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This circuit employs pulse frequency modulation (PFM) in a cell phone, to power a 1 V micro-processor (modeled by resistor R) from a single-cell lithium ion battery (modeled by voltage

source V, = 3.6 V). The wave forms below illustrate how this converter is operated in discontinuous conduction mode (DCM) with variable switching frequency. At the start of a switching period, MOSFET Q1 is turned on. At the time when the inductor current reaches Ipk, a control circuit turns Q1 off and synchronous rectifier Q2 is turned on.Transistor Q2 is turned off when the inductor current reaches zero. Both Q, and Q2 are off for the remainder of the switching period. To regulate the output voltage to 1 V, a controller (not shown) adjusts the switching frequency f; = 1/T,. The variable-frequency control method described here is know as peak-current controlled Pulse-Frequency Modulation (PFM). This approach causes the converter losses to vary in proportion to output power, leading to constant(high) efficiency over a wide range of load powers. This is an essential feature in portable battery-powered applications, where it is important not to drain the battery in sleep mode. You may assume that all components are ideal in this problem, and hence neglect losses. 5. Find an expression for the switching frequency fs as a function of • Inductance L • Peak inductor current Ipk • DC output current I, • DC output voltage V • Conversion ratio M where M = V/Vg. Enter your expression below. You should find that fs is proportional to Io.

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