Complex amplitudes and impedances.
The equation
$$
\frac{d^2I}{dt^2} + \frac{dI}{dt} + 5I = \frac{dV}{dt}
$$
governs the response of an RLC circuit with L = 1H, R = 10, C = 1/5 F to an externally imposed voltage
V(t). In the case of a sinusoidally varying voltage V(t) = cos(wt) the equation becomes
$$
\frac{d^2I}{dt^2} + \frac{dI}{dt} + 5I = -\omega sin(\omega t)
$$
One way to solve this equation would be to apply the method of undetermined coefficients in the form
I = A cos(wt) + B sin(wt). It is often easier to complexify the equation, and work instead with a complex
voltage V(t) = e^iwt. This is what we will do in this set of problems. We will use W to denote the solution to
the complexified equation
$$
\frac{d^2W}{dt^2} + \frac{dW}{dt} + 5W = i\omega e^{i\omega t}
$$
We can find the real current I by taking the real part of W: I = Re(W)
Solve the complexified equation by looking for a solution in the form W(t) = Ae^iwt. Find the complex
amplitude A. This will depend on the frequency w. Please use w for w and i for √-1. Be careful that i
does not autocorrect to I.
Find the complex amplitude A
A = symbolic expression
Recall that the absolute value of a complex number z = a + ib is defined to be the length of the
corresponding vector, |z| = √a²+b², and that |z| = a + ib. Find the magnitude of the amplitude A.
Assume that w is positive. You can use (expression)**(1/2) or sqrt(expression) to compute the square root.
Do not use (expression)**(0.5).
Find |A|.
|A| = symbolic expression
The next problem in the worksheet contains a Jupyter notebook. Please modify the notebook to plot the
absolute value of the amplitude as a function of w. What frequency maximizes the amplitude?
What frequency w gives the largest amplitude |A|?
w = number (2 significant figures)
An important measure of the quality of a resonant circuit is the FWHM -- the full width at half maximum. The
FWHM is a measure of the width of the resonance peak at an amplitude level that is half of the maximum
amplitude. The curve that you plotted should have a maximum of 1 and should equal 1/2 at two different
frequency values.
Find the smaller w value where the amplitude is 1/2.
w = number (2 significant figures)
Find the larger w value where the amplitude is 1/2.
w = number (2 significant figures)
The difference between these two quantities is the FWHM. Find the FWHM for your RLC circuit.
w = number (2 significant figures)
