HW3F09 - Problem 2.16, P. T. Krein, Elements of Power...

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Problem 2.16, P. T. Krein, Elements of Power Electronics . New York: Oxford, 1998. This problem defines integral cycle control. The switching function q(t) is on for a cycle of Vin, then off for a cycle, and so on. If the switching function is sketched by hand, we notice three things: 1. The frequency of q(t) is 30 Hz. 2. The duty ratio is 1/2. 3. The phase is not zero. Instead, the center of the pulse occurs at t = t 0 = 1/240 s. Since the switching function radian frequency is 2 π 30 rad/s, and since the phase is defined as φ 0 = ω t 0 , we can compute that φ 0 = 2 π 30/240 = π /4 radians, or a 45 degree delay. This can be checked by using it to plot the results: V0 5 := (Set to 5 for easier-to-read plots.) vin t ( ) V0 cos 2 π 60 t () := qt ( ) if cos 2 π 30 t π 4 0 > 1 , 0 , := vout t ( ) vin t ()qt := Now, some plots: tlast 0.1 := i 0 2000 .. := t i tlast 2000 i := 0 0.02 0.04 0.06 0.08 0.1 6 4 2 0 2 4 6 vin t i i t i 0 0.02 0.04 0.06 0.08 0.1 6 4 2 0 2 4 6 vout t i t i
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The correctness of the plots confirms that the right switching function was used. The switching function Fourier Series can be written directly, q(t) = 1 2 2 π 1 n sin n π 2 n cos n 2 π 30 t n π 4 = + Now, does this give the desired conversion? The symmetry suggests it does, but the non-zero phase raises some uncertainty. The question is whether a 30 Hz component appears in the output voltage. The dc term in q(t) does not do the job. What about various values of n? vout nt , ( ) cos 2 π 60 t () 2 π sin n π 2 n cos n 2 π 30 t n π 4 := This is not bad to check directly. We need the trig identity (A.5), which shows that the product of cosines gives a sum frequency term and a difference frequency term.
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This note was uploaded on 11/02/2009 for the course ECE ece464 taught by Professor Abc during the Spring '09 term at Jacksonville College.

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HW3F09 - Problem 2.16, P. T. Krein, Elements of Power...

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