MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Experiment 4: RL Circuits and Undriven RLC Circuits
OBJECTIVES
1. To explore the time dependent behavior of RC and RL Circuits
2. To understand how to measure the time constant of such circuits
3. To explore the time dependent behavior of Undriven RLC Circuits
PRE-LAB READING
INTRODUCTION
In the first two parts of this lab we w ill continue our investigation of DC circuits, now including, along
with our “battery” an d resisto rs, inductors (RL circuits). W e will m easure the very different
relationship between current and volt age in an inductor, and study the time dependent behavior of RL
circuits.
In the second three parts of the lab we will stud y a circuit that includes a “battery”, resistor, capacitor
and inductor (undriven RLC circuits).
As most children know, if you get a push on a swi ng and just sit still on it, you will go back and forth,
gradually slowing down to a stop. If , on the othe r hand, you m ove your body back and forth you can
drive the sw ing, making it swing higher and higher. Th is only works if you m ove at the correct rate
though – too fast or too slow and the swing will do nothing.
This is an exam ple of r esonance in a m echanical system. In the secon d two parts of this lab we will
explore its electrical analog – th e RLC (resistor, inducto r, capacitor) circuit – and better understand
what happens when it is undriven. In the next lab we will consider what happens when it is driven
above, below and at the resonant frequency.
The Details : Inductors
Inductors store energy in the for m of an internal magnetic field, and find their behavior dom inated by
Faraday’s Law. In any circuit in whic h they are placed they create an E MF  proportional to th e time
rate of change of current I through them :  = L dI/dt. The constant of proportionality L is the
inductance (measured in Henries = Ohm s), and determines how strongly the inductor reacts to curren t
changes (and how large a self energy it contains fo r a given current). Typical circuit inductors range
from nanohenries to hundreds of millihenries. The direction of the induced EMF can be determ ined by
Lenz’s Law: it will always oppose the change (inductors try to keep the current constant)
RL Circuits
Consider the circuit shown in Fi gure 1. The inductor is connected to
emf . At t = 0, the switch S is closed.
a voltage source of constant
Figure 1 RL circuit. For t<0 the switch S is open and no current flows
in the c ircuit. At t= 0 th e switch is closed and current I can begin to
flow, as indicated by the arrow.
.
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As we saw in clas s, before the switch is closed there is no curren t in th e circuit. When the switch is
closed the inductor wants to keep the same current as an instant ago – none. Thus it will set up an EMF
that opposes the current flow. At firs t the EMF is identical to that of the battery (but in the opposite
direction) and no current will flow. Then, as tim e passes, the inductor will gradually relent and cu rrent
will begin to flow. After a l ong time a constant current (I = V/R) will flow through the inductor, and it
will be content (no cha nging cur rent m eans no changing B f ield m eans no chang ing m agnetic f lux
means no EMF). The resulting EMF and current are pictured in Fig. 2.
(a)
If=/R
VResistor,f=
(b)
Inductor
I, VResistor
0 = 
Time
Time
Figure 2 (a) “EMF generated by the inductor” decreases with time (this is what a voltmeter hooked in
parallel with the inductor would show) (b) the current and hence the voltage across the resistor increase
with time, as the inductor ‘relaxes.’
After the inductor is “fully charged,” with the current essentially constant, we can shut off the battery
(replace it with a wire). Without an inductor in the circuit the current would instantly drop to zero, but
the inductor does not want this rapid change, and hence generates an EMF that will, for a moment, keep
the current exactly the same as it was before the battery was shut off. In this case, the EMF generated by
the inductor and voltage across the resistor are equal, and hence EMF, voltage and current all do the
same thing, decreasing exponentially with time as pictured in Fig. 3.
(b)
VR,0=L,0=; I0=/R
Inductor, VR, I
(a)
V0/e =
0.368 V0
t=
Time
Figure 3 Once (a) the battery is turned off, the EMF induced by the inductor and hence the voltage
across the resistor and current in the circuit all (b) decay exponentially.
The Details: Non-Ideal Inductors
So far we have always assum ed that circuit elem ents are ideal, for example, that inductors only have
inductance and not capacitance or re sistance. This is g enerally a decen t assumption, but in reality no
circuit element is truly ideal, and to day we will need to cons ider this. In particular, today’s “inductor”
has both inductance and resistance (real inductor = ideal inductor in series with resi stor). Although
there is no way to physically separate the inductor from the resistor in this circuit element, with a little
thought you will be able to measure both the resistance and inductance.
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The Details: Measuring the Time Constant 
In this lab y ou will be faced with an exponentially decay ing current I = I0 exp(-t/) from whic h you
will want to extract the tim e constant . We will do this in two dif ferent ways, using the “two-point
method” or the “logarithmic method,” depicted in Fig. 4.
(b)
(a)
Current
ln(Current)
(t1, I1)
(t2, I2)
Time
Time
Figure 4 The (a) two-point and (b) logarithmic methods for measuring time constants
In the two-point m ethod (Fig. 4a) w e choose tw o points on the curve (t 1,I1) and (t 2, I 2). Because the
current obeys an exponential decay, I = I0 exp(-t/), we can extract th e time constant  most easily by
picking I 2 such that I 2 = I 1/e. We should, in theory, be able to find this f or any t 1, as long as we don’t
switch the battery off (or on) before enough tim e has passed. In practice the current will eventually get
low enough that we won’t be able to accurately measure it. Having made this selection,  = t2 – t1.
In the log arithmic method (Fig. 4b) we f it a line to the natural log of the current plotted vs. tim e and
obtain the slope m, which will give us the time constant as follows:
I t 
1
rise ln I t2  ln I t1


ln  2 
m
run
t2  t1
t2  t1  I t1 
 I et2  
1
1   t2  t1 
1
1
 t t 

ln  0 t   
ln e  2 1  





t2  t1  I 0 e 1  t2  t1
t2  t1 

That is, from the slope (which you get from fitting a line) you can obtain the time constant:  = -1/m.
     






In using both of these methods you must take care to use points well into the decay (i.e. not on th e flat
part before the decay begins) and try to avoid tim es where the curren t has fallen clo se to zero, which
are typically dominated by noise.
(b)
X0
-X0
0T
1T
Time (in Periods)
Figure 5 Oscillating Functions
2T
Amplitude
(a)
Amplitude
The Details: Oscillations
In this lab y ou will be investiga ting current and voltages (EMFs) in RLC circuits. T hese oscillate as a
function of time, either continuously (Fig. 5a) or in a decaying fashion (Fig. 5b).
X0
-X0
0T
1T
2T
3T
4T
Time (in Periods)
. (a) A purely oscillating function
5T


x  x0 sin  t   has fixed
amplitude x0, angular frequency  (period T = 2/ and frequency f = /2), and phase  (in this case
 = -0.2 ). (b) The amplitude of a dam ped oscillat ing function decays exponentially (am plitude
envelope indicated by dotted lines)
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Undriven Circuits: Thinking about Oscillations
Consider the RLC circu it of Fig. 6 belo w. The capacitor h as an initial charge Q 0 (it was charged by a
battery no longer in th e circuit), but it can’t go anywhere because the switch is open. When the switch
is closed,
op plate of the capacitor, through the resistor and
(a) the positive charge will flow off the
(b)t
inductor, and on to the bottom plate of the capacitor. This is the sa me behavior that we saw in RC
circuits. In those circuits, however, the current flow stops as soon as all the positive c harge has flowed
to the n egatively ch arged plate, leaving both p lates with zero charge. The additio n of an ind uctor,
however, in troduces ine rtia into the circu it, kee ping the cu rrent flowing even when the capacitor is
completely discharged, and forcing it to charge in the opposite polarity (Fig 6b).
Figure 6 Undriven RLC circuit. (a) For t< 0 the switc h S is open and althou gh the capa citor is
charged (Q = Q0) no current flows in the circuit. (b) A half period after closing the switch the capacitor
again comes to a maximum charge, this time with the positive charge on the lower plate.
This oscillation of positive charge from the upper to lower plate of the capacitor is only one of the
oscillations occurring in the circuit. For the two tim es pictured above ( t=0 and t=0.5 T) the charge on
the capac itor is a m aximum and no current f lows in the circu it. At interm ediate tim es current is
flowing, and, for example, at t = 0. 25 T the current is a m aximum and the charge o n the capacitor is
zero. Thus another os cillation in th e circu it is be tween ch arge on the capacitor and curren t in the
circuit. This corresponds to yet another oscilla tion in the circuit, that of energy betw een the capacito r
and the ind uctor. W hen the capacitor is fully char ged an d the current is zero, the capacitor stores
energy but the inductor doesn’t ( U C  Q 2 2C; U L  12 LI 2  0 ). A quarter period later the current I is a
maximum, c harge Q = 0, and all the energy is in the inductor:
U C  Q 2 2C  0; U L  12 LI 2 . If there
were no resistance in the circuit this swapping of energy between the capacito r and inductor would be
perfect and the cu rrent (and voltag e acro ss the capacitor and EMF in duced by the inductor) would
oscillate as in Fig. 5a. A resistor, however, da mps the circuit, rem oving energy by dissipating power
through Joule heating (P=I2R), and eventually ringing the current down to zero, as in Fig. 5b. Not e that
only the resistor dissipates power. The capacitor and inductor both sto re energy during half the cycle
and then completely release it during the other half.
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