Question

Problem 31: (34 points) 1. (10 points) A pulse width modulated (PWM) signal fPwM(t) in Figure 2. The symbol D represents a duty cycle, a number between zero and one. Determine the compact trigonometric Fourier series coefficients (Co Cn, θ.) of the signal f(t) 2. (10 points) One use of PWM is to generate variable DC voltages. While the PWM signal is not DC, you should be able to see from your results in part 1 that it has a DC component that depends on the duty cycle, D. If the voltage is applied to a system that is low pass in nature, it will respond to the DC component of the signal PWM(t) DTo T 0 Figure 2: A pulse width modulated signal. while being largely unaffected by the other frequency components. Assume the PWM signal is applied to the low-pass system represented by the frequency response function where the time constant τ. Show that the sinusoidal steady-state output y(t) can be expressed as Cn (t) Co

Answer 1

Given: The Given fig. is. Pulse with modulation (PWM) is a modulation Process or technique used in most communication system for encoding the amplitude of a signal right into a Pulse width or duration of another signal, usually a Carrier signal, for transmission. A Pulse width modulated (PWM) signal f_PWM (t). \r\n The signal D Represents a duty cycle, a number between zero and one. One of PWM is to generate voltage variable DC voltages. If the voltage is Applied to a system that is low pass in nature. We have to determine compact trigonometric Fourier series coefficients. C_0 = a_0: C_n = squareroot a^2_n + b^2_n theta_n = tan^- 1 (- b_n/a_n) \r\n and: a_n = 2/T_0 integral^T_0_0 f (t) cos n omega_0 t dt = 2/T_0 integral^DT_0_0 A cos n omega_0 t dt = 2 A/T_0 [sin omega_0 t/n omega_0]^DT_0_0 = 2 A/T_0 [sin omega_0 DT_0/n omega_0] and, omega_0 = 2 pi/T_0 = > omega_0 T_0 = 2 pi