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1.3. Resistors in Series and in Parallel
R
1.3.1. Resistors in Series
Figure 1.4 shows three resistors in series, connected to an e.m.f. source.
We can see the polarity of the source; by convention, the arrow head is at the
positive end. The same convention is used for the potential difference across
each resistor.
R
A
1
I
A
I
V
S
R
3
B
Fig. 1.4. Three resistors in series and the equivalent circuit
R
2
V
S
B
R
At the left-hand end each resistor has a more positive potential than that
at the right-hand end because the direction of the current flowing in the
circuit is from left to right.
The same current I flows through each component in the circuit, so for
the voltage across each resistor we have:
VIR
11
The total potential difference across all three resistors is
this must be equal to the applied e.m.f. V
VIR
,
22
VIR
,
33
, since there are no other
S
.
VVV
123
, and
components in the circuit across which potential can be dropped. (We make
the assumption that the internal resistance of the source, and the resistances
of connecting cables, can be neglected because they are much smaller than
resistance values of the resistors.) Hence
V V V V IR IR IR IR R R
123123123S
.
In the equivalent circuit the equivalent resistance R has a value such that
the same current I flows, and
VIR
Hence,
S
RRR
123
.
.
Thus, the resistance equivalent to three resistors connected in series is
the sum of the individual resistances. The same argument can be applied to
any number of resistors in series, giving the general result that the
equivalent
resistance of n resistors in series connection is equal to the sum of the n
individual resistances
.
11
1.3.2. Resistors in Parallel
I
I
I
R
R
R
Figure 1.5 shows two resistors in parallel, connected across an e.m.f.
source. This time the current flowing through each resistor is different, but
the potential difference across each resistor must be the same because the
e.m.f. source V
is connected directly across each resistor.
S
V
A
I
I
1
S
B
R
1
I
2
R
2
V
A
I
R
S
B
Fig. 1.5. Parallel connections of resistors
We can write an expression for the current in each resistor:
VG
11S
,
VG
22S
.
The e.m.f. source supplies all the current in the circuit, so the current I is
the sum of the two currents I
Hence,
So the equivalent conductance is
and I2. In the equivalent circuit,
1
.
VG
S
VG VG VG
SS S
12
GG G
.
12
.
Thus, we can say that the equivalent conductance of the two resistors in
parallel is the sum of individual conductances. The same argument can be
applied to any number of resistors in parallel, giving the general result that
the equivalent conductance of n resistors in parallel is equal to the sum
of the n individual conductances
.
The equivalent conductance of a parallel connection of resistors should
always be greater than the largest individual conductance. The equivalent
resistance should always be smaller than the smallest individual resistor value.
For the case of two resistors in parallel, there is an expression for the
equivalent resistance. Here,
Hence,
GG G
11
12
1
R
GRR
RRR
12 12
R
12
12
R
12
.
.
Thus, ‘the equivalent resistance of two resistors in parallel is equal to
the product of their resistance values, divided by their sum’.
12
1.3.3. The Voltage Divider
I
I
A particular case of series connection where two resistors are connected
across the e.m.f. source is shown in fig. 1.6(a).
I
R
V
S
12V
1k
R
3k
1
V
1
V
S
2
V
2
12V
I
I
1
R
1
2k
(a) (b)
Fig. 1.6. Divider circuits: (a) the voltage divider; (b) the current divider
The voltage ratio V1/V2 is
VIRR
111
VIRR
222
.
I
2
R
3k
2
This expression illustrates the voltage divider rule. According to this
rule, the voltage across two resistors in series divides between them in the
ratio of their resistance.
For example, in the circuit, fig. 1.6(a), the voltage divides in the ratio
1:3 and V
and V
is ¼ of the applied e.m.f. VS, while V2 is ¾ of it. Hence V1 is 3 V
1
is 9 V.
2
The voltage divider is particularly useful to obtain a potential difference
which is a fraction of the voltage of an applied e.m.f. source.
1.3.4. The Current Divider
Figure 1.6(b) shows a particular parallel connection of two resistors
across the e.m.f. source. The source current I divides between the resistors R
and R
in proportion to their conductance.
2
V
S
112
221
RR
V
S
R
1
11
1
2
R
2
GR
.
GR
1
For example, in the circuit, fig. 1.6(b), the current divides in the ratio 3:2
and I
is 3/5 of the source current I while I2 is 2/5 of it. Hence the current of R1 is
1
V
I
1
R
12
S
2000
1
mA and the current of R
6
2
is
V
I
2
R
12
S
3000
2
4
mA.
13
1.3.5. The Variable Potential Divider
Figure 1.7 shows a variable potential divider or ‘pot’. These devices have
three terminals and, as shown in fig. 1.7(a), two of the terminals (A and B)
are connected to either end of a resistor. The third terminal, which is called
wiper, makes electrical contact with the resistor along its length.
the
Wiper turns
with Dial
A
Wiper
B
A
W
Resistive
Material
B
(a) (b)
Fig. 1.7. Variable potential divider: (a) rotary potentiometer construction;
(b) single turn potentiometer
As the shaft of the pot is turned, the wiper connection is moved along
the length of the resistor.
The pot circuit can be used as a part of the voltage divider circuit,
fig. 1.6, in which the ratio of R
to R2 can be varied by rotating the shaft of
1
the potentiometer. If the pot is used in a circuit, as shown in fig. 1.8, the
output voltage can be varied from zero to the input voltage level by adjusting
the position of the wiper contact.
+V
R
1
V
in
R
L
0V
V
out
Fig. 1.8. Potentiometer to vary the output voltage
Pots are made with many different types of resistor and values of
resistance. In some cases the resistance varies linearly and in other cases
logarithmically, with length. Pots are specified with their resistance value and
the maximum power that they can dissipate.
14
1.4. Kirchhoff’s Laws
I
I
I
Kirchhoff’s circuit laws are two equations that deal with the
conservation of charge and energy in electrical circuits first described in 1845
by Gustav Kirchhoff. At present they are widely used in electrical
engineering and are known as Kirchhoff’s rules or Kirchhoff’s laws.
1.4.1. Kirchhoff’s Current Law
Consider fig. 1.9. The points (such as , ) where two or more
components are connected together are called
nodes.
V
R
ABC
S
I
1
1
I
2
R
D
Fig. 1.9. Kirchhoff’s laws illustration
R
I
3
2
3
I
4
R
4
Kirchhoff’s current law states that the current cannot be accumulated at
a circuit node, so that the current flowing towards the node must be equal to
the current flowing away from it. This leads to the formal statement which is
Kirchhoff’s current law (sometimes called the first Kirchhoff’s law).
At any node of a network, at any instant of time, the sum of the currents
getting into the node is equal to the sum of the currents getting out of the node.
The application of this law to the node (the circuit in fig. 1.9) gives
the equation
123
node D we have
II
241
. For the node C we can say that
II
.
I
34
and for the
An alternative form of Kirchhoff’s current law (KCL) can be obtained
by considering currents directed into a node as positive, while currents
directed out of a node are considered negative. In this case, Kirchhoff’s
current law can be stated as
At any node of a network, at every instant of time, the algebraic sum of
the currents at the node is zero.
For the node (fig. 1.9), the law gives
123
0III
which is clearly
an alternative form of the previous equation.
15
1.4.2. Kirchhoff’s Voltage Law
This law is a formalization of the fact that the voltage drops around a
circuit add up to the voltage of the e.m.f. source. The law states that
The algebraic sum of the voltages across all the components around any
loop of a circuit is zero.
As the term ‘algebraic sum’ is used in the law, it is clear that we expect
the voltage in the circuit to be both positive and negative. Fig. 1.9 shows
which voltages are positive and which are negative. As it has been explained
previously, the arrow, next to the source, shows the direction in which the
source drives the current, and the arrowhead indicates the more positive
voltage. In the resistor, the voltage drop is proportional to the current
flowing. So the voltage polarity will be, as shown in fig. 1.9, plus-minus
according to the direction of the current flow (the voltage becomes less
positive within the resistor in the direction of the current flow).
Kirchhoff’s voltage law (also called the second Kirchhoff’s law) is
applied to the loops in fig 1.9. In the clockwise direction around the loop
ABDA we get
VV V
12SR R
VV V
or
SR R
12
0
.
The algebraic sum is zero, in accordance with Kirchhoff’s voltage law.
Going round the loop BCDB, we would get
VVV
234RRR
VVV
or
234
RRR
0
.
For the loop ABCDA, the result is
VV V V
134SR R R
or
VV V V
SR R R
134
0
.
Remember that the voltage drop in a resistor is the product of the current and
resistance. Therefore
VIRIRIR
11 3 3 4 4
S
0
.
Kirchhoff’s voltage law (KVL) can be used in the circuit analysis to
produce equations which enable us to evaluate an unknown voltage or
current.
Self-Assessment Questions
1. How do you describe current if positive and negative charges are
moving in opposite directions?
Write the expression for Ohm’s law.
2.
A resistor has 9 V across it and a current of 3 mA flowing through it.
3.
What is its resistance?
16
4. A resistor of 20 has a voltage of 10 V across it. What power is
being dissipated by the resistor?
A power supply produces a 60 W output with an input of 70 W.
5.
What’s the efficiency?
A conductor has a conductance of 2 S, and the potential difference
6.
across it is 0.5 V. How much current is flowing through it?
Three resistors, connected in series, have resistances of 1 k, 10 k
7.
and 500 each. What is the total equivalent resistance R
Three fixed resistors are connected in parallel. Calculate the total
8.
equivalent resistance R
Resistors R
9.
of the parallel circuit.
TP
and R2 are connected in series to the input direct voltage
1
source. If the input voltage is 15 V, and the resistance R
voltage across the resistor R
Kirchhoff’s current law and Kirchhoff’s voltage law: explain them
10.
2
?
of this circuit?
TS
= 2R2, what is the
1
using a simple circuit.
17
CHAPTER 2. SIGNALS, WAVEFORMS AND A.C. COMPONENTS
After studying this chapter you will understand the difference between
analogue and digital signal waveforms, periodic and non-periodic
waveforms. You will be able to calculate the average voltage and r.m.s.
voltage of periodic waveforms, to define the capacitance and inductance and
to calculate their reactance at a given frequency. You will be also able to
describe the behaviour of a capacitor, an inductor and a transformer in terms
of input and output currents and voltages.
2.1. Electrical Waveforms
The word ‘waveform’ is used in reference to the graph of the voltage, or
current, of a varying value against time. Consider a simple circuit shown in
fig. 2.1(a). If the switch is closed, the current almost instantaneously starts
flowing through the lamp, causing it to shine. If the switch is opened and
closed repeatedly at 1-second intervals, the graph of the current is as shown
in fig. 2.1(b). This waveform is called a
square wave. It is also the graph of
the voltage across the lamp and approximately the graph of the lamp
brightness. If the switch is opened and closed irregularly according to some
code, as shown in fig. 2.1(c), the circuit can be used to transmit information
from the message sender to users. The information is transmitted to people
watching the lamp if they know the code. So the waveform, fig. 2.l(c), can be
called a
signal. This type of on-off waveform is an example of a digital
waveform.
A voltage applied in one part of the circuit in fig. 2.1 appears across the
lamp almost instantaneously, though the electrons move quite slowly. This is
because all the electrons begin to move around the circuit almost
simultaneously. Similarly, if the voltage applied to a circuit is varied
continuously, by means of a signal generator or a microphone, for example,
the current in the whole circuit will vary in a similar way.
Figures 2.2(b) and 2.3(b) show waveforms of currents which
microphones can cause to flow. Consider first the circuit in fig. 2.2(a). Before
you make use of such a microphone, a steady direct current of perhaps 2 mA
flows in the circuit caused by the battery, as shown by the waveform in fig.
2.2(b). As soon as you speak this current is changed. The microphone
modifies the current, so that it varies around the average value (2 mA in this
case). It varies between 1 mA and 3 mA in this figure. Conventionally this
kind of the current variation is regarded as the sum of a d.c. and an a.c.
components of current waveforms.
18
(a)
V
m
battery
Voltage
Positive
Half
switch
Negative
Half
I
lamp
V
Square Wa ve
Amplitude
0
Period
T
T
2
time
(b)
V
0
(c)
time
Fig. 2.1. Simple switching circuit: (a) the circuit; (b) the ‘squarewave’ voltage
waveform across the lamp; (c) a digital signal produced by operating the switch
according to a code in order to transmit a message
The microphone in the circuit, fig. 2.3(a), does not need a battery, so
the quiescent current is zero as shown in fig. 2.3(b). The electromagnetic effect
is used to create a voltage whenever the diaphragm (similar to an eardrum)
in the microphone is disturbed by the sound waves impinging on it. For the
same speech input received by the other microphone, this one might produce
a similar waveform as shown in fig. 2.3(b). The resulting current in this case
is an alternating current and it alternates between perhaps 1 mA in one
direction and 1 mA in the opposite direction. Note that, in this graph, the
current flowing in one direction is called a positive current and in the opposite
direction it is called a negative current. The positive and negative parts of the
waveform tend to cancel out, producing the average current of zero value.
As regards the information communication, the waveforms in figs. 2.2(b)
and 2.3(b) are identical; the average current of 2 mA in fig. 2.2(b) might not
be there as far as their information-carrying capacity is concerned. If the two
currents represented by these two waveforms flow through an earphone, you
would not hear much difference between them because only changes in the
current produce audible sounds. Additional d.c. components of currents may
be necessary to make certain devices work properly, but they are not usually
used to transmit messages.
19
microphone
battery
(a)
current
earphone
alternating
current
microphone
earphone
(a)
i, mA
3
2
1
(b)
0
t
1
t
Fig. 2.2. A microphone, a battery and
an earphone in the circuit: (a) the
communication circuit; (b) the current in
the circuit varies around the mean value,
in this case, of 2 mA
The waveform in fig. 2.3(b) is called the
earphone: (a) the communication circuit;
i, mA
2
1
0
-1
(b)
Fig. 2.3. A circuit with the
electromagnetic microphone and
(b) the current varies around a zero mean
analogue waveform, because
it is an electrical analogue of the sound waveform.
2.2. Sinusoidal Waveforms and Frequency
t
1
t
value
The waveforms shown in figs. 2.4(a) and 2.4(b) are examples of
sinusoidal waveforms. Figure 2.4(a) is a sine wave because it is a graph of
. Figure 2.4(b) is a cosine wave because it is a graph of
sinyA
The difference between the waveforms is the value of y at
wave,
when
0y
, but for the cosine wave,
0
yA
when
sinusoidal waveforms have the same shape and are simply called
In electronics
y usually stands for either a voltage or a current. Suppose
. For the sine
0
. Both
0
sinusoids.
.
cosyA
y stands for a sinusoidal voltage, then A is the maximum value of the voltage,
called the
amplitude of the waveform. It is usually symbolized by the capital
V, often with a pair of subscripts to identify it. So in general the equation can
be written as
vV
sin
m
. However, in an electrical waveform, increases
uniformly with time and so is usually expressed as a constant multiplied by
t
time t. That is,
sinusoid and is the rate of increase of the angle
and 2.4(b),
is expressed both in degrees and radians, so could be
, where is called the angular frequency of the
per second. In figs. 2.4(a)
20
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