Two black boxes display the same impedance at all frequencies. The first contain
ID: 1535457 • Letter: T
Question
Two black boxes display the same impedance at all frequencies. The first contains a single 1 Ohm resistor. Each end is connected to a wire, so that two wires protrude from the box. The second box looks identical from the outside, but inside there are 4 components. A 1 F capacitor is in parallel with a 1 Ohm resistor, and a 1 H inductor is in parallel with the other 1 Ohm resistor. The RC combo is in series with the RL combo, as shown in the figure below:
The boxes are painted black, unbreakable, impervious to x-rays, and magnetically shielded.
a) Demonstrate that the impedance of each box is 1 Ohm at all frequencies.
What measurement would allow one to determine which box contains the single resistor?
Explanation / Answer
capacitive reactance of capacitor=1/(s*C)=1/s
inductive reactance=s*L=s
then total impedance of RC combo=(1/s)*1/((1/s)+1)
=1/(s+1)
total impedance of RL combo=s*1/(s+1)=s/(s+1)
total impedance of the series combination=(1/(s+1))+(s/(s+1))
=1
so impedance is 1 ohms for all the frequencies. (proved.)
part b:
by connecting a DC supply and looking at the current waveform
we can easily determine which box has single resistance
(in case of box containing RC components, there will be initial transient
before settling to steady state)