B1.1 Kirchhoff’s and Ohm’s Laws
This first ‘linked’ experiment examines networks of resistors, and the currents that flow in them and
voltages that occur when they are connected to batteries or d.c. power supplies. If the components are
well described by Ohm’s Law, and the networks obey Kirchhoff’s Laws, then experimental
measurements should be consistent with the predictions of these theories.
B1.1.1 Analysis of circuit networks
The theoretical analysis of the behaviour of a network of resistors, batteries and other components is
based upon two sets of physical laws or assumptions:
1. specific relationships, such as Ohm’s Law, between the current through and voltage across individual
components
2. general principles of the conservation of charge and conservation (or single-valued nature) of energy,
known as Kirchhoff’s Laws, that
• the sum of the potential differences across components forming a loop will be zero, and
• the sum of the currents flowing into and out of the node of a circuit is zero.
The expression of these principles to the components and connections of an electrical circuit will
produce a set of simultaneous equations that we may solve to give the currents and voltages around
the circuit. While solution of the simultaneous equations may sometimes be messy – particularly when
the circuit contains components such as transistors – it is straightforward in principle, and can easily be
performed automatically by a computer. Traditionally, there are two equivalent ways of arriving at the
set of simultaneous equations:
a. the identification of current loops, characterized by a single current, whereby the current flowing
through a component that shares two or more loops is the sum of the currents in those loops
b. the assignment of a separate current to each component (or part of a component), and the
application of Kirchhoff’s current law at each connection between components.
Generally, the first approach is adopted by electronic technicians as it produces fewer initial equations,
whereas the second is preferred by physicists as it avoids the confusion of having to identify the loops.
V2
V1
I4
V4
E1
R1
I1
I6
I1 I
3
I2
R2
R3
V3
I2
I5
E2
V5
I7
Figure 1 Network of three resistors R1, R2 and R3 and two batteries with EMFs E1 and E2.
We illustrate our analysis by considering the circuit shown in Figure 1 above. In addition to the resistors
and batteries forming the circuit, we have shown the voltages V1—V5 across the five components, and
the currents I1—I7 at various places in the circuit.
We shall eventually take the specific case E1 = 15 V, E2 = 5 V, R1 = 20 kΩ, R2 = 10 kΩ, R3 = 20 kΩ but, as
usual, we shall first solve the problem algebraically and only substitute these values as the final step.
B1.1
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Electric & Electronic Circuits
B1.1.2 Loop analysis of circuit networks
For our loop analysis, we identify the two current loops I1 and I2 shown in grey. I1 alone passes through
R1 and the battery E1, and I2 alone passes through R2 and battery E2 both currents pass through R3, but
they do so in opposite directions. The voltages across each resistor are in the same direction as the
current flow, and are given in each case by Ohm’s law.
=
(1)
=
(2)
=
−
(3)
For each loop, we now calculate the sum of the voltages, which by Kirchhoff’s voltage law must be zero.
It is important to note the directions of the potential differences.
+
−
=0
(4)
+
−
=0
(5)
Substituting for V1 – V3 from equations (1) to (3), setting V4 and V5 by definition to E1 and E2, we hence
obtain
+
−
+
−
−
−
=0
(6)
=0
(7)
which may be rearranged to give
+
=
+
+
+
(8)
=
(9)
Rearranging equation (9) to give I2, we find
−
=
(10)
+
and substituting this into equation (8) now gives
=
+
−
+
(11)
+
so that, collecting terms in I1,
=
+
+
−
(12)
+
I1 and the voltages V1—V3 may now be found via equations (1) —(3) and (10).
Substituting the specific component values given,
I2 = (1/8,000) A = 0.13 mA, V1 = 8.8 V, V2 = 1.3 V, V3 = 6.3 V.
we
obtain
I1 = (7/16,000) A = 0.44 mA,
Note that, instead of one of the current loops I1 or I2, we could have taken a current loop around the
perimeter (through E1, R1, R2 and E2). Any two of these three loops would have worked just as well.
B1.2
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Electric & Electronic Circuits
B1.1.3 Branch current analysis of circuit networks
With complex circuits, it may be difficult to identify the appropriate set of independent current loops to
solve the problem, but we arrive at the same initial equations if we begin by assigning separate currents
to each of the components. Considering each component separately, we obtain
=
(13)
=
(14)
=
(15)
=
+ 0
=−
(16)
+ 0
(17)
(since the voltages the batteries are assumed to be independent of the currents through them) and,
from Kirchhoff’s laws, we find
+
−
=0
(18)
+
−
=0
(19)
=
−
(20)
=
(21)
=
(22)
Solution then proceeds as above.
B1.1.4 Preliminary exercises
The following questions will check your understanding of basic concepts in d.c. circuit theory. Write your
answers in your logbook. If you have trouble with the questions, read the revision notes in the appendix.
1.1
Write down Kirchhoff’s Laws. Use them to find an expression for the current flowing in the circuit
shown in Figure 2 below, assuming E1, E2 and R to be known.
I
R
E1
E2
Figure 2 Two batteries, with EMFs E1 and E2, are connected to a resistor of resistance R.
1.2
Calculate the effective resistance of the network of resistors shown in Figure 3 (a) below by using
the rules for series and parallel resistors – i.e., find the single resistor R, as in (b), that has the
same resistance between the terminals X and Y. (It is usually helpful to work back from the
furthest point from the terminals, successively reducing the number of resistors by applying the
rules for series and parallel combinations.)
B1.3
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Electric & Electronic Circuits
X
X
R1
R2
R
R3
R4
Y
Y
(b)
(a)
Figure 3 Two batteries, with EMFs E1 and E2, are connected to a resistor of resistance R.
1.3
Calculate the potential difference across resistor R4 in the circuit shown in Figure 4 below. Take
the component values to be R1 = 10 kΩ, R2 = 20 kΩ, R3 = 15 kΩ, R4 = 5 kΩ, E1 = 10 V.
R1
R3
R2
R4
E1
Figure 4 Network of four resistors R1, R2, R3 and R4, connected to a battery of EMF E1.
1.4
1.5
If a car battery has an EMF of 12 V and delivers a power of 1 kW into a resistance of 0.1 Ω, what is
the internal resistance of the battery?
Find the current through the resistor R3 in the circuit shown in Figure 5 below. Take the
component values to be R1 = 10 kΩ, R2 = 10 kΩ, R3 = 20 kΩ, E1 = 10 V, E2 = 5 V, E3 = 5 V.
R1
R2
R3
E2
E1
E3
Figure 5 Network of three resistors R1, R2 and R3 and three batteries with EMFs E1, E2 and E3.
B1.4
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Electric & Electronic Circuits
B1.1.5 Laboratory: Apparatus
Your apparatus for this experiment comprises a number of electronic components, and a ‘breadboard’
that, together with suitable wires, allows them to be mounted and connected to form electrical circuits.
You may be provided with wire ‘jumpers’ that have already been cut and stripped, or you may simply
have been given some wire and tools to make your own. The breadboard has four 4 mm terminal posts
to allow robust connections to other apparatus. Suitable cables may be found in the racks elsewhere in
the laboratory. If you require any further apparatus, just ask one of the demonstrators or technicians.
To supply power and signals to your circuits, you will generally use a d.c. power supply and a signal
generator; multimeters and an oscilloscope allow signals to be measured.
All of these instruments, together with the resistor colour code, are described in part B1.4 of these
notes.
Connect the d.c. power supply to the mains, and turn on the power at both the mains socket and
at the instrument itself.
Measure and note the fixed voltage outputs provided by the power supply.
Measure and record the range of voltages available from the variable output of the supply.
Connect the 4 mm terminal posts your breadboard to the 0 V, +5 V, +15 V and variable voltage
outputs of your d.c. power supply, and wire them to separate bus rails of the breadboard.
B1.1.6 Laboratory: Resistor networks
A first test of the theory of component networks is whether the combination of resistors gives the
resistance predicted. The next stage is to determine whether, when connected to a supply of known
voltage, the voltages across the components are consistent with those calculated.
Take four 10 kΩ resistors and measure their resistances using a multimeter. Note whether the
values are within the labelled tolerance – and whether your measurements are sufficiently precise
to determine this.
Construct the circuit shown in Figure 6 below, using resistors with a 1% tolerance. You may use
the cutters and pliers provided to shape the resistors to fit neatly into the breadboard.
X
R1 20 k
R2
R3
10 k
20 k
Y
Figure 6 Network of three resistors R1, R2, R3.
Measure the resistance between points X and Y, and compare this with the value predicted by
applying Kirchhoff’s and Ohm’s laws. (Use your results from the preliminary exercises if you wish.)
Change the multimeter to its voltage reading, connect X and Y to the variable power supply, and
connect a second multimeter to measure the voltage across resistors R2 and R3.
Record the multimeter readings for a range of applied voltages, plot your results, and determine
whether the measured behaviour is consistent with your theoretical predictions.
B1.5
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Electric & Electronic Circuits
B1.1.7 Laboratory: The potential divider
Having developed and tested a scientific theory, the next step is often to apply it for a particular
purpose. It is common in electronic circuits to require voltages – for example, to bias active components
such as transistors and operational amplifiers – that lie somewhere between those provided by the
power supply. A simple and often-used arrangement is the potential divider, shown in Figure 7 below,
whereby the power supply is connected between terminals X and Z, and the voltage required is
provided at terminal Y.
X
R2
Y
R1
Z
Z
Figure 7 The potential divider, used to provide voltages that lie between those provided by the power supply.
Derive a formula describing how the output voltage of a potential divider depends upon the input
voltage and resistors R1 and R2. Rearrange your formula to give an expression for R1 in terms of R2
and the input and output voltages.
Design a potential divider, using resistors from among those provided, that will produce a
potential difference of 2.5 V between terminals Y and Z when terminal X is connected to +15 V
and terminal Z to 0 V.
Construct your potential divider on the breadboard, connect it to the power supply and measure
the input and output voltages to check its operation. Determine whether the measured behaviour
is consistent with your theoretical predictions.
B1.1.8 Laboratory: Multi-loop circuits
You may now test whether Ohm’s and Kirchhoff’s laws hold for multi-loop circuits.
Use the breadboard to set up the circuit shown in Figure 5, using component values R1 = 20 kΩ,
R2 = 10 kΩ, R3 = 20 kΩ, E1 = 15 V, E2 = 5 V, E3 = 0 V. Use 1% resistors throughout.
Measure the potential differences E1 and E2, and the voltage across resistor R3.
Hence determine the currents through the three resistors.
With reference to sections B1.1.1-3, calculate the theoretical values for the voltages and currents,
and compare them with your experimental measurements, summarizing your results in a table.
Determine whether the measured behaviour is consistent with your theoretical predictions.
B1.1.9 Laboratory: The Wheatstone bridge
A common problem with voltage measurement is that the finite internal resistance of the measuring
device causes it to affect the circuit being measured. A good solution is to use a ‘bridge’ circuit, which
compares the component or circuit under test with one that is known, in such a way that the measured
voltage difference is zero: the current through the meter, and hence its effect upon the circuit, is then
also zero. The Wheatstone bridge uses this principle to measure an unknown resistance.
B1.6
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Electric & Electronic Circuits
The Wheatstone bridge, shown in Figure 8 below, relies upon having a pair of known fixed resistors R1
and R2 and a known variable resistance Rs. R1 and Rs form one potential divider, while R2 and the
unknown resistor Rx form a second, and a voltmeter or ammeter is connected between the two
potential divider outputs. When no voltage or current is measured, the two divider outputs A and B
must be at the same potential, and the bridge is said to be balanced. The divider resistors must then be
in the same ratio, and the unknown resistor is hence given in terms of R1, R2 and R2 by
(23)
=
With an appropriate choice of R2, the bridge circuit can measure a wide range of resistance values.
R1
R2
M
E1
Rs
Rx
Figure 8 The Wheatstone bridge.
Calculate the value of Rs that will balance the bridge circuit if R1 = 10 kΩ, R2 = 10 kΩ and Rx = 7 kΩ.
Calculate the current through the meter if E1 = 10 V and the meter has a resistance of 1 kΩ.
Construct the Wheatstone bridge circuit, using a 10 kΩ potentiometer as Rs and with a 1 kΩ
resistor in parallel with the meter (set to measure voltage) to simulate a lower internal resistance,
and adjust the potentiometer until the meter reading is zero.
Without changing the potentiometer, disconnect it from the circuit and use the multimeter to
measure its resistance. Disconnect the resistor Rx and measure its resistance using the same
multimeter. Compare and comment upon your results.
B1.1.10
Laboratory: A simple audio-frequency volume control
A common domestic application of the potential divider is as the volume control in a radio or other
audio device.
In place of the power supply, connect the signal generator to provide the input signal across the
outer terminals of the 10 kΩ potentiometer. Connect one input channel of the oscilloscope to
monitor this voltage.
By using the oscilloscope to measure the waveform, set the signal generator to provide a
sinusoidal signal with a frequency of 1 kHz and a peak-to-peak voltage of 1 V. (If in doubt, ask a
demonstrator to show or check the operation of the signal generator and oscilloscope.)
Connect the second input channel of the oscilloscope to monitor the output of the potentiometer
from its middle connection. Check that the signal amplitude increases as you turn the shaft
clockwise; if this is not the case, reverse the connections of the potentiometer to the signal
generator. Note which terminal of the potentiometer is connected to the ground connection to
the signal generator.
By using the slot in the end of the potentiometer as a rough pointer, measure and plot the
amplitude of the output signal as a function of the shaft angle.
The potentiometer is of the type known as ‘logarithmic’ (as opposed to ‘linear’). Suggest why this
might be useful in audio applications.
B1.7