Electric Current and Voltage

1. 1
Week 1
Electric Current and Voltage

2. Electric Current
Consider a medium with cross-
section of A m2 with charges moving with
a velocity v from left to right, as pictured in
Figure 1.1. If in a period of time ∆t, ∆q
coulombs cross A in the indicated
direction, we define the average current I
generated by the charge flow to be
Definition of Current: Current is charge in motion.
t
q
I



Notes
1. The physical dimension of current is coulomb per second (C/s).
2. The SI unit for current is the ampere (A).
3. The direction of the current I is the same as the direction of the
charge motion
cross section A
velocity v
Figure 1.1
current I

3. Specifically, if only positive
charges are continuously crossing the
cross-section A, then the resulting
current is solely due to the flow of
these positive charges. If in a period of
time ∆t, ∆q+ coulombs cross A in the
indicated direction, we define the
average current due to the flow of
positive charges as
Current due to flow of positive charges
Figure 1.2
t
q
I





cross section A
velocity v
current I+
Note
1. The direction of the current I+ is the same as that of the positive
charge motion.

4. If only negative charges are
continuously crossing the cross-section
A, then the resulting current is solely
due to the flow of these negative
charges. If in a period of time ∆t, ∆q-
coulombs cross A in the indicated
direction, we define the average
current due the flow of negative
charges as
Current due to flow of negative charges
Figure 1.3
t
q
I






cross section
A
velocity v
current I-
Note
1. The direction of the current I- is opposite to that of the negative charge
motion.

5. If both positive and negative
charges are continuously crossing the
cross section A in opposite directions,
then the resulting total current I (the
current measured by an external
ammeter) is given by
Current due to flow of both positive and negative charges
Figure 1.4
Note
1. The direction of the current total current I is taken to be the same as
that of the current I+ (conventional current flow).



 I
I
I
cross section
A
current I+
current I-
total current I

6. On the other hand, if both positive
and negative charges are continuously
crossing the cross section A in the
same direction, then the total average
current flow I is given by
Current due to flow of both positive and negative charges
Figure 1.5
Note
1. The direction of the current total current I is taken to be the same as
that of the current I+.



 I
I
I
cross section
A
current I+
current I-
total current I

7. In practice, current is measured
using an ammeter. Direct currents are
measured using dc ammeters. A direct
current (abbreviated as dc current) is a
current that flows in one direction only.
The magnitude of a direct current can
be constant or fluctuating but the sign
is unchanged over time.
Current is measured by connecting
an ammeter in the path of the current
flow.
Measuring Current
Figure 1.6
current I
DC Ammeter

8. A dc ammeter is a polarised
measuring instrument in that it has both
a positive and a negative terminal for
connection to a circuit. The reading
displayed by an ammeter will depend
on how it is connected in the circuit. A
dc ammeter might display a positive
reading when connected one way, but
will display a negative reading when
the meter connections are reversed, or
vice-versa.
Measuring Current (continued)
Figure 1.7
current I
DC Ammeter
current I
DC Ammeter

9. In the conventional view of current
flow, the positive reading displayed by an
ammeter measuring a current i is
attributed to the flow of positive charges
through the ammeter from the positive
terminal to the negative terminal. A
negative reading displayed by the
ammeter, on the other hand, is attributed
to positive charges flowing through the
ammeter from the negative terminal to
the positive terminal.
Conventional view of current flow
Figure 1.8
current i
DC Ammeter
current i
DC Ammeter
(a) A positive ammeter reading is
attributed to positive charges flowing
from the positive terminal to the negative
terminal of the ammeter.
(b) A negative ammeter reading is
attributed to positive charges flowing
from the negative terminal to the positive
terminal of the ammeter.

10. In the modern view of current flow,
the positive reading displayed by an
ammeter measuring a current i is
attributed to the flow of electrons through
the ammeter, entering from the negative
terminal and exiting via the positive
terminal. A negative reading displayed by
the ammeter, on the other hand, is
attributed to electrons flowing through
the ammeter from the negative terminal
to the positive terminal.
Modern view of current flow
Figure 1.9
current i
DC Ammeter
(a) A positive ammeter reading is
attributed to electrons flowing from the
negative terminal to the positive terminal.
current i
DC Ammeter
(b) A negative ammeter reading is
attributed to electrons flowing from the
positive terminal to the negative terminal.

11. While it has now been conclusively
proven that the current flowing in the
metallic conductors of an ammeter is
due to the motion of electrons, to avoid
the confusion of dealing with negative
signs, we will use the conventional view
of current flow when analysing circuits.
Which one to use: the conventional
or the modern view of current flow ?
Figure 1.10
current I
DC Ammeter
current I
DC Ammeter
(a) In the conventional view of current
flow, a positive ammeter reading is
attributed to positive charges flowing
from the positive terminal to the
negative terminal of the ammeter.
(b) A negative ammeter reading is
attributed to positive charges flowing
from the negative terminal to the positive
terminal of the ammeter.

12. In circuit analysis, we use arrows to
represent ammeters measuring the
currents of interest. The direction of the
arrow points from the point where the (+)
terminal of the ammeter is connected
toward the point where the (-) terminal of
the meter is connected. The direction
pointed to by the arrow is defined as the
current’s reference direction.
Reference direction for current
Figure 1.11
(a) Physical circuit showing how the
ammeter is connected to measure the
current flowing through the circuit element.
Ammeter
Circuit
element
i
Circuit
element
(b) The arrow in the schematic circuit represents
the location and relative connection of the
physical ammeter used to measure the
current flowing through the circuit element.

13. The reference direction of a current
follows the ammeter connection. If the
ammeter connection is reversed, the
direction of the arrow must also be
reversed.
The direction of the arrow does not
change with the current, even though the
current might be reversing its flow with
time.
Reference direction for current (continued)
Figure 1.12
(a) The ammeter connection is now reversed.
Ammeter
Circuit
element
i
Circuit
element
(b) When the ammeter connection is reversed,
then the direction of the arrow in the
schematic circuit follows.

14. We can connect the ammeter in any
direction we choose. There is no right or
wrong direction for the meter connection.
While the old analogue technology only
measures the positive current, the new
digital ammeters can measure both the
positive and negative currents
Reference direction for current (continued)
Figure 1.13
Positive ammeter reading only means
the meter has been connected in a
way that measures the positive
current, and vice versa.
Ammeter 2
Circuit
element
Ammeter 1
(a) Two ammeters are used to measure the
same circuit current.
Circuit
element
i1 i2
(b) The two ammeters give the same
magnitude reading but their signs are
opposite to one another.
i1 = - i2

15. Example
Figure 1.14
For each of the hypothetical
conductors shown in Figure 1.14,
determine the magnitude and sign of
the ammeter reading. Assume for
questions (c) to (e) that one positive
or negative charge is equal to 0.5 C.
(b)
1023 electrons/min
DC Ammeter
(c)
10 ‘+’ charges/sec
DC Ammeter
(a)
1023 electrons/min
DC Ammeter
(e)
6 ‘+’ charges/sec
DC Ammeter
(d)
6 ‘+’ charges/sec
DC Ammeter
12 ‘-’ charges/sec
12 ‘-’ charges/sec

16. Solution (Figure 1.14a)
Figure 1.14(a)
The ammeter gives the average
current I
1023 electrons/min
DC Ammeter
t
n
e
t
q
I
I









 











 
60
10
10
609
.
1
23
23
mA
27


I
Therefore,
I

17. Solution (Figure 1.14b)
Figure 1.14(b)
The ammeter gives the average
current I
t
n
e
t
q
I
I








 










 
60
10
10
609
.
1
23
23
mA
27

I
Therefore,
(b)
1023 electrons/min
DC Ammeter

18. Solution (Figure 1.14c)
Figure 1.14(c)
The ammeter gives the average
current I
t
n
e
t
q
I
I







 
10
5
.
0 

A
5

I
Therefore,
10 ‘+’ charges/sec
DC Ammeter

19. Solution (Figure 1.14d)
Figure 1.14(d)
The ammeter gives the average
current I
A
9

I
Therefore,
6 ‘+’ charges/sec
DC Ammeter
12 ‘-’ charges/sec
t
q
t
q
I
I
I












   
12
5
.
0
6
5
.
0 





20. Solution (Figure 1.14e)
Figure 1.14(e)
The ammeter gives the average
current I
A
3


I
Therefore,
t
q
t
q
I
I
I












   
12
5
.
0
6
5
.
0 




(e)
6 ‘+’ charges/sec
DC Ammeter
12 ‘-’ charges/sec

21. Instantaneous current
Figure 1.15
If the time t gets smaller and smaller,
then, in the limit t goes to zero, the ratio
q/t approaches the slope of the curve
at point t; that is,
i
dt
dq
t
q
t





 0
lim
i is called the instantaneous current and
is time-dependent. To explicitly show the
time dependence, we sometimes write
dt
dq
t
i 
)
(
time
q(t)
q
t
Slope = dq/dt
t

22. Example
Figure 1.16
Find and plot i(t) if q(t) is given by the graph in Figure 1.16.
t, s
1 2 3 4 5 6 7 8
1
2
3
4
5
- 1
- 2
- 3
- 4
- 5
q(t)

23. Solution
Figure 1.17
t, s
1 2 3 4 5 6 7 8
1
2
3
4
5
- 1
- 2
- 3
- 4
- 5
q(t)
For 0 ≤ t ≤ 2 s, slope of line is
2
0
2
0
4
1 






t
q
m
0
2
4
4
4
2 






t
q
m
7
4
5
4
3
3 








t
q
m
0
2
4
4
4
4 






t
q
m
C/s = 2A
C/s = 0 A
C/s = - 7 A
C/s = 0 A
For 2 ≤ t ≤ 4 s, slope of line is
For 4 ≤ t ≤ 5 s, slope of line is
For 5 ≤ t ≤ 6 s, slope of line is
For 6 ≤ t ≤ 8 s, slope of line is
C/s = 1.5 A
5
.
1
6
8
)
3
(
0
5








t
q
m

24. Solution (continued)
Figure 1.18
t
q
dt
dq
i




s
s
s
s
s
t
t
t
t
t
A
A
A
A
A
t
i
8
7
7
5
5
4
4
2
2
0
5
.
1
0
7
0
2
)
(












For the case where the charge q(t) varies linearly with time, we can write
Hence, the current i(t) is given by the piecewise function
t, s
1 2 3 4 5 6 7 8
1
2
3
4
5
- 1
- 2
- 3
- 4
- 5
i(t)

25. Relationship between current and charge
We can determine the charge that passes through the cross-
section A of the medium in Figure 1.19 in the time interval -∞
to t if we integrate current with respect to time; that is
cross section A
Figure 1.19
current i



t
d
i
t
q 
)
(
)
(
where i is current in amperes
q is charge in coulombs
t is time in seconds

26. Relationship between current and charge
By breaking the integration into two time segments, namely,
from  to 0, and from 0 to t, we can write Eq.() as
cross section A
Figure 1.19
current i

 




t
t
q
d
i
d
i
t
q
0
)
0
(
)
(
)
(
)
( 



where



0
)
(
)
0
( 
 d
i
q

27. Example
The current flowing through a circuit element is given as 50 mA. Compute
the amount of charge transferred in 100 ns.
Solution
To solve this problem using calculus, we assume that the given current is
the instantaneous current i(t) that has a constant value of 50 mA for times
t 0 and a zero value for times t < 0. Mathematically, we write
0
0
mA
50
0
)
(






t
t
t
i
The graph for i(t) is shown in Figure 1.20.
i(t)
50 mA
t
Figure 1.20

28. Solution (continued)
i(t)
50 mA
t
Figure 1.20



ns
d
i
ns
q
100
)
(
)
100
( 


 



ns
d
i
d
i
100
0
0
)
(
)
( 



The amount of charge transferred between times t = 0 and t = 100 ns is








ns
d
d
100
0
3
0
10
50
0 

  ns
100
0
3
10
50 


 nC
5


29. Example
The current in a wire is given by
otherwise
s
6
t
0
A
0
2
sin
10
)
(
















t
t
i

Find the total charge passing a cross section of the wire from t = 0 and t = 6 s.
i(t)
Figure 1.21

30. Solution
A plot of the current i(t) is shown Figure 1.22.
i(t) A
t (s)
10
0 2 4 6
Figure 1.22
)d
i(
)
0
(
)
6
(
6
0


 

q
q
)d
i(
)
0
(
0
-


 

q
The total charge up to time t = 6 is given by
the expression
where

31. )d
2
10sin(
)
6
(
6
0

 

q
C
12.73

Figure 1.23
Since i(t) = 0 for t < 0, we have
0
d
0
)
0
(
0
-

 


q
1
2
cos
10
6
0

















Hence
A plot of q(t) is shown in Figure 1.23
0 1 2 3 4 5 6
2.12
4.24
6.37
8.49
10.61
12.73
12.732
0
q t
( )
6
0 t

32. Charge transferred by a constant current flow





T
0
0
I
0 
 d
d
IT

If the current flow is constant, that is if the current i(t) = I for times
between t = 0 and t = T, where I and T are both constants, then
the charge transferred up to time T is given by



T
d
i
T
q 
)
(
)
(
If we write q(T) ≡ Q, then we obtain
IT

Q

33. IT

Q
Example
The current flowing through a circuit element is given as 50 mA. Compute
the amount of charge transferred in 100 ns.
Solution
Since the current is constant we can simply obtain Q using the relationship
Hence
nC
5
ns
100
mA x
50
IT 


Q

34. Definition
Potential Difference and Voltage
The work w done by the electrical system in moving a charge q from
a point A to another point B is determined by the potentiaI difference
(or simply, voltage) that exists between A and B. Quantitatively, the
potential difference between A and B (indicated by the voltage vAB) is
defined to be
q
w
q
vAB





moved
charge
of
amount
B
A to
from
charge
moving
in
done
work
The unit for potential difference is energy/charge, joules per coulomb in
the MKS system, but to honour Count Alessendro Volta, we use the
special name volt (V) for this unit. Thus, we say that the potential
difference between point A and point B is 1 volt if 1 joule of work is
done in moving a unit charge (+1 C) from A to B.
I =q/t
vAB
A
B
Figure 1.24

35. Measuring voltage
For convenience we simplify the circuit
drawing as shown in Figure 1.25b. The
symbol v represents the voltmeter reading,
and the (+) and (-) signs associated with the
voltage v correspond to the location of the
(+) and (-) terminals of the voltmeter in the
measurement. The location of the +/- signs
defines the reference direction for the
voltage.
In practice, we measure voltage with a
voltmeter (VM), as shown in Figure 1.25a.
The meter will indicate the potential
difference between A and B.
Voltmeter
Circuit
element
A
B
(a)
v
Circuit
element
A
B
(b)
Figure 1.25

36. +/- Notation for voltage
In Figure 1.25(b) we have
expressed the potential difference
across the circuit element by
marking both ends of the circuit
element with polarity symbols: a ‘+’
at one end and a ‘–’ at the other
end. This is called the +/- notation
for voltage labelling. With this
notation, the convention is that the
‘+’ represents the first subscript and
the ‘–’ the second subscript of the
voltage. (a) (b)
Figure 1.26
C
v2
B
v1
A
#1
#2
v1
B
A
#1
#2
C
v2
Figure 1.26b shows the voltage across the two circuit elements marked in
this manner. To differentiate between the two voltages we can label them use
either a numerical or an alphabetic subscript.

37. Arrow notation for voltage
It is sometimes convenient to use arrows
to define voltage reference directions.
Then, as in Figure 1.27, the head of the
arrow is the point of measurement and
the tail is the point of reference. With this
notation, the convention is that the
‘arrowhead represents the location of the
(+) terminal of the measuring voltmeter
and the tail represents the (-) terminal.
Figure 1.27
+ 15 V
-10 V 5 V
v1
A B C D
-15 V
v2
E
#1 #2 #3
By using the arrow notation, we can treat
the voltages as vectors; hence the rules
for vector addition and vector subtraction
can be applied for the voltages.
0 V

38. Arrow notation for voltage (continued)
Example
Referring to Figure 1.27, find vA, vB, vC,
v1, and v2.
Answer
vA = + 15 V ; vB = - 10 V
vC = + 5 V ; vD = - 15 V
v1= vB – vA
= (-10 V) – (15 V)
= - 25 V
Figure 1.27
+ 15 V
-10 V 5 V
v1
A B C D
-15 V
v2
E
#1 #2 #3
0 V
v2 = vC – vD
= ( 5 V) – (- 15 V)
= 20 V

39. Double subscript notation
We often use double subscripts to indicate the
voltmeter connections. The first subscript indicates
where the (+) terminal is connected in the circuit,
and the second subscript indicates where the (-)
terminal is connected. vAB
vCB
B
A
#1
#2
C
C
B
VM1
A
#1
#2
Figure 1.28
(a) (b)
VM1
Using the double-subscript notation, the
voltages measured by voltmeters VM1 and
VM2 in Figure 1.28(a) would be written as vAB
and vCB, respectively. Thus, according to the
double-subscript notation, vAB in Figure
1.28(b) is the potential difference measured
between points A and B, and vCB is the
potential difference measured between points
C and B.

40. Rule on changing the order of voltage subscripts
vAB = + 10 V
A
B
vBA = - 10 V
Changing the order of the subscripts changes the sign of the voltage
Example
What is the voltage vBA for the circuit shown in Figure 1.29?
Answer
Figure 1.29

41. Rule on intermediate points for voltage subscripts
Voltage at intermediate points in a circuit when measured in the same
direction adds algebraically.
Thus, in the circuit shown in Figure 1.30 we can write, for example, the voltage
vAE as follows:
vAE = vAB + vBC + vCD + vDE
Figure 1.30
A C
B
#1 #2 #3 #4
D E
or
vAE = vAC + vCD + vDE
vAE = vAB + vBD + vDE
or

42. Rule on intermediate points for voltage subscripts (continued)
Example
Figure 1.31
Find the voltage vAC in Figure 1.31
C
B
A
#1
#2
vAB = 10 V
vCB = -5 V

43. Rule on intermediate points for voltage subscripts (continued)
Answer
Figure 1.31
From the figure, we can write
vAC = vAB + vBC
= vAB + (-vCB)
Hence, C
B
A
#1
#2
vAB = 10 V
vCB = -5 V
= vAB - vCB
vAC = (10 V) – (- 5 V) = 15 V

44. Exercise
Refer to Figure 1.32. If vAB = + 2 V and vCB = - 1 V, find vBA and vCA.
Rule on intermediate points for voltage subscripts (continued)
C
B
A
#1
#2
Figure 1.32

45. Answer
vBA = - vAB = - 2 V
vCA= vCB+ vBA = (- 1 V) + (-2 V) = - 3 V
Rule on intermediate points for voltage subscripts (continued)
C
B
A
#1
#2
Figure 1.32

46. Double subscript notation (continued)
Figure 1.33
Note
Avoid using arrows to show
voltage reference direction
when using the double-
subscript notation.
vAB
vCB
B
A
#1
#2
C
(a)
vAB
vCB
B
A
#1
#2
C
(b)
Question
Which of the two figures on
the right shows the correct
application of the arrow
notation for voltage reference
direction, based on the double
subscripts written for the
voltages?