PHASORS AND ALTERNATING CURRENTS

To supply an alternating current to a circuit, a source of alternating emf or is required. An example of such a source is a coil of wire confrontation angular velocity in a magnetic field, which we discussed in Example 30-4 (Section 30-3). This develops a sinusoidal alternation emf and is the prototype of the commercial alternating-Current generator alternator.

we will the term ac source for any device that supplies a sinu-soidally varying voltage (potential difference) u or current i. The usual circuit-diagram symbol fO£an ac source is

Alternating Current
Alternating Current

A sinusoidal voltage might be described by a function such as

Alternating Current
Alternating Current

In this expression, u (lowercase) is the instantaneous potential difference; V (uppercase) is the maximum potential difference, which we call .the voltage amplitude; and OJ is the angular frequency, equal to 21f times

the frequency f In the United States and Canada, commercial electric-power distribution systems always use a frequency of f = 60 Hz, corresponding to w = (21r rad)(6O s-I) = 377 radls; in much of the rest of the world.!= 50 Hz (w= 314 radls) is used. Similarly,
a sinusoidal current might be described as

Alternating Current
Alternating Current

where i (lowercase) is the instantaneous current and I (uppercase) is the maximum current or current amplitude.

To represent sinusoidal varying voltages and currents, we will use rotating vector diagrams similar to those we used in the study of simple harmonic motion in Section 13-3 (see Fig. 13-3). In these diagrams the instantaneous value of a quantity that varies sinusoidal with time is represented by the projection onto a horizontal axis of a vector with a length equal to the amplitude of the quantity.The vector rotates counterclockwise with constant angular velocity w. These rotating vectors are called phosphors, and diagrams containing them are called phosphor diagrams. Figure 32-1 shows a phosphor diagram for the sinusoidal current described by Eq. (32-2). The projection of the phosphor onto the horizontal axis at time t is I cos wt; this is why we chose to use the cosine function rather than the sine in Eq. (32-2).

A phosphor is not a real physical quantity with a direction in space, such as velocity, momentum, or electric field. Rather, it is a geometric entity that helps us to describe and analyze physical quantities that vary sinusoidal with time. In Section 13-3 we  used a single phosphor to represent the position of a point mass undergoing simple harmonic motion. In this chapter we will use phosphors to add sinusoidal voltages and currents. Combining sinusoidal quantities with phase differences then becomes a matter of vector addition. We will find a similar use for phosphors in Chapters 37 and 38 in our study of interference effects with light.

How do we measure a sinusoidal varying current? In Section 27 -4 we used a d’ Arson val galvanometer to measure steady currents. But if we pass a sinusoidal current through a d’ Arson-val meter, the torque on the moving coil . . ‘dally, with one  direction half the time and the opposite direction the beige- . Thee needle may wiggle a little if the frequency is low enough. but i av ‘ is zero. Hence a d’ Arson val meter by itself isn’t very useful for g currents. To get a measurable one-way current through the can use diodes, which we described in Section 26-4. A diode (or rectifier ‘a ‘a: that conducts better in one direction than in the other; an ideal diode has zero for one direction of current and infinite resistance for the other. One lent is shown in Fig. 32-2a. The current through the galvanometer G is rared, regardless of the direction of the current from the ac source (i.e., which pan cycle the source is in). The current through G is as shown by the graph in Fig. 32- It but always has the same direction, and the average meter deflection is not zero, This arrangement of diodes is called eyeful-wave rectifier circuit.

The rectified average current I•••is defined so that during any whole number of cycles, the total charge that flows is the same as though the current were constant with a value equal to prov’ The notation 1m and the name rectified average current emphasize  that this is not the average of the original sinusoidal current. In Fig. 32-2b the total charge that flows in time t corresponds to the area under the curve of i versus t (recall that i = dq/dt, so q is the integral of 0; this area must equal the rectangular area with
height l.w’ We see that I rev is less than the maximum current I; the two are related by

Alternating Current
Alternating Current

(The factor of 2I1ris the average value of [cos wtl or of [sin wtl; see Example 30-5 in Section 30-3.) The galvanometer deflection is proportional to 1m, The galvanometer scale can be calibrated to read I, l.w’ or, most commonly,/rmr.(discussed below).

Alternating Current
Alternating Current

A more useful way to describe a quantity that can be either positive or negative is the mce mean -square (rms} value. We used rms values in Section 16-4 in connection with the speeds of molecules in a gas. We square the instantaneous current i, take the aver- age(mean) value of i2, and finally take the square root of that average. This procedure denies the root-mean-square current, denoted as IrIm• Even when i is negative, l is always positive, so Irms is never zero (unless i is zero at every instant).

Here’s how we obtain lmu’ If the instantaneous current is given by i =1 cos cut, then

Alternating Current
Alternating Current

Using a double-angle formula from trigonometry,

Alternating Current
Alternating Current

we find

Alternating Current
Alternating Current

The average of cos 2wt is zero because it is positive half the time and negative half the time. Thus the average of i2 is simply 12/2. The square root of this is 1ms:

Alternating Current
Alternating Current

In the same way, the root-mean-square value of a sinusoidal voltage with amplitude (maximum value) V is

Alternating Current
Alternating Current

We can convert a rectifying ammeter into a voltmeter by adding a series resistor, just as for the de case discussed in Section 27 -4. Meters used for ac voltage and current measurements are nearly always calibrated to read rms values, not maximum or rectified average. Voltages and currents in power distribution systems are always described in terms of their rms values. The usual household power supply, “120-volt ac,” has an rms voltage of 120 V. The voltage amplitude is

Alternating Current
Alternating Current
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