A Resistive Load
shows a circuit containing a resistance element of value R and an ac generator with the alternating. emf of . By the loop rule, we have resistor is connected across an alternating-current generator. (h) The current iR and the potential difference “R across the resistor are plotted on the same graph. both versus time I. They are in phase and complete one cycle in one period T. (e) A phase diagram shows the same thing as (h). From the definition of resistance (R = VIi), we can now write the current iR in the resistance as where IRis the amplitude of the current iR in the resistance. Comparing and 33-32, we see that for a purely resistive load the phase constant 4> = If. We also see that the voltage amplitude and current amplitude are related byAlthough we found this relation for the circuit of Fig. 33-80, it applies to any resistance in any ac circuit. By comparing Eqs. 33-30 and 33-31, we see that the time-varying quantities VR and iR are both functions of sin wdt with 4> = 00 • Thus, these two quantities are in phase, which means that their corresponding maxima (and minima) occur at the same times. Figure 33-8b, which is a plot of vR(t) and ;R(t), illustrates this fact. Note that VR and iR do not decay here, because the generator supplies energy to the circuit to make up for the energy dissipated in R.The time-varying quantities VR and iR can also be represented geometrically by phasors. Recall from Section 17-10 that phasors are vectors that rotate around an origin. Those that represent the voltage across and current in the resistor of are shown in c at an arbitrary time t. Such phasors have the following properties.
A”gular speed: Both phasors rotate counterclockwise about the origin with an angular speed equal to the angular frequency Wd of VR and iR• Length: The length of each phase represents the amplitude of the alternating
quantity: VR for the voltage and lR for the current. Projection: The projection of each phase on the vertical axis represents the value of the alternating quantity at time t: VR for the voltage and iR for the current.RoUllion ang”: The rotation angle of each phase is equal to the phase of the alternating quantity at time t. In Fig. 33-&, the voltage and current are in phase, so their phasors always have the: same phase wdt and the same rotation angle,
and thus they rotate together. Mentally follow the rotation. Can you see that when the phasors have rotated
so that wdt = 900 (they point vertically upward), they indicate that just then vR = VR and iR = lR? Equations 33-30 and 33-32 give the same results