i L = VT L (16.34) 918a i D v OUT i L v C V - S 1 FIGURE A switched power supply circuit with diode and a switch.
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1 A SWITHED POWER SUPPY USINGA DIODE In his example, we will analyze he behavior of he diodebased swiched power supply circui shown in Figure Noice ha his circui is similar o ha in Figure 12.41, wih he swich S 2 replaced wih a diode. As before, he purpose of he circui is o conver he D inpu volage o a differen D oupu volage. The MOSFET in he circui operaes as a swich, and he squarewave inpu o he MOSFET is shown in Figure As before, we are ineresed in deermining he behavior of over ime. As we will see shorly, he diode in he circui also acs a swich, and resuls in an oupu waveform ha is largely he same as ha of he circui in Figure We will assume ha he swich has zero resisance associaed wih is ON sae, and ha he diode is ideal, so ha he model in Figure 16.6 applies. Specifically, his means ha he diode urns on and behaves like a shor circui when a posiive curren (i D ) flows hrough i. The diode urns off and behaves like an open circui when he volage (v D ) across i is negaive. When he swich is closed, i shors he erminal connecing he diode and he inducor o ground. Assuming ha is nonnegaive, he diode being reverse biased is off. The D volage appears direcly across he inducor as illusraed in Figure 16.17, and he inducor curren i ramps up. Since is he on for ime T, he inducor curren builds up o i = T (16.34) as shown in Figure Meanwhile, if here is no applied load a, he capacior volage remains consan. Nex, when is opened, he inducor curren canno insananeously go o 0. Insead, he curren finds a pah hrough he diode (hereby urning i on) and ino he capacior. In is ON sae, he diode behaves like a shor circui, and so he driven circui shown in Figure resuls. The curren i in he i i D v D FIGURE A swiched power supply circui wih diode and a swich. 918a
2 Sae T P One cycle OSED (ON) OPEN (OFF) T Sae OFF ON OFF FIGURE Swiched power supply operaion. i T 0 [n 1] [n] 0 circui follows a sinusoidal paern as illusraed in Figure Because of he flow of curren ino he capacior, is volage sars o increase, and i oo follows a sinusoidal paern. As i follows is sinusoidal paern, i soon reaches zero and he posiive volage on he capacior aemps o drive i negaive. A his insan, he diode urns off and disconnecs he capacior from he res of he circui, so in he absence of a load, he capacior mainains is volage. 918b
3 i FIGURE The equivalen circui when is closed and he diode is open. i i D FIGURE The equivalen circui when is open and he diode is ON. This cycle repeas, dumping some amoun of charge ino he capacior each cycle. We can compue he increase in very quickly using an energy argumen similar o ha used in Example 12.4 as follows: A he end of he ramp, he inducor curren is given by Equaion 16.34, and so he energy sored in he inducor is given by: w M = 2 T 2 2. Since he capacior is charged by he inducor unil i becomes zero, he energy (w M ) sored in he inducor is ransferred compleely o he capacior in each cycle. Afer n cycles, he energy sored in he capacior becomes n imes he energy ransferred in a single cycle, plus any energy iniially sored on he capacior (say w E [0]): w E [n] =n 2 T 2 2 w E[0]. Unlike Example 12.4, he capacior mus sar wih =, since i is conneced by a diode insead of a swich o a volage source. Unlike he swich, which can 918c
4 be forced o say off, he (ideal) diode urns on if is greaer han. Therefore, w E [0] = Since w E [n] = [n] 2 /2, we can derive he volage afer n cycles as: nt [n] = 2 1. Subsiuing, ω o = 1/, we have [n] = nt 2 ωo 2 1. If nt 2 ω 2 o 1, we ge [n] =Tω o n. Finally, when a load is added o he circui as shown in Figure 16.19, he capacior begins o discharge hrough he load. Suppose we wish o mainain he volage a a specified average value, say v REF, hen in each cycle, we mus arrange o have he capacior charged up by he same amoun of charge ha i supplies o he load. This can be accomplished by using a feedback sysem as shown in Figure In he circui in Figure 16.20, he conroller compares o v REF, and if falls below v REF, i increases he duraion T for which he swich is kep ON, hereby increasing. onversely, he conroller decreases he duraion T if increases pas he value of v REF. Thus, is kep close o v REF hroughou. i FIGURE Adding a load. R 918d
5 i R v REF onrol change T FIGURE Feedback sysem o mainain a volage v REF a he load. T T P 918e
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