Tetrahedron impedance planar network -- What is the phase angle between nodes 1 & 2?

This is an AC circuit operating at a frequency of 1 MHz 2 π \frac{1\text{MHz}}{2\pi} . The inductors are 1 mH and the capacitors are 1 nF.

Simplifying assumptions: the components are pure, ideal components, there is no mutual inductance between the inductors and the wiring is negligible.

The device impedance formulae are simple and the frequency was chosen to give integer impedance magnitudes, in the sense that a unit circle in the complex plane has a magnitude of 1 1 around the entire circle. The standard unit for inductors is a Henry and for capacitors is a Farad. Convert the units used above to those units before computing the device impedances.

Kirchhoff's circuit laws and some matrix algebra (simultaneous equations) work should yield the result.


The answer is 0.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

With the impedances entered as conductances, the graph is:

The computation method: resistance = With [ { Γ = PseudoInverse [ With [ { wam = WeightedAdjacencyMatrix [ $#$1 ] } , DiagonalMatrix [ Tr/@ wam T ] wam ] ] } , Outer [ Plus , Diagonal [ Γ ] , Diagonal [ Γ ] ] Γ Γ T ] & ; \text{resistance}=\text{With}\left[\left\{\Gamma =\text{PseudoInverse}\left[\text{With}\left[\{\text{wam}=\text{WeightedAdjacencyMatrix}[\text{\$\#\$1}]\},\text{DiagonalMatrix}\left[\text{Tr}\text{/@}\text{wam}^T\right]-\text{wam}\right]\right]\right\}, \\ \text{Outer}[\text{Plus},\text{Diagonal}[\Gamma ],\text{Diagonal}[\Gamma ]]-\Gamma -\Gamma ^T\right]\&; .

The result: ( 0 500 250 + 250 i 250 250 i 500 0 250 250 i 250 + 250 i 250 + 250 i 250 250 i 0 500 250 250 i 250 + 250 i 500 0 ) \left( \begin{array}{llll} 0 & 500 & 250+250 i & 250-250 i \\ 500 & 0 & 250-250 i & 250+250 i \\ 250+250 i & 250-250 i & 0 & 500 \\ 250-250 i & 250+250 i & 500 & 0 \\ \end{array} \right) .

The impedance between nodes 1 and 2, the 500 in the second entry of the first row is resistive, the phase angle is 0.

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...