Engineering Applying symmetry to an asymmetrical circuit

[Baluncore]

I have a broad-band solution, where the load is a voltage source, the negative, (or a 180° phase shift), of the input voltage.
I have a narrow-band solution, where the load is simply an equal but negative inductance.
I await the KL solution.

 

[The Electrician]

I have a broad-band solution, where the load is a voltage source, the negative, (or a 180° phase shift), of the input voltage.
I have a narrow-band solution, where the load is simply an equal but negative inductance.
I await the KL solution.

By "negative inductance" do you mean capacitance? What is the negative inductance "equal" to? Do you have an expression for it at some particular frequency, or an expression involving frequency as a variable?

Show us some details of the solution.

 

[DaveE]

I await the KL solution

What's KL? Kirchhoff's Laws?

The phase inversion you are stipulating is only valid for a specific load impedance that makes the L2 voltage zero. Otherwise it's a mess.

Here's my worksheet. Once I started with the Z parameters, I didn't check the results very carefully, so there could be a mistake in the algebra. Also I haven't included the rough sketches of the network under the various Z-parameter conditions (I₁=0, I₂=0, etc.), ask if you need to see those; I'll throw something together. Frankly, I have lost interest at this point. The original question about that specific load is easy, the general solution for any load is hard to simplify, and not the way real EEs would have to solve this.

 

[Baluncore]

By "negative inductance" do you mean capacitance? What is the negative inductance "equal" to? Do you have an expression for it at some particular frequency, or an expression involving frequency as a variable?

Yes. Negative inductance, has negative reactance that makes it a capacitance.

Since L1 is connected between the zero impedance source, and the load, I just wondered what would happen if the load canceled all of L1. When I tried it with -L1, I found a deep null, so I transformed the -L into +C at the frequency of the null, but made a numerical error and got a +C causing a deep null at a different frequency. Fixing the numerical value put the null back in place, revealing that for a point frequency, the load solution would probably be a pure capacitance.
I was expecting that a KL solution would produce that equation.

Now I am also left wanting to know what that frequency to capacitance relationship is, and if there are bounds to the range of frequency that might hide the relationship in a complexity. I do expect it will be a simple frequency transformation, when I have time to look at it.

And yes, KL is Kirchoff's Laws, KVL & KIL.

 

[The Electrician]

Since the voltage across L2 is zero, the current in L2 is also zero. Since the junction of C1 and C2 is at zero volts, the voltage across each capacitor individually must be the same. That's where symmetry comes in. Obviously the voltage across C1 is just V1, the supply voltage. So the voltage across C2 must be -V1 in order that the voltage at the junction be zero. So we reason like this:

Pick a frequency and get values for the two impedances, evaluate the expression and get the impedance of a capacitor.

 

[The Electrician]

A full bore nodal analysis could also be done. Pick a frequency and evaluate the expresion for ZL and get the impedance of a capacitor. It also just so happens that value of ZL = infinity is a solution.
This all assumes the the voltage V1 is 1 volt RMS.

The very last line above should say that the ZL port voltage equals -V1.