 7 V
B] 1 V
E] 12 V
C] 5 V
If we change frequency, we change
the reactance X of the cap and
the inductor.
If we increase the frequency:
A)
B)
C)
D)
Both X’s go up
XL goes up, but Xc goes down
Xc goes up, but XL goes down
Both X’s go down.
Recall V=iX
If we change frequency, we change
the reactance X of the cap and
the inductor.
If we increase the frequency:
A)
B)
C)
D)
Both X’s go up
XL goes up, but Xc goes down
Xc goes up, but XL goes down
Both X’s go down.
Recall V=iX… if we raise the frequency, we can
make the amplitude of the voltage across the
inductor = amplitude of the voltage across the cap.
At the special frequency where the
the reactance of the inductor =
reactance of the capacitor:
A] the current will be zero
B] the current will be infinite, for any
finite applied voltage
Recall V=iX… if we raise the frequency, we can
make the amplitude of the voltage across the
inductor = amplitude of the voltage across the cap.
This is called resonance.
Real circuits must have a little
resistance, so the current,
though large, remains finite.
But at the resonance frequency, a
very small driving voltage gives
a very large current.
The directions of the voltage drop in
the cap and in the inductor are
opposite.
The directions of the current in the
cap and the inductor are the
same. (in series.)
Power in AC
Phi is the
phase
angle
between
current
and
voltage.
Electromagnetic “Discontinuities” Must Propagate at a speed
c

1
00
Our wavefront
satisfies both
“Gauss’s laws”
because there is no
enclosed charge or
current, and fields
on opposite sides of
the box are the
same.
There is a changing B flux
as the wavefront moves
by. This changing flux
must be equal to the line
integral of the E field. Only
the back edge (gh)
contributes to this line
integral.
There is a changing E flux
also. This gives another
reqd relation between E
and B.
Accelerating Charges Radiate
http://www.its.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html#
Accelerating Charges Radiate
Coulomb’s law can’t describe the “kinked” E field.
We got it from connecting field lines (Gauss’ law!) + geometry.
So, while Gauss “derived” his law from Coulomb, Gauss’ Law is better.
It’s always true, while Coulomb’s law is only true for unaccelerated charges.