23.1 ALTERNATING CURRENTS
23.1.1 Alternating Current
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| Figure 23.1 A resistor connected to an alternating voltage supply. The current flowing through the resistor is an alternating current that reverses direction periodically with a constant frequency. |
1. Meaning of alternating current
Figure 23.1 shows a resistor which is connected to an alternating voltage supply. The current flowing through the resistor is an alternating current. An alternating current is a current which reverses its direction at a constant frequency. Since an alternating current flows in a to-and-fro manner, it must have period and frequency.
2. Sign to indicate direction of current
The value of an alternating current is given either a plus sign or a minus sign to indicate the flow direction. For example, the current in Figure 23.1 is positive when it flows in the clockwise direction, like I = +2.0 A. Conversely, it is negative when it flows in the anti-clockwise direction, like I = −1.5 A.
3. Graphical representation
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| Figure 23.2 The figure shows different waveforms of alternating current, illustrating both positive and negative values at different times and that the waveform can take various shapes. |
A graph of alternating current I against time t can show the waveform of the current. The graphs in Figure 23.2 show examples of three different current waveforms. Notice the following:
(a) Each graph shows the positive value and negative value of I at different time.
(b) The waveform of an alternating current can take any shape.
23.1.2 Sinusoidal Current
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| Figure 23.3 The figure shows a sine curve representing a sinusoidal current, from which the maximum current and period can be determined. |
If the current waveform takes the shape of a sine curve, the current is known as a sinusoidal current. Figure 23.3 shows a graph which has a sine curve.
We can obtain the following information from this graph:
(a) The maximum current I0.
(b) The period of oscillation of the current T.
From the value of T we can obtain
(i) the frequency f, where
f = 1 / T
(ii) the angular frequency ω, where
ω = 2πf
23.1.3 Equation Representing Sinusoidal Current
A sinusoidal current is a sine function of time t. Hence, its instantaneous value I can be expressed by the following equation:
I = I0 sin (ωt + φ0)
or,
I = I0 cos (ωt + φ0)
where I0 is the maximum current, ω is the angular frequency and φ0 is the initial phase angle.
If we let φ0 = 0, then we have
I = I0 sin (ωt)
or,
I = I0 cos (ωt)
EXAMPLE 23.1
An alternating current I is represented by the following equation:
I = (2.0 A) sin (100πt)
where time t is measured in second. Determine
(a) the maximum current
(b) the frequency of oscillation of the current
(c) the current at time (i) t = 2.5 ms (ii) t = 12.5 ms
Answer
(a) Maximum current I0 = 2.0 A.
(b) Given
ωt = 100πt
2πf = 100π
f = 50 Hz
(c) (i) At t = 2.5 ms,
I = (2.0) sin (100π(2.5 × 10−3))
= (2.0) sin 45° = +1.4 A
(ii) At t = 12.5 ms,
I = (2.0) sin (100π(12.5 × 10−3))
= (2.0) sin 225° = −1.4 A
EXERCISE 23.1
A current I is represented by the following equation:
I = (4.0 A) sin (100πt)
where time t is measured in second. Determine
(a) the current at time t = 10 ms
(b) the time when the current has a value of +3.464 A.
23.2 ALTERNATING VOLTAGE
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| Figure 23.4 The figure shows the reversal of polarity between two terminals over time, producing an alternating voltage. |
23.2.1 Meaning of Alternating Voltage
A voltage supply like a battery normally has two terminals. Each terminal is charged. For an alternating voltage supply, at a particular instant one terminal, A, is positively charged while the other, B, is negatively charged, as shown in Figure 23.4(a). After half of a period has passed, the polarity of each terminal is reversed, as shown in Figure 23.4(b). An alternating voltage is produced across the terminals of this alternating voltage supply.
23.2.2 Graphical Representation of Alternating Voltage
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| Figure 23.5 The figure shows a graph of alternating voltage against time, illustrating the change in polarity where the voltage is negative at one instant and positive at another. |
Refer to the circuit shown in Figure 23.4. We may plot the potential of terminal A (or B) relative to the potential of terminal B (or A) against time t. Figure 23.5 shows one example of a V–t graph for the alternating voltage across AB. At time t1, terminal A is negative, but at time t2 the terminal is positive.
23.2.3 Sinusoidal Voltage
A sinusoidal voltage is a sine function of time. Hence, the instantaneous voltage V at time t can be represented by the following equation:
V = V0 sin (ωt + φ)
or,
V = V0 cos (ωt + φ)
where V0, φ and ω are the maximum voltage, the initial phase angle and the angular frequency respectively.
23.3 POWER
23.3.1 Instantaneous Power
Suppose an alternating voltage is applied to a circuit element. At time t, the voltage across the element is V, producing a current I flowing through the element. At that instant, the electrical power P received by the element is given by
P = IV
23.3.2 Sign of Power
The value of power may be positive or negative. Refer to the table below.
| Sign of Power | Meaning |
|---|---|
| + | Power is supplied to / absorbed by the element |
| − | Power is expended / supplied by the element |
EXAMPLE 23.2
The voltage V applied across a circuit element is given by
V = (20 V) sin (100πt)
The current flowing through the element is given by
I = (2.0 A) sin (100πt)
Determine the electrical power delivered to the element at time t = 2.0 ms.
Answer
The instantaneous power P is given by
P = IV
= [2.0 sin (100πt)][20 sin (100πt)]
= 40 sin² (100πt)
At t = 2.0 ms,
P = 40 sin² (100 × 180° × 2.0 × 10−3)
= 40 sin² (36°) = +13.8 W
At that instant, power is delivered to the element.
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