Problems

1. In the following circuit, the input signal is , and the components have been chosen such that and . The output is at the terminals AB.

   

   1. What is the transfer function H( )?
   2. Find the current in the circuit.
   3. What is the voltage drop across each element of the circuit?
   4. Show algebraically that at any instant the potential difference around the circuit is zero.
   5. At time , where N=0,2,4,... make a sketch showing the voltage across each element in the complex plane and show that the vector sum of the voltage drops is equal to the voltage supplied.
   6. Write an expression for .
   7. What is the limit of as goes to zero?
   8. What is the limit of as goes to infinity?
   9. What is the corner frequency?
   10. What is at the corner frequency?
   11. Make a sketch showing the characteristics of on a log-log plot. Label the slope of the curve where possible, the corner frequency, and the value of at the corner frequency.
   12. Describe the high and low frequency behavior in dB/octave.
2. 1. Draw a passive LCR low-pass filter and write down the transfer function of your four-terminal network.
   2. Determine approximations to the transfer function and filter corner frequency(s).
   3. Write the resonance frequency, , in terms of the corner frequency(s).
3. Write down the transfer function, , for the network shown below, and from it find:
   1. the corner frequency(s) and
   2. the value(s) of at the corner frequency(s).
   3. Sketch and the voltage phase-shift as a function of .
   4. What type of filter is this?

   

4. 1. What is the transfer function for the following circuit?

      

   2. Describe the frequency response at low and high frequencies?
   3. Sketch the magnitude on a log-log plot. (label slopes, the corner frequency(s), and )
   4. What is the signal attenuation for .
5. 1. Show that the transfer function of a single-pole RC filter drops by 3 db at the corner frequency. This is often refereed to as the 3 db down-point.
   2. Design a bandpass RC filter with the transfer function shown below.  rad/s and  rad/s are the 3 db down-points of the RC filter sections. Choose impedances so that the first section is not much affected by the loading of the second section.
   3. Are and the 3 db down-points of a bandpass filter?