Alternating CurrentAC Voltage Applied to a Resistor
Fig. 7.1 shows a resistor R connected to a source ε of ac voltage. The symbol for an ac source in a circuit diagram is a circle with a small wave inside it. The source produces a potential difference that varies sinusoidally with time,
where is the amplitude of the oscillating potential difference and ω is its angular frequency. Applying Kirchhoff's loop rule (Section 3.12) to the circuit, , so
where the current amplitude is
Equation (7.3) is Ohm's law: for a resistor it works equally well for ac and dc. Throughout this chapter, voltages are blue, currents green and power amber; resistors are purple, inductors orange and capacitors teal. Below, V across R = 3 Ω, so A.
Move ωt and find these instants.
ωt =
The voltage and the current reach zero, their maximum and their minimum at the same times (Fig. 7.2): they are in phase with each other. Like the voltage, the current takes positive and negative values in every cycle, so the sum of its values over one complete cycle is zero, and so is its average.
Zero on Average, Yet Still Hot
An average current of zero does not mean that no energy is used. Joule heating is , which depends on , and is positive whether i is positive or negative. The instantaneous power dissipated in the resistor is
and its average value over a cycle is
where the bar over p denotes its average and ⟨…⟩ means "the average of the quantity inside". Average both quantities over whole cycles.
Now the algebra. Since and R are constants,
Using the identity , we have , and because the cosine is positive for exactly as long as it is negative. So and
Tip
The average of F(t) over a period T is . For example, . Over whole cycles, sin and cos average to 0, and sin² and cos² average to ½.
The Root Mean Square: a Fair Value
We would like to write ac power in the same form as dc power, . So we ask: what steady current heats a resistor just as fast as this ac does? Two identical 10 Ω immersion rods sit in identical buckets of water. The left one runs on ac with A; the right one on a dc supply you control. Find the dc current that heats its bucket at the same rate.
dc current: I =
The purple line, for the dc, now sits on the dashed line, the average of the green curve, for the ac. That dc current is the root mean square (rms) or effective value of the ac, written or simply I (Fig. 7.3). It is the square root of the mean of the square:
In terms of I, the average power P is
Similarly, the rms voltage or effective voltage is
From Eq. (7.3), , so , that is,
With rms values, the power equation and the relation between current and voltage in ac circuits look exactly like the dc ones; Eq. (7.7) can also be written . That is the advantage of rms values, and it is customary to measure and specify ac quantities by them. The household line voltage of 220 V is an rms value; its peak is ≈
Note
The rms current is the dc current that would produce the same average power loss as the alternating current. Ac meters read rms values, and the power rating of an appliance is its average power. The power used in an ac circuit is never negative.
Example 7.1: A light bulb is rated at 100 W for a 220 V supply. Find (a) the resistance of the bulb, (b) the peak voltage of the source and (c) the rms current through the bulb.
Solution: we are given P = 100 W and V = 220 V.
(a) The resistance of the bulb is =
(b) The peak voltage of the source is =
(c) Since P = IV, the rms current is =
Notice how every formula looks just like its dc version, because we used rms values throughout.
Speed Round: rms Values
, and . Get five right to move on.