Brandeis University | Physics 29a |

Fall 2016 | Kevan Hashemi |

**Note:** You may derive all necessary equations outside the laboratory sessions (see Parts 4 and 5).

**Part 1:** Construct the following circuit. Use your function generator to generate a square wave for *X*. Use 10 kΩ for R1 and a photo-resistor for R2. Its resistance decreases as we shine light upon it.

Connect a ×10 probe to *X* and display *X* on your oscilloscope. Use a function generator to apply a 10-kHz, 0-5 V square wave to *X*. For what fraction of the period is the square wave at its high-level voltage? This fraction is the *duty cycle* of the square wave. Adjust the duty cycle to 50%. With another ×10 probe, display *Y*. Measure the amplitude of *Y* with the photo-resistor in light and dark. What is the resistance of the photo-resistor in light and dark?

**Part 2:** Construct the following circuit. Use 1 kΩ for R and choose C to give RC ≈ 100 μs.

Drive *X* with a 100-Hz square wave and sketch the rising and falling edges of *Y*. How long does it take *Y* to move through 63% of its full range after one of the square wave edges? This is the *rise time* of *Y*. Increase the frequency to 1 kHz. What is the amplitude of waveform on *Y*? Sketch one complete cycle of this waveform. Do the same at 10-kHz. Is there any difference between the response of your circuit and a 10-kHz sine wave?

**Part 3:** Continuing Part 2, choose C for RC ≈ 1 μs. Sketch the rising edges of *X* and *Y* and measure their rise times. Does the rise time of *Y* agree with your calculation? What does the rise time of *X* imply about the maximum frequency square wave the function generator can produce?

**Part 4:** Consider the following ciruit. In your laboratory notebook, derive an equation for the step-response of *Y*. You may assume all currents and voltages are zero before the step, and no current leaves the circuit through the *Y* output.

Construct the circuit with 10 mH for L and 1 kΩ for R. Calculate the time constant of the step response for these values. Apply a 100-Hz square wave to *X* and sketch the step responce of *Y*, marking amplitude and fall time. Change R to 100 Ω. Sketch and measure *Y* again. Try to understand any significant difference between your measurements and your calculations.

**Part 5:** Consider the following circuit for the special case R1 = R2 = R and C1 = C2 = C. In your laboratory notebook, derive an equation for the step response of *Y*.

Build the circuit with R1 = R2 = 1 kΩ, and C1 = C2 = 10 nF. Sketch the step response of the circuit. How does your measurement compare to your calculation?

**Part 6:** Build the following circuit with L = 10 mH, C = 10 nF, and R1 = R2 = 1 kΩ.

Measure the step response at *Y*. Why does the step response overshoot its final value?