Node Voltage Calculator: Find Circuit Node Voltages

A node voltage calculator applies nodal analysis to find the voltage at each circuit node relative to a chosen reference node. Enter the branch conductances and current-source injections for a linear resistive circuit, along with its node-voltage coefficients; the calculator returns the unknown node voltages in volts.

Formula

For two unknown node voltages, nodal analysis forms a linear system:

aV₁ + bV₂ = J₁
cV₁ + dV₂ = J₂

Solving gives V₁ = (J₁d − bJ₂)/(ad − bc) and V₂ = (aJ₂ − cJ₁)/(ad − bc).

• V₁ and V₂: the unknown node voltages, in volts (V), measured from the reference node.
• a, b, c and d: coefficients in siemens (S), where 1 S = 1 A/V. They describe the conductance connections in the node equations.
• J₁ and J₂: net current injected into each node, in amperes (A). A source directed into a node is positive; a source directed out is negative.
• ad − bc: the determinant of the coefficient matrix. It must not be zero; otherwise these equations do not have a unique solution.

This calculator solves a two-node equation system. It does not infer a circuit diagram or calculate the coefficients from component values. You must derive the coefficients and net current injections from your circuit first.

Worked Example

Using the calculator’s default values, the equations are 0.03V₁ − 0.01V₂ = 0.1 and −0.01V₁ + 0.02V₂ = 0. The determinant is (0.03 × 0.02) − (−0.01 × −0.01) = 0.0005 S².

Therefore, V₁ = [(0.1 × 0.02) − (−0.01 × 0)] / 0.0005 = 4.000 V. V₂ = [(0.03 × 0) − (−0.01 × 0.1)] / 0.0005 = 2.000 V. These are the node voltages relative to the selected reference, rounded to three decimal places.

What Node Voltage Analysis Does

Node voltage analysis, also called nodal analysis, is a systematic way to solve circuit voltages using Kirchhoff’s Current Law (KCL). KCL says that the algebraic sum of currents at a node is zero: current flowing into a junction must equal current flowing out. A node is a continuous region of ideal conductor shared by circuit elements.

Choose one node as the reference, often called ground, and assign it 0 V. The voltage at every other node is then measured relative to that reference. For a circuit with N essential nodes, the usual method needs N − 1 unknown node voltages. An essential node is a junction where three or more circuit elements meet.

Ground is a mathematical reference in the circuit model; it does not automatically mean earth ground or a safe connection to touch. Changing the reference changes the reported voltage values, but not the voltage difference between two nodes.

How to Build the Equations

For a resistor between unknown nodes i and j, Ohm’s law gives the current leaving node i toward node j as (Vᵢ − Vⱼ)/R. Its conductance is G = 1/R. In the node-i equation, that branch contributes G to the coefficient of Vᵢ and −G to the coefficient of Vⱼ. A resistor from node i to the reference contributes G to the coefficient of Vᵢ because the reference voltage is zero.

For each unknown node, add the branch currents and set their sum equal to the net current injected into that node. Collecting terms produces a matrix equation GV = I. The diagonal coefficient for a node is generally the sum of conductances connected to it; off-diagonal coefficients are the negative sum of conductances directly connecting the corresponding pair of nodes. Independent current sources contribute to the right-hand side according to their direction.

For example, a 100-ohm resistor has conductance 1/100 = 0.01 S. If it joins two unknown nodes, it adds 0.01 S to each corresponding diagonal and −0.01 S to each matching off-diagonal. This sign pattern is a useful check when setting up the calculator inputs.

When to Use It and What the Result Means

Use this calculator after reducing a circuit to a two-unknown-node linear system. It is useful for checking a hand calculation or solving a small resistive network. The output is voltage relative to the chosen reference, not the voltage across every component. To find the voltage across a part connected between nodes 1 and 2, calculate V₁ − V₂, respecting the polarity you chose.

Conductance is the reciprocal of resistance: siemens (S) = 1/ohm (Ω). Keep units consistent. If you enter conductance in millisiemens, convert it to siemens first: 1 mS = 0.001 S. Likewise, current in milliamperes must be converted to amperes for these equations: 1 mA = 0.001 A. The resulting volts follow from A/S = V.

Common Mistakes and Limitations

• Reversing current-source direction: define positive injection as current entering a node, then use negative values for current leaving it.
• Mixing resistance and conductance: the coefficients are conductances, not resistances. Convert each resistor using G = 1/R, then assemble the node equations.
• Using inconsistent units: do not mix ohms with millisiemens or amperes with milliamperes without conversion.
• Forgetting the reference: the answers are not absolute potentials; they are measured relative to the selected 0 V node.
• Rounding too early: retain extra digits while deriving coefficients, and round only the displayed results. Small rounded values can noticeably affect a system with nearly dependent equations.

The provided two-by-two formulas apply to linear equations with a nonzero determinant. A zero determinant means there is no unique solution for the specified system; a determinant close to zero can make calculated voltages highly sensitive to small input or rounding errors. Nonlinear components, such as diodes operating outside a linearized model, require nonlinear circuit analysis. Circuits containing ideal voltage sources between unknown nodes often require a supernode equation or modified nodal analysis; they cannot always be represented by simply inserting a current injection into this form.

Frequently Asked Questions

What is a node voltage calculator?
It solves node-voltage equations derived from Kirchhoff’s Current Law. This version solves a linear system for two unknown node voltages when you supply its coefficients and current injections.

How do I calculate node voltage by hand?
Choose a reference node, write a KCL equation at each other node, express resistor currents as voltage differences divided by resistance, and solve the resulting simultaneous equations.

Is node voltage the same as voltage across a resistor?
No. Node voltage is measured from a reference node. The voltage across a resistor between two nodes is the difference between those nodes’ voltages, with the sign set by the chosen polarity.

Why is my calculated voltage negative?
A negative result means the node is below the reference potential under your chosen sign convention. It can be a valid result, or it can indicate a reversed source direction or incorrect equation sign.

Can this calculator solve a circuit with more than two unknown nodes?
No. It handles two node-voltage unknowns. Larger circuits need a larger system of simultaneous equations, which can be solved with matrix methods or suitable circuit-analysis software.

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in a | Coefficient a11 | S | 0.03 | 0
in b | Coefficient a12 | S | -0.01 | -
in c | Coefficient a21 | S | -0.01 | -
in d | Coefficient a22 | S | 0.02 | 0
in j1 | Current injection at node 1 | A | 0.1 | -
in j2 | Current injection at node 2 | A | 0 | -
out v1 | Node 1 voltage | V | (j1*d-b*j2)/(a*d-b*c) | 3
out v2 | Node 2 voltage | V | (a*j2-c*j1)/(a*d-b*c) | 3