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So my question is about the type of relationship in a particular circuit. I would like to know if it is linear or something else like a power.

The circuit in question is a pair of adjacent, mutually inductive coils placed in OPPOSITE directions, and the discharge of a capacitor through them.

schematic

simulate this circuit – Schematic created using CircuitLab

With an initial voltage on the capacitor, what would the voltage function look like at Point A? (linear, summation of logs, etc.) What would the voltage function look like if a turns ratio of something other than 1:1 was used? (a mathematical explanation would also be appreciated.)

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If there is perfect coupling between the two coils they will cancel out any inductance so it will act as if it has none. If it has no resistance either there will be a short across the capacitor.

In this ideal circuit there will be infinite current and no losses which does not give a meaningful result.

This type of inductor is typically called a common-mode choke and most devices powered from AC use them to minimize conductive interference.

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It depends on Mutual Coupling Coefficient (0.99xx?) and primary inductance even if perfectly balanced windings.

  • This coupling k controls the "Leakage Inductance" in each winding.
  • Shorting one winding and measuring the other gives you the other winding leakage inductance

Since the primary and secondary inductance cancels out with perfect matched winding , it will still have a leakage inductance from imperfect coupling that results in an impedance acting like a LPF at some high frequency.

f-3dB 45 deg breakpoint will be;

k = coupling factor 
L = 10 mH, C = 1 uf    L = 10 uH, C = 1 uF  L = 10 uH, C = 1 nF
 k    f-3dB 45 deg     f-3dB                f-3dB 
===   =======          =======              =======
0.9   200 Hz           200 kHz              200 MHz
0.99  2000 Hz          2 MHz                N/A
0.999 20 kHz           20 MHz

Now changing the turns ratio

0.999 3.3 kHz @ 1 : 1.1 ratio or 1 : 0.9

Other

  • This is like taking a long magnet wire of some 10nH/cm and folding in half then twisting it and using the end points. So the mutual coupling is high and primary secondary inductance cancels out but the leakage inductance is much lower.

  • The results are different for a resistive load where the impedance matching is possible on primary and secondary sides and then the null inductor has a wider BW.

    • For an RF directional coupling made from a hybrid balanced transformer like those used in 3dB TV splitters for 75 Ohm cable to give isolation between the 2 ports with matched impedances.

enter image description here or enter image description here

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