Single and Double Subscript Notation

What is Single Subscript Notation

Single subscript notation is a point node which identifies a voltage value between a reference point. There must be a reference point while applying single or double subscript notation to some nodes in the circuit diagram shown in fig.no.3(a) and (b).
reference designation
(a)
alternative reference designation
(b)
Fig.no.3: alternative reference designation
(a)
(b)
Fig.no.4: Conventions for specifying voltages on a circuit diagram.
For example if node c in Fig.no.4(a) specified as the reference node, then single subscript notation at point node a is $V_a =V_{ac}$ and $V_b = V_{bc}$.
In Fig.no.4(b), voltages are specified using plus minus notation, where the minus sign identifies the reference point, where the positive sign identifies the point at which V is specified with respect to the reference. The +/- are reference marks, that define what is meant by a positive value of v just as a and b in Fig.no.4(a) are reference marks that define what is meant by a positive value of $V_{ab}$. Where $V_{ab}$ is the double subscript notation of voltage in the circuit diagram.
The single-subscript notation $V_a$ specifies the voltage at point a with respect to ground (zero volts). If the voltage is less than zero volts, a negative sign must be associated with the magnitude of $V_a$.
When a single subscript notation is used that is when one point in a circuit is chosen as a reference and other voltages in the circuit are specified with respect to that point, we are in fact treating the reference point as if its potential were zero, but in fact we usually do not know that - nor do we care. Again we are concerned with potential difference not absolute potential.

What is a Double Subscript Notation

The fact that voltage is an across variable and exists between two points has resulted in a double-subscript notation that defines the first subscript as the higher potential. In Fig.no.5(a), the two points that define the voltage across the resistor R are denoted by a and b. Since a is the first subscript for $V_{ab}$, point a must have a higher potential than point b if $V_{ab}$ is to have a positive value. If, in fact, point b is at a higher potential than point a, $V_{ab}$ will have a negative value, as indicated in Fig.no.5(b).
Defining the sign for double subscript notation
Fig.no.5: Defining the sign for double subscript notation
The double-subscript notation $V_{ab}$ specifies point a as the higher potential. If this is not the case, a negative sign must be associated with the magnitude of $V_{ab}$.
In other words,
The voltage $V_{ab}$ is the voltage at point a with respect to (w.r.t.) point b.
A particularly useful relationship can now be established that has extensive applications in the analysis of electronic circuits. For the above notational standards, the following relationship exists:
$$ \bbox[5px,border:1px solid red] {\color{blue}{V_{ab} = V_a - V_b}}$$Eq.(1)
Example 1:
Find the voltage $V_{ab}$ for the conditions in Fig.no.6.
Fig.no.6.
Solution:
$$\begin{array} {rcl} V_a & =& 10V\\ V_b& = &4V\\ V_{ab}& = &V_a - V_b\\ & = &10v - 4V = 6V \end{array}$$