What is the ratio between the impedance of a wireless radio receiver when receiving a signal with frequency 𝑓 and the impedance of the same radio receiver when receiving a signal with frequency two 𝑓?
Recalling the general formula for the impedance, 𝑍, of a circuit, we see it has to do with the circuit’s resistance, 𝑅, as well as its inductive and capacitive reactance. However, there’s an important clue in our problem statement that helps us simplify the scenario.
We’re working with a wireless radio receiver, which means that this receiver can be and is tuned to particular frequencies it wants to receive. When the receiver is tuned to resonate at the same frequency as the incoming radiation, that means the frequency running through the receiver’s circuit is equal to one over two 𝜋 times the square root of the inductance times the capacitance of the circuit. When the receiver is at resonance, that is, at 𝑓 sub zero, mathematically, that means that the inductive reactance is equal to the capacitive reactance.
Therefore, this term in parentheses for the impedance is equal to zero, and the overall circuit impedance is simply 𝑅, the resistance. We can say that when the radio receiver is tuned to receive a signal with frequency 𝑓, at that point, its impedance is equal to its resistance. And then when the receiver is tuned once more now to receive a frequency of two 𝑓, once again, it experiences resonance at a different frequency such that its impedance is equal to its circuit resistance. All this implies that the ratio of circuit impedance in the receiver when the frequency is 𝑓 to when the frequency is two 𝑓 is equal simply to one. That’s this ratio from 𝑓 to two 𝑓.