Question

Resonant wireless charging is an emerging technology where a powered transmitting circuit induces a current in a receiving circuit by the magnetic interaction between the circuits' inductors. Similar to how transformers work, what makes this process unique is that the receiving circuit can be several centimeters to a few meters away from the transmitting circuit, depending on the system. This induction with distance works only when both circuits, which can be treated like effective LRC circuits, operate at the same resonant frequency. One medical benefit of this technology is the ability to remotely charge an internal battery in a medical implant. The figure shows a graph of the induced current in such a medical implant (the receiving LRC circuit) that is operating at its resonant frequency. What is the rms value irms of the current? 15 18 21 24 27 30 33 36 39 42 -3- -6 t (ms) irms = 4.24 nA What is the implant's resonant frequency 00? 00= rad/s

Answer 1

The following data can be obtained from the graph.

Maximum current, I₀ = 6 mA

Time period of the signal, T = 24 ms = 0.024 s

1.

Rms current, I_rms = I₀ /√2 = 6/1.41 = 4.25 mA

2.

Frequency, f = 1/T = 1/0.024 = 41.67 Hz

Angular frequency, ω₀ = 2πf = 2 x 3.14 x 41.67 = 261.7 rad/s

3.

Resonant frequency, ω₀ = 1/( LC)^1/2 = 1/(0.055 x 3.25 x 10^-6)^1/2 = 2365.7 rad/s

4.

Maximum current at resonance, I_max = V/R = 84/855 = 0.098 A

(impedance is equal to resistance at resonance)

5.

Resonant frequency, ω₀ = 1/( LC)^1/2

Inductance, L = 1/ C ω₀² = 1/ 4.25 x 10^-6 x 39700² = 0.000149 H = 0.149 mH