[Physics] Resistors parallel to short circuit


I have a trouble solving this problem.
(Actually, each resistor is not parallel with the bottom wire since the voltage across each of them is different with that of wire.)

In the manual, the bottom wire(which is red circled) is simply ignored and thus can be easily solved.

I don't understand why the bottom wire can be ignored. At first, I thought it's because the node between the wire has the same voltage, making voltage difference across the wire to be 0 resulting current flowing through the wire to be 0.

I used mesh current method by myself, not ignoring the bottom wire and i got different answer, which indicated that the wire has some non-zero current flowing through it.

Best Answer

You can consider that wire as being one big node. I suggest you redraw the circuit in such a way that you replace the encircled wire with a node.

I don't understand why the bottom wire can be ignored.

It is not short circuit since it does not directly connect two terminals of the same element.

To solve for the output voltage $v_o$ I suggest you use the Millman method:

$$\varphi_1 - \varphi_3 = 480$$

$$\varphi_2 - \varphi_3 = -168$$

$$\varphi_+ (\frac{1}{5} + \frac{1}{4} + \frac{1}{20}) = \varphi_1 \frac{1}{5} + \varphi_2 \frac{1}{4} + \varphi_- \frac{1}{20}$$

$$\varphi_- (\frac{1}{7} + \frac{1}{1} + \frac{1}{20}) = \varphi_3 \frac{1}{7} + \varphi_3 \frac{1}{1} + \varphi_+ \frac{1}{20}$$

The above equations can be written in matrix form:

$$ \left[ \begin{array}{cc} \frac{1}{2} & -\frac{1}{20} & -\frac{1}{5} & -\frac{1}{4} & 0 \\ -\frac{1}{20} & \frac{167}{140} & 0 & 0 & -\frac{8}{7} \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & -1 \end{array} \right] \left[ \begin{array}{c} \varphi_+ \\ \varphi_- \\ \varphi_1 \\ \varphi_2 \\ \varphi_3 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 480 \\ -168 \end{array} \right] $$

There are 5 unknowns and 4 equations, which means the system of equations is underdetermined. If we choose $\varphi_3$ as independent variable, the solution to the above equation is:

$$ \left[ \begin{array}{c} \varphi_+ \\ \varphi_- \\ \varphi_1 \\ \varphi_2 \end{array} \right] = \left[ \begin{array}{c} \frac{180360}{1663} \\ \frac{7560}{1663} \\ 480 \\ -168 \end{array} \right] + \left[ \begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array} \right] \varphi_3 $$

The output voltage is then

$$v_o = \varphi_+ - \varphi_- = 103.9 \text{ V}$$

Although the potential $\varphi_3$ is independent variable, it does not affect voltage calculation since it cancels out when subtracting potentials. For this reason we usually set one potential (any one!) to zero which mimics ground connection and makes matrix manipulation much easier. I suggest you solve this problem by taking some other potential as independent variable, you should get the same final result.

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