Objective: In this lab, you will add to your understanding of analog filters by designing a circuit to meet a frequency response specification. We will build the circuit and compare the measured frequency response with the analytical frequency response and the specification. We will also work with circuits that perform the operations of integration and differentiation. All of the circuits in this lab may be useful in the design project that we will do in April.

Filter Design Problem: (Please see Section 15.1 in the textbook for useful circuits and the Insights for Filter Design notes.)

• Design a first-order, low-pass filter with a passband gain of -5 volt/volt at low frequencies and a cutoff frequency of approximately 500 Hz.

Please do the following.

1. Derive the analytical expression for the magnitude of the frequency response of the filter. Explain how this formula is used to choose the circuit component values to achieve the design specification.

2. Build the circuit and measure the frequency response. Be sure to measure the cutoff frequency and compare it with the design specification. Make a (computer-generated) plot that shows the analytical and measured frequency response magnitude as a Bode plot with gain in dB and logarithmic frequency axis (see Section 15.1). Choose the frequency range so that your plot shows the passband, cutoff frequency, and -20 dB per decade slope in the stopband.

3. Prepare a one-page summary that shows the circuit diagram with analytical frequency response magnitude (these can be handwritten) and the Bode plot. Indicate the cutoff frequency on the Bode plot (also can be handwritten).

Integrator and Differentiator Circuits:

1. Analyze the integrator and differentiator circuits shown below. That is, determine the relationship between the output voltage v_o(t) and the input voltage v_i(t). Please refer to the Chapter 6 Review Notes if necessary.
   

2. Simulate the integrator circuit in PSpice using values R₁ = 4.7 k ohm, C = 0.1 micro F, and the uA741 op amp. Use a square pulse for the voltage source (VPULSE) with period 0.5 ms and values 0 V and 1 V.

   Does the circuit work? If not, what is the problem, and how might you fix it? For what range of frequencies does your circuit perform integration? You may want to test the circuit by connecting the input voltage to ground (0 V).

3. Set up the integrator circuit using values R₁ = 4.7 k ohm and C = 0.1 micro F. Test the circuit with the following input signals, each with frequency 2,000 Hz:
   Square wave, triangle wave, sine wave, and sawtooth wave.
   Use your analysis to decide what range of input amplitudes will work.

4. Set up the differentiator circuit with R₂ = 4.7 k ohm and C = 0.1 micro F. For what range of frequencies does your circuit perform differentiation? Do you need to modify the differentiator circuit to make it work? What happens at high frequencies?

5. No lab report is required for the integrator and differentiator circuits, but please discuss your results with the instructor or lab assistant. We will revisit these circuits in future labs, so you may want to leave them connected on your breadboard.