# How do I use the given voltage drop across $R_3$

I can't figure out how to solve this circuit if I use mesh I get these 3 equations what is the fourth equation? $$24E3*i_1 - 12E3*(i_2+i_3) = 24-v_{52} \\ -12E3*i_1 + 25E3*i_2 - 10E3*i_3 = 0 \\ -12E3*i1 - 10E3*i_2 + 46E3*i_3 - v_{52} = 0 \\$$

As this appears to be homework, so I'll push you a little way along the way I'd do it.

Those mesh equations may or may not be correct. However, I'm going to ignore them as they don't offer any insight into the circuit.

Consider the node joining R3 and Rs. It has three 'voltage source with known impedance' connected to it.

• The first voltage source is the 24 V in series with the top 12k.
• The second voltage source is V52 in series with Rs.
• The third voltage source consists of the 24 V supply, Thevenin with the 3k and 24k to get an effective lower voltage and output impedance, in series with R3.

The voltage drop across R3 gives you the current through it, which must be the sum of the currents flowing in the other two resistors connected to the node being considered.

I'll stop there.

... except to comment on the problematic specification of voltage across R3, which is -2.524 V. The problem is not that the voltage is negative, the problem is that the polarity of R3 is not specified in the question. Resistors are not polar components, so there is no 'natural' polarity associated with them. You have to choose a polarity, arbitrarily is fine, as long as you then stick with that polarity through the rest of the calculation. Choosing the polarity of R3 amounts to choosing a 'measurement' end and a 'reference' end.

The ideal solution is to go back to the original question source and see if it's specified anywhere. Another solution is to state your polarity assumption, solve using that assumption, and hope that the marker realises they have set an ambiguous question, and gives you the marks even if you guessed wrongly. A better solution is to solve using both polarities and present both answers. Maybe one of them will result in an infeasible temperature and can be rejected, which could be part of the point of the question.

If I had to guess, I'd say that the temperature sensor end of R3 is probably the measurement end, while the 3k/24k end has a feel of 'reference' about it. But I'd also solve for the other polarity just in case.

• What does the negative sign for R3 voltage indicate? Commented Aug 9 at 19:05
• That the assumed current direction is incorrect. Commented Aug 10 at 0:09