Convention for expressing voltage between two points

Question

Voltage is always related to two points, because it means a difference in electric potential, between two points.

If we want to express the voltage between A and B, we could say:

The potential at A is 3 V higher than the potential at B.

And there is no place to doubt.

The problem is, if we want to write it using less text, as follows:

$$V_{AB} = 3 V$$

The reader could be confused and ask:

But hey, is A the end with higher potential or is it B?

Sign might matter or not, but I think it does, since it gives a consistent understanding of what's going on, if we see a represented diagram and somewhere it says: $V_{AB} = 3 V$

There are a lot of similar questions regarding this topic, like this one I pasted recently.

But here and now I want to ask a very specific thing that deserves its own question:

What does the shorthand $V_{AB}$ actually mean?

I was taught in university it means: "potential you get from going from A to B".

But that now doesn't make sense to me anymore, since A or B could have the highest potential and it does not matter if we go from one to the other. What makes it clear is, which of both is at highest potential? And that is not clear at all by just saying I go from one to the other.

What I got from the answer at a related question is that when you write $V_{AB}$ you explicitly consider that A has a higher potential than B. It's not about going from A to B at all, but from taking the voltage between A and B and saying:

Hey, in this case, A has a 3 V higher potential than B

That makes a lot of sense, but it also turns the reverse into something that must be considered.

Example:

I have a circuit. And somebody asks me to calculate $V_{KS}$, where K and S are two nodes in the circuit.

This is not telling me to calculate the voltage "going from K to S".

It's saying:

Please suppose that K is at a higher potential than S. Calculate the value.

So you do it, and if the sign happens to end up being negative, it means that the supposition was incorrect, and S is at a higer potential than K.

Of course, maybe nothing of this is that important because in real life, one knows what one is doing and everything is fine.

But in a situation where one is learning and is asked to solve circuits with messy values that randomly jump between the positive and the negative, it is very helpful to know once for all what the real meaning of the $V_{AB}$ shorthand is, and if it's standard or if it also has multiple interpretations.

Is there any reference where this is stated clearly or is it just something that everybody agrees with and just knows?

Answer 1

You are correct is saying that the labelling is all down to convention and one hopes that a convention is clearly stated before one attempts to start labelling.

$V_{\rm AB}$ is often defined as the potential at position $A$ relative to / with respect to the potential at position $B$ which is the same as the change in potential in moving from position $B$ to position $A$.

As you have pointed out $V_{\rm A'B'}$ also can be defined as the change in potential in moving from position $A$ to position $B$.

So it does not help to find that using both definitions produces the equality $V_{\rm AB} = - V_{\rm A'B'}$.

Once circuit diagrams are available then life can become a bit easier but again the conventions used must be stated beforehand.

The potential of position/node $A$ relative to node $C$ can be written as $V_{\rm AC}$.
However what is called a reference node is often chosen and labelled with an inverted T and taken to be at a potential of zero.
In the diagram this is node $C$.
Then the potential of node $B$ relative to node $C$ can be written as $V_{\rm BC}$ or more simply $V_{\rm B}$ or even just $b$.

So $V_{\rm AB} = V_{\rm A} - V_{\rm B} = a -b = V_{\rm AC} - V_{\rm BC}$

The plus and minus sign by the side of circuit elements also have a meaning.

The potential of the node labelled $+$ eg node $B$, relative to the node labelled $-$ eg node $C$, is equal to $v_1$ in the diagram.

So $B_{\rm BC} = V_{\rm B} - 0 = V_{\rm B}= v_1$

I am afraid that you will always need to find out the convention which is being used at the start.

PS There is the same need for a clearly stated convention in other fields eg when forces are labelled.
$F_{\rm AB}$ could mean the force on body $A$ due to body $B$ or the force due to body $A$ on body $B$.

Answer 2

Rather than try to respond to your questions point by point, perhaps the following will help you to make sense of this. If after considering the following you still have questions, please comment further.

Let's start with the following definition of voltage, or potential difference, from the NCEE reference handbook for the FE exam for Electrical and Computer Engineering.

Voltage : The potential difference $V$ between two points is the work per unit charge required to move the charge between the points.

Let's apply this definition to the term $V_{AB}$ by substituting it for $V$ in the definition of voltage:

Voltage $V_{AB}$: The potential difference $V_{AB}$ between the points $A$ and $B$ is the work per unit charge required to move the charge from point $A$ to point $B$

Next we have the following conventions pertaining to the electric field and current.

1. The direction of an electric field, $E$, is the direction of the force that a positive charge would experience if placed in the field.

2. Conventional current is defined as the flow of positive charge.

Regarding 2, even though we now know that in most cases current is the flow of negative charge (electrons), this convention was established during Benjamin Franklin's day when electricity was not fully understood and we have been stuck with it since.

Now, with respect to $V_{AB}$ which point is considered high potential and which point low potential, $A$ or $B$? That will depend on the direction of the electric field between the points.

If the direction of the electric field is from point $B$ to point $A$ an external force will be needed to push positive charge from $A$ to $B$ against the direction (force) of the electric field (convention 1). The external force will be doing positive work. At the same time the electric field is doing negative work taking the energy provided by the external force and storing it as electrical potential energy of the charge/field system. The electrical potential energy of the positive charge increases. Thus point $B$ is considered to be at higher potential than point $A$.

Bottom line: Point $B$ is designated as high potential relative to point $A$ if it takes external (to the field) work to move positive charge from $A$ to $B$.

Hope this helps.