Problems

1. Consider the circuit below. (You may assume that the op-amps are ideal.)

   

   1. Write an expression for the transfer function . Express your result in terms of the amplitude of the output and the phase relative to the input. Let  k ,   ,   F and   F. Do not simplify the algebra.
   2. What are the (real) zeroes in the transfer function, if any?
   3. What are the (real) poles in the transfer function if any?
   4. Sketch the transfer function as a function of on a log-log plot. Your sketch should show the slope of in the large and small limits, the corner frequencies, and the value of at the corner frequencies.
   5. Describe the dependence of the output on frequency at small and large frequencies in dB/octave.
2.  
   1. Write the two rules for the analysis of circuits which utilize ``ideal'' op-amps.
   2. Write an expression for the potential for the following circuit.

      

   3. Let  k and  k . Sketch a log-log plot showing the function for the above circuit assuming that a general purpose op-amp such as the 741 is used.
   4. The 741 op-amp has a corner frequencies of 4 Hz, DC open-loop gain of and a fall off at high frequency of 6 dB/octave. What is the frequency domain over which the amplifier defined in part (c) will have constant gain? What is the gain of the amplifier in this frequency domain?
   5. Suppose now that the impedance is replaced with a capacitor with a capacitance C (  k ). For frequencies much greater than 4 Hz, the 741 op-amp will attenuate the signal if the product RC is greater than some maximum value. For a frequency of 50 kHz, what is the maximum value for C for which the op-amp will not attenuate the output signal at high frequencies? Hint: If you sketch and , you will be able to see the constraint on .
3. Set up an operational amplifier circuit to solve the equation

   

   Hint: the input is .

4. Draw a schematic diagram of a circuit for which . Specify the limitations on the input voltage range, if any.

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