Homework on Circuits and Impedance.

Note. Graphing, figure captions, labels and scales are an important part of this assignment.

1.  Consider a resistor and a capacitor in series driven by a voltage source.
a) First suppose that the system is in equilibrium with no current flowing and no charge on the capacitor for t less than zero. Then at t=0 the voltage jumps up to 5 volts in a very short time. Calculate and graph the voltage across the resistor and the charge on the capacitor as a function of time. Please do a sketched graph drawn by hand with sufficient care to capture what is essential and important, but ideally without excessive detail or a million points calculated. Include a figure caption for your graph as well as labels and scales on all axes.
b) Alternatively, suppose that instead of looking at the response to a step voltage, we imagine that the system is driven by the voltage source with an applied voltage of \(v(t) = V_o cos (\omega t)\). In this driven case calculate the current and the charge as a function of time for the steady state solution (ignore possible transient effects).
c) Graph the driving voltage and the current (through the resistor) as a function of time for a frequency that seems relatively low to you. Indicate the phase relationship for these in your graph.
d) Graph the driving voltage and the current (through the resistor) as a function of time for a frequency that seems relatively high to you. Indicate the phase relationship for these in your graph.
e) Do the same for an intermediate frequency. (Please draw all graphs by hand with sufficient care to capture what is essential and important.)
f) Graph the magnitude of the impedance of this system as a function of driving frequency.
Graph the phase relationship between the driving voltage and the current as a function of driving frequency. Please include a figure caption for your graph as well as labels and scales on all axes.
g) Thinking back to part a) what do you think would constitute a "very short time". Explain. You can use phrases like "characteristic time scale" in your explanation.

2. Consider a resistor and an inductor in series driven by a voltage source.
Do all the same things as in problem 1, parts a-g.

3. Discuss/explain the similarities and differences between your results from 1. and 2.