Calculating Induced EMF in Wireless LED Circuit I have built the following circuit to power a wireless LED and my calculations and measurements do not give the same values. I am measuring a voltage nearly 6 times larger than I would expect in my receiver circuit.


I measured the AC frequency of the circuit to be $\frac{\omega}{2\pi} = f = 350 \ kHz$. Assuming the magnetic field is given by
\begin{align}
B &= \frac{N \mu_0 I}{2R}, \quad I = I_0\cos(\omega t) \approx 0.052A \cos(\omega t),
\end{align}
and I use faradays law of induction, the induced EMF in the receiver circuit should be given by
\begin{align}
\mathcal{E} &= - \frac{d\Phi}{dt} = - A \frac{dB}{dt} \\ 
&= A \frac{N \mu_0 I_0}{2R} \omega sin(\omega t), \quad A = \pi R^2 \\
& = \frac{N  \pi R I_0 \mu_0}{2} \omega \sin(\omega t)  \\ 
& = \frac{(30)\pi (0.05m)(0.052 A) (4\pi \times 10^{-7}H/m)}{2} (2\pi \times 350 \times 10^3 Hz) \sin(\omega t) \\
& \approx 0.339 \sin(\omega t) \text{ Volts }
\end{align}
However when I read the voltage in the receiver, I am actually getting a much larger value of over $1.9 V$.

From searching, I believe this has something to do with resonant frequencies, but I do not understand this. If someone can show me why my calculation is wrong and provide the right one, I would appreciate it!
 A: Take a look at your circuit with an oscilloscope. You will most likely find that the waveforms are not sinusoidal. Every time the transistor turns off, there might be an inductive voltage spike across the transmitter coil with an amplitude that is higher than the supply voltage. Should the circuit run in a sinusoidal mode because your coils have a resonance frequency of 350kHz, then your circuit drawing is wrong. You would instead have to draw the coils as resonance circuits with an inductance L in parallel to a capacitor C. If you have an LCR-meter, measure the inductance L at a frequency far below resonance. Based on the size of your coils it will probably be in the 100uH range (give or take an order of magnitude, I can't tell any better based on what I am seeing). The resonant capacitance value can then be calculated from that inductance and the resonance frequency.
Another way to understand your circuit is with a circuit simulator. LTSpice is free and works very well. It will be able to simulate your entire setup with fairly good precision, assuming that you can extract the inductance and resonance capacitance value of your coils.
