PHY 111
Class #31 , 3/30/16
Spring 2016
1. Inductors in series and parallel
As with resistors and capacitors, you can derive simple relationships that show how inductors combine in series and
parallel.
Series
Consider 2 inductors in series with a voltage source. Using the fact that the current
is the same in each inductor in series, and that the voltage across the inductors adds
up to the total voltage of the source, we showed that the equivalent inductance is
Leq = L1 + L2
Parallel
Consider 2 inductors in parallel with a voltage source. Using the fact that the voltage
is the same across each inductor in series, and that the current through each inductor adds
up to the total current drawn from the source, we showed that the equivalent inductance is
1
Leq
=
1
L1
+
1
,
L1
or Leq =
L1 L2
L1 +L2
.
2. Mutual Inductance
We revisited the “double solenoid example, done in an earlier class, in which two sets of wires are wound on the same
cylindrical core, one with N1 windings and the other with N2 . We had a voltage source V1 in circuit 1 and a resistor R2
connected to the #2 set of windings. If the voltage source has some time dependence, e.g. we turn it on by closing a
switch, then a time dependent B1 field is produced in the solenoid. By Faraday’s Law, a voltage V2 is induced in the
second windings:
V2 = -−N2
ⅆΦ
ⅆt
= -−N2  μ0 Nd 1 A 
ⅆI1
ⅆt
where A is the cross-sectional area and d is the length of the solenoid (d assumed large compared with the radius).
Thus, we have related the voltage in the second circuit to the changing current in the first. Defining a new quantity M,
the mutual inductance, by
V2 = -−M21
ⅆI1
,
ⅆt
we can read M21 off the first equation: M21 =
μ0 N1 N2 A
.
d
Flipping the problem around and asking how the voltage in
circuit #1 depends on the changing current in circuit 2, you would calculate the mutual inductance M12 from
V1 = -−M12 ⅆIⅆt2 and you’d discover that M12 = M21 . This is common enough that we will assume this symmetry and
just write a single M for both situations. We can then write
M=
-−Vi
ⅆIj /∕ⅆt
where M describes how changing flux in the jth circuit induces a voltage in the ith , for any set of circuits.
PHY 111 Spring 2016
3. An L-C circuit
R. Martin
circuit #1 depends on the changing current in circuit 2, you would calculate the mutual inductance M12 from
V1 = -−M12 ⅆIⅆt2 and you’d discover that M12 = M21 . This is common enough that we will assume this symmetry and
just write a single M for both situations. We can then write
M=
2
-−Vi
ⅆIj /∕ⅆt
where M describes how changing flux in the jth circuit induces a voltage in the ith , for any set of circuits.
3. An L-C circuit
Putting a charged capacitor C in series with an inductor L, gives the Kirchoff loop equation
-− Q
-− L
C
ⅆ2 Q
ⅆt2
ⅆI
ⅆt
= 0, or, using I = ⅆ Q/∕ⅆ t,
+  L1C  Q = 0
If you memorize any physics equations, this might be one to consider. You’ve already seen it once when you studied
the mass on a spring, where instead of charge the variable was the distance from the equilibrium position of the mass.
No matter what the relevant variable is, this equation tells you that it undergoes harmonic oscillation with period equal
to coefficient of the linear term. In other words, for any physical variable z(t), if
ⅆ2 z
ⅆt2
+ ω20 z = 0,
then the variable z undergoes harmonic oscillations at period ω0 ! In our case the charge oscillates at the frequency
given by the square root of the coefficient of the ‘Q’ term: ωLC = 1 L C . We can even write down the equation
for it, assuming, Q(t = 0) = Qmax and I(t = 0) = 0:
Q (t) = Q max cos (ωLC t) , with ωLC = 1
LC
4. Time scales
When physicists and engineers approach a new problem, they often ask the question “what are the relevant time
scales?” By this we mean, those characteristic time quantities that govern the behavior of the system. For the LC
circuit, there is one obvious time scale: the oscillation period τLC = 2 π/∕ωLC . We looked at the LR circuit earlier and
saw that it obeyed an exponential behavior with terms like ~∼ ⅇ-−t/∕τRL , where τRL is the RL circuit time constant. The
RC circuit had similar time dependence like ~∼ ⅇ-−t/∕τRC , with the RC time constant. These are summarized below.
We’ll be seeing more time scales as we study more complicated situations, if you keep your eye out for them, you can
quickly learn the time behavior of those systems.
Summary of time scales to-date:
τRL =
τRC =
τLC =
L
R
1
RC
2π
ωLC
RL exponential time constant
RC exponential time constant
=2π
L C LC oscillation period
Recall that the exponential time constant is the time it takes for the exponential function to get to 1/ⅇ of it initial value.
Hence we usually say “after a few time constants”, the exponential is “essentially zero”. So a charged capacitor
discharges “nearly completely” in a time equal to a few times τRC . Similarly, if you close a switch in an RL circuit,
the current starts at zero and increases like 1 -− ⅇ-−t/∕τRL , so after t ~∼ "a few" ⨯ τRL , the exponential is essentially zero
and the current is “essentially at it’s final value”. The LC period is simply the period of oscillation for the L circuit.
PHY 111 Spring 2016
R. Martin
What happens if you combine all 3 elements R, L, and C in series? Which time scales are important? That depends on
the actual values of R, L, and C, and you’ll begin to build some intuition for it in the final simulation lab, CL-4.
τLC =
2π
ωLC
=2π
L C LC oscillation period
Recall that the exponential time constant is the time it takes for the exponential function to get to 1/ⅇ of it initial value.
Hence we usually say “after a few time constants”, the exponential is “essentially zero”. So a charged capacitor
discharges “nearly completely” in a time equal to a few times τRC . Similarly, if you close a switch in an RL circuit,
the current starts at zero and increases like 1 -− ⅇ-−t/∕τRL , so after t ~∼ "a few" ⨯ τRL , the exponential is essentially zero
3
and the current is “essentially at it’s final value”. The LC period is simply the period of oscillation for the L circuit.
What happens if you combine all 3 elements R, L, and C in series? Which time scales are important? That depends on
the actual values of R, L, and C, and you’ll begin to build some intuition for it in the final simulation lab, CL-4.
PHY 111 Spring 2016
R. Martin