Mesh-Analysis, Ohm's Law & Passive Sign Convention

I am using Fundaments of Electric Circuits 5ed and I have a few questions about Passive Sign Convention with respect to Ohm's Law & PSC

FYI passive sign current (PSC) is based on this diagram: https://en.wikipedia.org/wiki/Passive_sign_convention#/media/File:Passive_sign_convention.svg

On page 40, it states: to find the voltage of the circuit (i.e. V1 and V2): simulate this circuit – Schematic created using CircuitLab

I get what to do above, however, once I get to page 83 for mesh analysis, ohm's law somehow changes and PSC is no longer used (I think): simulate this circuit

So my question is: why did we change from ±i in ohm's law (page 40) to V_higher - V_lower (page 83) in ohm's law?

• They didn't go away from passive sign convention. They are just telling you how to be smarter about choosing the reference direction for each of the resistors. – The Photon Apr 5 '17 at 18:25
• Higher includes polarity not |magnitude| so same rules apply to differential voltages. – Tony Stewart Sunnyskyguy EE75 Apr 5 '17 at 18:47
• In the first circuit, the - sign in the V2 calculation indicates that your guess for the voltage polarity was wrong. – Peter Bennett Apr 5 '17 at 18:48
• But that doesn't explain why we go from ±i to V_higher - V_lower – Edward Apr 5 '17 at 18:55
• How does that explain why we went from ±i to V_higher - V_lower? – Edward Apr 5 '17 at 18:55

Positive current (conventional current/positive charge) flows into the + terminal comes out through the - terminal in a resistor. That's why Ohm's law will yield a positive result for the current when going from a higher potential to a lower potential:

$$I=\dfrac{V_{higher}-V_{lower}}{R}$$

Since the numerator is $\geq$ 0, current is positive.

Positive sign convention is used so that you don't have to deal with the signs as you do in the first example you showed, where you 'manually' had to place a minus sign in Ohm's law. Now, what is higher or lower potential is an assumption that could turn out to be wrong in the end, but it helps to keep the math consistent.

So if in your analysis you assume that the current flows into the positive terminal and flows out of the negative terminal (which is the same as from higher to lower potential), the ohm's law relationship is positive, otherwise you need to add a negative sign to the relationship (like you did in your first example).

This is an excerpt from the Nilsson-Riedel electric circuit textboot I think what they want to emphasize with that statement is that if you're better off assuming the current to be in the direction of the voltage drop, from + to -, as it'll make things less confusing with the signs

• That clears it up! Thank you! So, in formulas, when I see V, it should really be V_higher - V_lower, like in ohm's law? – Edward Apr 5 '17 at 20:03
• Ohm's Law is usually written V=IR. V must always be the voltage across the resistor. Whether it is given as a single value or some calculated value like V1-V2 depends on the particular situation. Ohm's Law may also be written as R = V/I or I = V/R, depending on what value we need to calculate. – Peter Bennett Apr 5 '17 at 20:11