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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.

My circuit Diagram My 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 $1.9 V$.

Voltage in Receiver

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!

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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.

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  • $\begingroup$ The fact that it is not a sine wave may indeed be the culprit in my faulty calculation. A voltage spike would make $|\frac{dB}{dt}|$ larger. If I was able to measure the inductance, 'L' of my coils, is there a way that I could obtain an analytical solution of the waveform without an oscilloscope? $\endgroup$
    – Dayton
    Oct 21, 2022 at 21:00
  • $\begingroup$ Analytical as in closed form? Probably pretty close, if you are willing to make the approximation of the transistor as an ideal switch, but since your supply voltage is low that may not be too good an approximation. I would simply do a circuit simulation. You can get the desired result in a few minutes in numerical form which for electronics is almost always sufficient. $\endgroup$ Oct 21, 2022 at 23:30
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    $\begingroup$ @Dayton Just for completeness, I did a circuit simulation and the waveforms are not even close to sinusoidal. I am getting short voltage spikes at the collector of the transistor that are tens to hundreds of Volts, depending on the feedback inductance and resistance. With these component values the circuit does not behave like a harmonic oscillator, at all. $\endgroup$ Oct 22, 2022 at 10:17

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