Assignment: Signals and Systems

1. The transfer function of a circuit is given by

a. Express the transfer function in a form in which the coefficients of the highest power of s are unity in both numerator and denominator.

b. What is the characteristic equation of the system? (Hint: see this week's lecture for a definition of characteristic equation.)

c. Determine the order of the transfer function.

d. Determine where the poles and zeroes of the system are located.

2. The transfer function of a circuit is of the form

a. Express the transfer function in descending powers of s with the coefficients of the highest power of s unity in both the numerator and denominator.

b. Determine where the poles and zeroes of the transfer function are located.

c. Is this the transfer function of a causal system? Justify your answer.

3. A circuit is shown below.

a. Determine its transfer function Vout/(Vin )using the voltage divider method.

b. Using the methods outlined in the lecture, determine the numerical value of the transfer function,as a complex number, at the following three frequencies: DC or 0 radians/sec; 1 radian/sec; finally, as frequency approaches infinity. Express the results in polar form of complex numbers.

c. Express the gain of the circuit at 1 radian/sec in decibels and its phase angle at 1 radian/sec in degrees.

d. Based on the three answers found in part (b), determine what kind of filter (low pass, high pass, bandpass or band stop) the circuit is. (Hint: The value of the transfer function at DC and infinite frequency can give a quick estimate of the type of filter.)

e. Now using MATLAB, create a Bode plot of the transfer function of the circuit and paste it below. Does the Bode plot produced by MATLAB confirm your finding of the type of filter the circuit is? Briefly justify your answer.

4. A RLC circuit is shown below.

a. Determine the transfer function of the circuit Vout/Vin using the voltage divider rule and write it with the coefficient of the highest power of s in the denominator, unity.

b. Determine the numerical value of the transfer function, as a complex number, at DC; at 1 radian/sec, and as frequency approaches infinity.

c. Determine what kind of filter (low pass, high pass, bandpass or band stop) the circuit represents. Briefly justify your answer. Confirm by computing the Bode plot using MATLAB and paste the graph below.

d. Using Laplace transform methods you learned last week, determine the response of the circuit, Vout(t), if Vin(t) is a unit step, and calculate the numerical value of Vout(t) at t =1 second.

e. Is this a stable system? Why or why not? (Briefly Justify your answer.)
System is stable

5. Determine the location of the poles and zeroes and the stability of the following two transfer functions. Express the location of the poles and zeroes in both rectangular and polar form.

a. H(s)= (s^2- √2 s +1 )/(s^2+ √2 s +1)

b. H(s)= (s^2+ √2 s +1 )/(s^2- √2 s +1)

Using MATLAB, plot the pole zero map and the Bode plot of the two transfer functions and paste the graphs below. Identify and briefly discuss the differences between the Bode plot of the two transfer functions.

c. Which of the two is a stable system and why?