Switching amplifiers provide an efficient way to generate a time-varying output voltage from a fixed supply...

 

Transcribed Image Text:

Switching amplifiers provide an efficient way to generate a time-varying output voltage from a fixed supply voltage. They are particularly useful in high-power applications. The basic idea is that an output voltage v is generated by rapidly switching between zero and the supply voltage A> 0. The output is then lowpass-filtered to remove the harmonics generated by the switching operation. For our purposes, consider the lowpass filter as an integrator over many switching cycles, so the output voltage is the average value of the switching waveform. Varying the switching rate varics the output voltage. In this problem, we only consider the case where the desired output voltage is constant. We can apply the Fourier series to analyze this system. If the output pulses are spaced by To, the waveform the amplifier generates immediately before the lowpass filter is ało -2To -To To 2To t Denote the Fourier coefficients of this waveform by Xk, k integer. The duty cycle of the switching amplifier is a, and the width of the pulses is aTo. When a = 1, the amplifier is constantly on and produces its maximum output A for all t. (a) Reducing the duty cycle reduces the output voltage. After the lowpass filter, only the DC coefficient (time average mean) xo remains, which will be the output voltage. Find the value of xo as a function of the duty cycle a. If the desired output voltage is v and the supply voltage is A, what should a be? (b) The lowpass filter must suppress (average out) the harmonics generated by the switching waveform. The first harmonics X1 and X₁ are the most difficult to sup- press because they tend to be large and are closest in frequency to the desired DC component. Find an expression for the complex amplitude of the first harmonic X₁ as a function of the duty cycle a. (c) What duty cycle a results in the first harmonic having the largest magnitude, i.c., largest x₁]? Keep in mind that 0 ≤ a ≤ 1. Switching amplifiers provide an efficient way to generate a time-varying output voltage from a fixed supply voltage. They are particularly useful in high-power applications. The basic idea is that an output voltage v is generated by rapidly switching between zero and the supply voltage A> 0. The output is then lowpass-filtered to remove the harmonics generated by the switching operation. For our purposes, consider the lowpass filter as an integrator over many switching cycles, so the output voltage is the average value of the switching waveform. Varying the switching rate varics the output voltage. In this problem, we only consider the case where the desired output voltage is constant. We can apply the Fourier series to analyze this system. If the output pulses are spaced by To, the waveform the amplifier generates immediately before the lowpass filter is ało -2To -To To 2To t Denote the Fourier coefficients of this waveform by Xk, k integer. The duty cycle of the switching amplifier is a, and the width of the pulses is aTo. When a = 1, the amplifier is constantly on and produces its maximum output A for all t. (a) Reducing the duty cycle reduces the output voltage. After the lowpass filter, only the DC coefficient (time average mean) xo remains, which will be the output voltage. Find the value of xo as a function of the duty cycle a. If the desired output voltage is v and the supply voltage is A, what should a be? (b) The lowpass filter must suppress (average out) the harmonics generated by the switching waveform. The first harmonics X1 and X₁ are the most difficult to sup- press because they tend to be large and are closest in frequency to the desired DC component. Find an expression for the complex amplitude of the first harmonic X₁ as a function of the duty cycle a. (c) What duty cycle a results in the first harmonic having the largest magnitude, i.c., largest x₁]? Keep in mind that 0 ≤ a ≤ 1.