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or VB VC t 10:2 L From (10.1) and (10.2), it is seen that the inductor voltage is constant and that the inductor current is a straight line segment as shown by Fig. 10-1(b). i t i 0
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Example 10.1. Let VB 0 and inductor current i have a nonzero initial value for the circuit of Fig. 10-1(a). Assume that vC changes insigni cantly over the interval of interest, and determine the nature of vL and i. By KVL, vL t VC L From (1), it follows that i t
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or VC t 2 L From (1) and (2) it is concluded that the inductor voltage is constant and that the inductor current is a straight line segment, as shown by Fig. 10-1(c). i t i 0
Average Inductor Voltage Consider the case of an inductor L that carries a periodic current iL t , so that over a period Ts , iL 0 iL Ts . The average value of inductor voltage is given by   1 Ts 1 Ts diL L iL Ts VL hvL i vL dt L diL 0 10:3 dt Ts 0 Ts 0 Ts iL 0 dt As long as the inductor current iL is periodic, the average value of voltage across the inductor is zero.
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Average Capacitor Current Consider the case of a capacitor C for which the capacitor voltage vC t is periodic over Ts so that vC 0 vC Ts . The average value of current through the capacitor is found as   1 Ts 1 Ts dvC C vC Ts IC hiC i i dt C dvC 0 10:4 dt Ts 0 C Ts 0 Ts vC 0 dt As long as the capacitor voltage vC is periodic, the average value of current through the capacitor is zero.
BUCK CONVERTER
The SMPS circuit of Fig. 10-2, known as a buck converter, produces an average value output voltage V2 hv2 i V1 . The duty cycle D is de ned as the ratio of the ON time of transistor Q to the switching period Ts 1=fs (switching frequency). When the ideal transistor Q is ON, vD V1 . Conversely, when Q is OFF, continuity of current through inductor L requires that diode D be in the forward conducting state; thus, vD 0 for the ideal diode. Thus, vD is a rectangular pulse of duration DTs with period Ts .
Fig. 10-2 Buck converter
If capacitor C is large, reasonable approximations are that the time-varying component of iL ows through C and that the voltage across the load resistor RL is constant. Since Q is switched periodically, voltage vL and current iC are periodic once initial transients die out. As a consequence of the above approximations, vL and iL can be appropriately determined by (10.1) and (10.2), respectively, when Q is ON and by (1) and (2) of Example 10.1 when Q is OFF. Figure 10-3 displays the resulting waveforms for diode voltage vD , inductor current iL , inductor voltage vL , and capacitor current iC . The positive volt-second area of vL must be equal in value to its negative voltsecond area so that hvL i 0. As a result of (10.4), current iC must be the time-varying component of iL . Based on (10.3) and the vL waveform of Fig. 10-3, V1 V2 DTs V2 1 D Ts Rearrangement gives the buck converter voltage gain as V GV 2 D V1
10:5
The common case of continuous current iL exists only if L is sized su ciently large. Let L Lc , the critical inductance that results in marginally continuous iL . For this case, Imin 0 in Fig. 10-3. Since iL 0 0, application of (10.2) yields V V2 iL t 1 t 0 t DTs Lc Evaluate for t DTs and use (10.5) to nd iL DTs Imax V1 V2 V 1 D Ts DTs 2 Lc Lc
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