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# Circuit Behavior

Any circulating flow constitutes a circuit. Learn how to model the logic boards in your computer, the flow of nutrients in your blood, or the daily fluctuations in the temperature of your house.

Suppose there is a box containing an ideal sinusoidal AC voltage source (RMS magnitude of \(V\)) in series with a resistance \(R\) and an inductive reactance \(X\) (for a total complex impedance \(Z = R + jX)\). The circuit is incomplete, and a pair of terminals are brought outside the box. Aside from the availability of the terminals, the box is closed and its contents are inaccessible.

You want to find out the values of the resistance and inductive reactance, so you devise a clever strategy to determine them. You take three measurements, each time connecting an ideal AC ammeter in series with a test impedance \(Z_T\) and connecting the series combination across the box terminals.

For the first measurement, \(Z_T\) is a perfect short-circuit with zero impedance, and the measured RMS current magnitude is \(\Large{\frac{V}{\sqrt{\alpha}}}\).

For the second measurement, \(Z_T\) is a \(1 \Omega\) resistance, and the measured RMS current magnitude is \(\Large{\frac{V}{\sqrt{\alpha+5}}}\).

For the third measurement, \(Z_T\) is a \(1 \Omega\) inductive reactance, and the measured RMS current magnitude is \(\Large{\frac{V}{\sqrt{\alpha+9}}}\).

What is the value of \(R + X\)?

**Note:** In electrical engineering, the letter \(''j''\) is commonly used to denote \(\sqrt{-1}\), since the letter \(''i''\) is commonly used to represent current.

A \(\pi\)-circuit consists of two shunt capacitive impedances of \(-j \Omega\) and one series inductive impedance of \(j \Omega\).

An ideal AC voltage source feeds the circuit with \(\vec{V}\) volts and \(\vec{I}\) amps.

What is the value of the load impedance \(\vec{Z}?\)

**Note:** Regardless of which expression you pick, assume that \(\vec{Z}\) has units of ohms.

**Clarification:** \(j = \sqrt{-1}.\)

You have each of the five Platonic solids, with edges made out of \(1\Omega\) resistors. For which solid can you pick two vertices that have an effective resistance between them that is greater than \(1\Omega\)?

**Definitions**:

*Tetrahedron*is a polyhedron with 4 sides.*Octahedron*is a polyhedron with 8 flat faces.*Dodecahedron*is a polyhedron with 12 flat faces.*Icosahedron*is a polyhedron with 20 flat faces.*Oblate spheroid*is a quadric surface obtained by rotating an ellipse about its minor axis.

**Clarification**: "All of these" and "None of these" refer only to the geometrical answers not to each other!

A useless wire having a total resistance of \(48 \space \Omega\) is cut into 48 equal pieces. Then, a regular Deltoidal Icositetrahedron as shown below.

If the equivalent resistance between two opposite points, where four edges meet together is \(R \space \Omega\), then enter your answer as the value of \(100R\).

A uniform wire of resistance \(R\) is cut into four circular rings of equal radius. Rings are then connected such that their centers lie on the vertices of a square (as illustrated in the figure above).

If the equivalent resistance between \(A\) and \(B\) can be expressed as \(\frac{aR}{b}\), where \(a\) and \(b\) are coprime positive integers, determine the value of \(a+b\).

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