Single and Double Subscript Notation in Electrical Circuit Analysis

Introduction

In circuit analysis, accurate and consistent notation is crucial. For voltages (and often currents), engineers commonly use single-subscript or double-subscript notation to clearly indicate how potentials are measured and between which points. These notations help avoid ambiguity and make it easier to apply fundamental circuit laws (e.g. Kirchhoff's Voltage Law, Ohm's Law).

Single-Subscript Notation

Definition

: Single-subscript notation refers to the voltage at a single node with respect to a reference node, typically ground (zero-volt reference) as shown in Fig. 1(a) and (b).

How it's used

: Suppose you choose a particular node (call it c) as the reference (ground). Then the voltage at node a is denoted $$𝑉_𝑎=𝑉_{𝑎𝑐} $$ meaning the potential at a relative to reference c. meaning the potential at a relative to reference c.

Interpretation:

If 𝑉𝑎 is negative, it means the potential at a is below reference (ground). The sign conveys whether the node is above or below the reference potential.

Purpose:

Using single-subscript notation simplifies descriptions when most voltages in the circuit are referenced to a common ground — common in many DC (and AC) circuits. For example if node c in Fig. 1(a) specified as the reference node, then single subscript notation at point node a is $V_a =V_{ac}$ and $V_b = V_{bc}$. 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.

Double-Subscript Notation

Definition:

Double-subscript notation expresses a voltage difference between two arbitrary points (nodes) in the circuit. The notation $𝑉_{𝑎𝑏}$ denotes the voltage at point a with respect to point b.

Polarity convention:

If point a (the first subscript) has higher potential than point b, then $V_{ab}$. If point b is actually at a higher potential than a, then Vab is negative.

Mathematical relation:

$$𝑉_{𝑎𝑏} = V_a - V_b$$ That is, the voltage difference between a and b equals the difference of their potentials with respect to a common reference (ground) — if those potentials are known.

Why this matters:

Since voltage is fundamentally a “difference in potential”, double-subscript notation correctly captures this “across-element” nature. It is especially useful when dealing with elements (resistors, capacitors, sources) that connect two non-ground nodes, or when ground is inconvenient or absent. In Fig. 2(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. 2(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 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. 2(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. 2(b). In other words,

Why Notation Matters

Voltage is always relative:

Absolute potential (at a single point) has no standalone meaning; what matters is the difference between two points. The reference (ground) is often chosen arbitrarily for convenience.

Clarity in complex circuits:

In larger networks — especially with multiple nodes, sources, and components — double-subscript notation reduces confusion compared to multiple ground-referenced single-subscripts or ambiguous polarity markings.

Consistency with conventions:

By stating that the first subscript is the “higher potential” and using the same convention everywhere, you can reliably determine sign (positive/negative) of voltages or voltage drops across elements.

Link to circuit laws:

Once node voltages (e.g. 𝑉𝑎, Va) are known, calculating voltage differences (𝑉𝑎𝑏) becomes easy: $𝑉_{𝑎𝑏}=𝑉_𝑎−𝑉_𝑏$. This simplifies application of methods like node-voltage analysis, mesh analysis, or use of Kirchhoff’s Laws.