B1.2 R-C and L-C Circuits
The second ‘linked’ experiment considers the behaviour of capacitors and inductors in place of – or in
combination with – the resistors of last week’s circuits. The inclusion of capacitors and inductors means
that the voltages around and currents within a circuit will in general depend upon time: the circuit will
respond to any transient in the applied voltages or signals. Our experimental investigations will
therefore use signal generators and oscilloscopes.
We shall see that the circuit voltages and currents eventually settle down to steady-state values, but
only after a period of the order of the time constant for the circuit, which is typically given by the
product RC of a resistance and capacitance, or the ratio L/R of an inductance and a resistance.
B1.2.1 Reading
Read about capacitors, inductors and a.c. circuits in, e.g., Univ of Southampton’s University Physics vol
1, pp 282—290, 324—328, 372—379 and 387—390 (from R. Wolfson, Essential University Physics vol 2,
pp 384—392; 426—430, 474—481 and 489—492). You may also like to read about Gauss’s, Ampère’s
and Faraday’s laws (University Physics vol 1, pp 249—250, 349—355, 362--366; R. Wolfson, Essential
University Physics vol 2, pp 351—352, 451—457, 464—468).
B1.2.2 Capacitors
Although most capacitors that you will meet will be sealed to form a rigid device, at heart all capacitors
are formed of pairs of metal plates that are separated by vacuum or by a dielectric medium that may be
air, oil or a solid such as polyester, polystyrene or a ceramic. The construction is most obvious in tuning
capacitors, such as those shown in Figure 1 below, whereby one plate or set of plates rotates relative to
another so as to vary the overlapping area. Fixed-value devices may retain this planar geometry, or be
rolled into cylinders for compactness and rigidity.
Figure 1 Air-spaced variable capacitors (left, centre) and a dielectric-spaced trimming capacitor (right).
[Images: www.g3npf.co.uk; www.electronics-tutorials.ws]
If one plate (or set of plates) is charged with respect to the other, an electric field will exist between
them, and therefore a potential difference will occur. The capacitance is defined as the constant of
proportionality relating the potential difference V to the charge Q stored, assuming a linear relationship,
= (1)
Our circuit analysis determines the voltages and currents, however, so we must refer to the rate of
change of the charge, or the integral of the current,
= =
d
d
d = V
(2)
(3)
We may use either equation (2) or (3) in our circuit analysis, where for resistors we used Ohm’s law.
B1.1
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Electric & Electronic Circuits
B1.2.3 Inductors
A friend who’s in liquor production
Owns a still of astounding construction.
The alcohol boils
Through old magnet coils;
She says that it’s “proof by induction.”
This maybe needs some explanation,
For it refers to experimentation:
What we call an inductor
Is a coiled conductor,
Here hollow, for refrigeration.
The construction of inductors, unlike capacitors, is usually quite clearly apparent: they are formed by
winding conducting wire around a cylinder or ring to form a coil of toroid that efficiently confines the
magnetic field that is created when a current flows along the wire. Some typical examples are shown in
Figure 2 below.
Figure 2 Inductors: toroidal (left), linear (centre), high inductance/low current and low inductance/high current (right).
[Images: MPS Industries; Ali Express, Windell Oskay]
When the current through an inductor changes, the resulting change in magnetic field causes a voltage
to be induced in the coil. The inductance is defined as the constant of proportionality relating the
induced voltage V to the rate of change of the current I,
= or, equivalently,
d
d
d
(4)
I
(5)
We may use either equation (4) or equation (5), as convenient, where in our circuit analysis for resistors
we used Ohm’s law.
Note that the induced voltage acts in the direction that allows it to oppose the change of current. If, for
example, we connect an inductor through a resistor to a battery, then the initial increase in current will
produce a voltage that reduces the voltage across the resistor, and hence the current through it. As
shown in Figure 3 below, the voltages across resistors, capacitors and inductors are in the same
direction if the current, integrated current and rate of change of current are also all in the same
direction.
Figure 3 Relationships between currents I and voltages V for resistors, capacitors and inductors.
B1.2
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Electric & Electronic Circuits
B1.2.4 The transient response of RC and RL circuits
The analysis of circuits that contain combinations of resistors with capacitors or inductors follows the
same general principles as for networks of resistors alone: we use Ohm’s law and equations (2), (3), (4),
or (5) to define the relationships between the voltages across and currents through individual
components, and Kirchhoff’s laws to account for the way the components are connected in the circuit.
Such analysis is important not just in electronics, where such networks form parts of almost all devices
in the laboratory and home, but more widely in physics where directly analogous techniques are useful
when modelling, for example, the radioactive decay of nuclei or the dynamics of a pulsed laser.
Figure shows a basic combination of a resistor with a capacitor, where a switch connected to a battery
or ground (0 V) allows the creation of voltage pulses at the input to the network.
Figure 4 Simple RC circuit – a single-pole low-pass filter.
Our analysis proceeds as follows. First, we write the relations between the individual voltages and
currents, taking VR to be the voltage across the resistor and IR and IC to be the currents through the
resistor and capacitor respectively.
d
R
=
out
(6)
1
(7)
d
(In principle, we might also have an expression for Vin in terms of the battery emf but, as we need also to
take into account the switch, we shall for the time being leave things in terms of Vin itself.) We then
apply Kirchhoff’s laws, assuming that negligible current flows out of the output terminal, giving
out
R
C
0
in
(8)
R
(9)
R
(10)
We therefore have five equations in 9 unknowns (Vin, VR, Vout, IR, IC, I, R, C, t), where we hope to derive
an expression for Vout as a function of Vin, R, C and t. Combining equations (6) to (10) to eliminate VR, IR,
IC and I, we obtain
d
1
out
d
in
(11)
out
This first-order differential equation fully describes the general behaviour of the circuit; to obtain a
specific solution, we must first know the specific way in which the input voltage Vin varies with time.
For the situation illustrated, we may assume the input voltage to be either 0 V or the battery emf E, and
that it will be steady while the switch remains in a given position. During such periods, we may therefore
treat Vin as a constant. To solve equation (11) most neatly, we note that under such conditions,
d
d
R
d
d
out
in
d
out
d
(12)
B1.3
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Electric & Electronic Circuits
and hence, from equation (11), that
d R
1
R
d
This has well-known solutions (which you may check by substitution)
R
exp
R
!
(13)
"
(14)
and hence
out
#
in
in $exp
out
!
"
(15)
If, for example, we move the switch from B to A at time t = 0, when the output voltage Vout = Vout(0), the
output voltage at subsequent times while the switch remains in the same position will be
#
%
out
out
0
E$exp
!
"
(16)
The effect of any transient – caused by moving the switch – hence dies away with a time constant of
τ = RC, and the output voltage eventually reaches a steady state value equal to the input voltage. Note
that we may find the charge Q(t) on the capacitor at any time by applying equation (1) to give
out
(17)
If the switch is later moved back to position B, the output voltage afterwards will be given by applying
equation (15) again with new values of Vin, t0 and Vout(t0). Figure 5 below shows a typical result.
0
1
2
3
Figure 5 Output voltage (solid line) from the RC circuit of Figure 4 when the input voltage (dashed line) corresponds to connecting the
input to the battery between t = 0 and t = 1 s. The capacitor is here taken to be initially uncharged, and the time constant here is 1 s.
B1.2.5 The ac response of RC and RL circuits
Another situation of interest is when the input to a circuit is sinusoidal, of the form
' cos
in
*
(18)
For a linear differential equation such as equation (11), it turns out that the solution will then always be
of the form
cos *
out
+
(19)
where V0 and φ depend upon the details of the circuit and remain to be found. Substituting equations
(18) and (19) into equation (11), we hence obtain
*
B1.4
sin *
+
1
#
' cos
*
cos *
+ $
(20)
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Electric & Electronic Circuits
This may be solved by using the double-angle formula cos , − - ≡ cos , cos - + sin , sin -, but it is
tidier first to make the substitution * / ≡ * + +. After application of the double-angle formula,
equation (20) then becomes
−*
1
sin * ′ =
1 ' #cos * ′ cos + + sin * ′ sin + $ −
cos * ′ 2
(21)
We now collect together the cosine terms and, separately, the sine terms, giving
3*
+
'
sin + 4 sin *
/
= 3
'
−
cos + 4 cos * ′ (22)
This may only be solved for all values of t’ if the amplitudes of the sine and cosine terms are both zero,
i.e.,
'
'
sin + = −*
(23)
cos + = (24)
Taking the ratio of these equations gives the phase shift of the output waveform relative to the input
tan + = −*
(25)
while summing their squares gives the output waveform’s relative amplitude
!
'
6
6
" =
3*6 +
1
64
(26)
Combining these results into equation (19) gives the overall output waveform
out
=
71 + *
6
cos#* − tan8' *
$
(27)
At low frequencies * ≪ 1⁄
, the output voltage follows the input; at high frequencies, however,
the relative output amplitude falls by a factor of *
, and lags the input waveform by a quarter of a
cycle.
phase /deg
The theoretical relative amplitude and phase are shown in Figure 6 below, for a circuit with
RC = 0.001/(2π). Note that at a frequency of 1 kHz, when ωRC = 1, the output lags behind the input
waveform by 45° and is half the amplitude of the input (-6 dB).
V0/V1
frequency /Hz
frequency /Hz
Figure 6 Bode (pron. Bodey) plot of the amplitude and phase of the output for the RC circuit of Figure 4, measured relative to the input,
for RC = 0.001/(2π).
B1.5
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Electric & Electronic Circuits
B1.2.6 Preliminary exercises
1 By modifying the analysis of section B1.2.4 above, determine the first-order differential equation that
describes the variation with time of the output voltage Vout of the circuit shown in Figure 7 below.
2 Solve this differential equation for periods of constant Vin to obtain a solution similar to that of
equation (16).
3 Find the charge Q(t) on the capacitor in the circuit above, and the current I(t), at time t.
4 Taking the capacitor to be initially discharged when, at time t = 0, the switch is moved from B to A,
determine the output voltage and charge on the capacitor at subsequent times, if the switch is
moved back from A to B at a time t = T.
5 Sketch your results, over the interval 0 ≤ t ≤ 4T, for the cases (a) T << τ, (b) T = τ and (c) T >> τ, taking
τ = RC.
6 Suggest where, in everyday or laboratory electronic devices, you might find something similar to this
RC circuit, and the purpose that it serves.
Figure 7 Alternative arrangement of capacitor and resistor – a single-pole high-pass filter.
7 Repeat parts 1—6 for the inductor-resistor circuit shown in Figure 8 below, taking for the final part
τ = L/R.
Figure 8 A single-pole low-pass filter formed from an inductor and a resistor.
8 Determine how the natural logarithm of the current, ln I, will vary with time after the switch is
moved from B to A.
B1.6
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Electric & Electronic Circuits
B1.2.7 Laboratory: Transient response of an RC circuit
We may investigate the accuracy of our theoretical analysis by constructing the circuit of Figure 4 on a
breadboard and noting the voltage VR across the resistor as a function of time. For precise voltage
measurements we shall use a multimeter, and to allow time to read it we shall require a time constant
of many seconds – and hence a large value of RC. Unless we use a large capacitor, this means that we
must use a large resistance; the internal resistance of the multimeter (around 10 MΩ) is a convenient
way of achieving this.
Select a 4.7 µF capacitor and measure its value with a hand-held capacitance meter. Make sure
that you do not use an electrolytic capacitor, as these suffer from relatively high leakage currents
because of defects in the thin oxide layer that separates the conductors. Electrolytic capacitors,
which must be connected so that the voltage is always of the right polarity, are normally
cylindrical, with a plastic sleeve over aluminium casing and markings to identify the negative
connection.
Build the circuit shown in Figure 4 on a breadboard. Connect two of the 4 mm terminal posts to
the 0 V and +15 V outputs of a lab power supply, and wire the terminal posts into two of the
breadboard’s bus rails. Use the multimeter set to the 20 Vdc scale to measure the voltage
supplied.
Next, plug the capacitor into the breadboard, wire one side of the capacitor to the 0 V bus rail,
and connect the other side to a third 4 mm terminal post. To this, connect the multimeter, which
will serve as the resistor. Connect a lead from the other side of the multimeter: you can use this
as the pole of the switch by plugging it into the 0 V or +15 V terminals.
Connect a further lead across the capacitor via the 0 V and multimeter/capacitor terminals to
discharge the capacitor.
Prepare to measure the voltage across the (multimeter) resistance using a watch with a resolution
of 1 s or better; position it next to the multimeter so that you can see both.
Use your ‘switch’ to connect the ‘resistor’ to the +15 V supply.
With the supply connected, current should begin to flow through the resistance (multimeter), and the
multimeter should show the voltage across it. As long as the capacitor is shorted out, however, the
voltage across it and charge stored should be zero.
Remove disconnect the discharge lead to start the experiment, and record the multimeter voltage
every 5 s for 90 s as the capacitor charges up.
Discharge the capacitor by reconnecting the lead across it, and repeat the experiment to check
the reliability of your readings. Repeat the experiment a few times until you are confident of the
reliability of your measurements, and you have enough data to allow the experimental
uncertainties to be determined.
You may now apply the usual data analysis techniques to your measurements.
Plot the voltage VR, and hence the capacitor voltage VC = (E-VR), against time. Calculate the
uncertainties in your measurements, and show these as error bars on your graph.
Next plot the natural logarithm of the multimeter voltage, ln Vr as a function of time. Again plot
your uncertainties – noting that they will now be uncertainties in ln Vr rather than in Vr.
Determine whether your measured data are consistent with your theoretical predictions and, if
you can, deduce a value for the time constant RC.
Comment upon whether this is consistent with your measured value of the capacitance and
assumed value of the internal resistance of the multimeter.
B1.7
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Electric & Electronic Circuits
B1.2.8 Laboratory: Response of RC and LC circuits to periodic pulses
While the multimeter and stopwatch allow quite precise measurement of the waveform voltages for
signals that vary slowly, many signals of interest change rather more quickly and most practical circuits
are designed for rather higher frequencies. In such cases, we may use a signal generator to produce the
input waveforms, and an oscilloscope to measure the circuit voltages. The pulsed signals that will be
examined here are typical of those in a range of control applications, and will also be encountered in
computers and other digital electronics, and in the outputs of particle detectors and the like.
Use your breadboard to construct the circuit of Figure 7 using components with values R = 1—
10 kΩ and C = 10—100 nF. In place of the power supply and switch, connect the input and ground
to the ‘TTL/CMOS’ output and ground of a signal generator, and arrange for the input waveform
to be displayed as Channel 1 on your oscilloscope. Various leads and BNC T-pieces are available.
Connect an oscilloscope probe between the circuit output and ground, and arrange for the signal
to be displayed as Channel 2 on your oscilloscope. Set both channels of the oscilloscope to
operate in the ‘dc coupled’ mode, and adjust the triggering to achieve a stable trace.
Adjust the signal generator to give rectangular pulses with an output frequency of 1 kHz and unity
‘mark-space ratio’ – i.e. equal times spent at high and low voltages.
Sketch the waveforms displayed on the oscilloscope, noting key parameters such as the period
and voltage levels, as well as any unusual features, and estimating the uncertainties in your
experimental measurements.
Measure the vertical shift in each displayed waveform when the corresponding channel is
changed from dc coupling to ac coupling.
Repeat your observations over the range of frequencies of which your signal generator is capable.
Comment upon whether your observations are consistent with your theoretical analysis.
Now investigate a simple inductor-resistor circuit.
Modify your circuit to resemble that of Figure 8, using components with values L = 1—10 mH and
a R = 1—10 kΩ.
Repeat the experimental procedure described above.
Comment upon whether your observations are consistent with your theoretical analysis.
B1.2.9 Laboratory: Response of an RC circuit to sinusoidal signals
Audio signals, as well as a wide range of radio and radar waveforms, are often to a good approximation
sinusoidal, and the circuits that we have considered are common for the control of, for example, the
tone/bass/treble of an audio system.
Reassemble the RC circuit that you explored in section B1.2.8 above.
Set the signal generator to produce a sinusoidal output, and connect your circuit to the 50 Ω
output rather than the ‘TTL/CMOS’ output. Adjust the oscilloscope triggering if necessary to
obtain a stable trace.
Measure the amplitudes and relative phase of the input and output waveforms at a range of
frequencies, and calculate the ratio of the output amplitude to the input amplitude.
Plot how the relative phase and relative amplitude vary with frequency, using a linear scale for
the phase and logarithmic scales for the relative amplitude and frequency axes.
Comment upon whether your observations are consistent with your theoretical analysis.
With sinusoidal waveforms, it can be instructive to plot the output voltage against the input voltage.
Change the oscilloscope display format to the ‘X-Y’ mode, so that it shows a plot of Channel 2
against Channel 1.
Observe and note, with sketches, how the trace varies with frequency, and comment on your
findings.
B1.8