The main objective in lab this week is to learn about the impulse response of continuous-time, linear, time-invariant systems. You will determine the impulse response of an RC circuit two ways, analytically and experimentally. Then you will use the analytical unit impulse response expression in Matlab (or another computer programming environment) to simulate the unit step response of the RC circuit, using the properties of linearity and time invariance. Please work in pairs on these lab exercises.

1. Consider a series RC circuit driven by a voltage source, with the output voltage measured across the capacitor. We will use R = 10 kohms and C = 0.1 microfarads.
   • Analyze this circuit and derive the expression for the unit impulse response, h(t), which is defined as the output that occurs when the unit impulse function, delta(t), is applied at the input. In your analysis, consider applying a rectangular pulse that gets briefer and briefer while maintaining a constant area. (Recall that Prof. Kozick did this in lecture, but you should repeat the analysis and make sure you understand all of the steps.)
   • Devise a procedure to experimentally measure the impulse response of the circuit.
     Hint: Use the pulse-mode of the function generator and adjust the amplitude, duty cycle, offset, and frequency appropriately to make pulses that are much shorter than the time constant. Try using pulses with area on the order of 0.0001 volt-seconds.
   • Measure the impulse response of the circuit using the oscilloscope. What happens to the measured impulse response when you reduce the input duration by one-half and double the amplitude, so that the area remains constant?

2. Next we consider the unit step response of the RC circuit.
   • Obtain the analytical formula for the unit step response. You did this in ELEC 225-226 in the context of "closing a switch" and solving the differential equation. This will give you a chance to review how to solve those problems.
   • Measure the unit step response by applying a square wave input to the circuit, and compare with your analytical formula.
   • Write a Matlab (or other) program to simulate the unit step response using the unit impulse response. You can do this by dividing the unit step function into a sum of rectangles, where each rectangle is approximated by an amplitude-scaled and time-shifted impulse function. Then use the fact that the RC circuit is a linear and time-invariant system. In Matlab (or Excel), plot the numerically approximated unit step response and compare it with the analytical unit step response on the same graph.

     You may use a programming environment other than Matlab (such as Java, Excel, or something else) if you prefer, but we may not be able to provide as much help with other environments.

Thank you.