Node Analysis Calculator

Node Analysis Calculator

function calculateNodeVoltages() { var vs1 = parseFloat(document.getElementById("vs1").value); var r1 = parseFloat(document.getElementById("r1").value); var r2 = parseFloat(document.getElementById("r2").value); var r3 = parseFloat(document.getElementById("r3").value); if (isNaN(vs1) || isNaN(r1) || isNaN(r2) || isNaN(r3)) { document.getElementById("resultV1").innerHTML = "Node Voltage V1: Please enter valid numbers for all fields."; document.getElementById("resultV2").innerHTML = "Node Voltage V2: –"; return; } if (r1 <= 0 || r2 <= 0 || r3 <= 0) { document.getElementById("resultV1").innerHTML = "Node Voltage V1: Resistor values must be positive."; document.getElementById("resultV2").innerHTML = "Node Voltage V2: –"; return; } var g1 = 1 / r1; var g2 = 1 / r2; var g3 = 1 / r3; var denominator = (g1 * g2) + (g1 * g3) + (g2 * g3); if (denominator === 0) { document.getElementById("resultV1").innerHTML = "Node Voltage V1: Cannot calculate (denominator is zero). Check circuit configuration."; document.getElementById("resultV2").innerHTML = "Node Voltage V2: –"; return; } var v2 = (g1 * vs1 * g2) / denominator; var v1 = (g1 * vs1 * (g2 + g3)) / denominator; document.getElementById("resultV1").innerHTML = "Node Voltage V1: " + v1.toFixed(4) + " Volts"; document.getElementById("resultV2").innerHTML = "Node Voltage V2: " + v2.toFixed(4) + " Volts"; }

Understanding Node Analysis in Circuit Theory

Node analysis, also known as nodal analysis, is a powerful method used in electrical engineering to determine the voltage at each principal node in an electrical circuit relative to a chosen reference node (often called ground). This technique is fundamental for analyzing complex circuits and is based on Kirchhoff's Current Law (KCL), which states that the algebraic sum of currents entering a node must be zero.

How Node Analysis Works

1. Identify Nodes: First, all principal nodes in the circuit are identified. A principal node is a point where three or more circuit elements connect.
2. Choose a Reference Node: One node is selected as the reference node, typically assigned a voltage of 0 Volts (ground). This simplifies the calculations.
3. Assign Node Voltages: Unknown voltage variables (e.g., V1, V2, V3) are assigned to the remaining non-reference nodes.
4. Apply KCL: For each non-reference node, Kirchhoff's Current Law is applied. This involves writing an equation that sums all currents leaving (or entering) that node to zero. Currents are expressed in terms of node voltages and component values (using Ohm's Law, I = V/R).
5. Solve System of Equations: The KCL equations form a system of linear equations. This system is then solved simultaneously to find the unknown node voltages. Once the node voltages are known, other circuit parameters like branch currents and power dissipation can be easily calculated.

The Circuit for This Calculator

This calculator is designed to solve for the node voltages (V1 and V2) in a specific two-node circuit configuration. The circuit consists of:

• A voltage source (Vs1) connected in series with a resistor (R1). This combination is connected to Node 1.
• A resistor (R2) connecting Node 1 to Node 2.
• A resistor (R3) connecting Node 2 to the reference node (ground).
• The negative terminal of the voltage source Vs1 is also connected to the reference node (ground).

This setup allows us to derive two KCL equations for Node 1 and Node 2, which can then be solved to find V1 and V2.

Formulas Used

Based on Kirchhoff's Current Law, the following equations are derived for the specified circuit:

Let G1 = 1/R1, G2 = 1/R2, G3 = 1/R3 (where G represents conductance).

KCL at Node 1:
(V1 – Vs1) / R1 + (V1 – V2) / R2 = 0
G1(V1 – Vs1) + G2(V1 – V2) = 0
(G1 + G2)V1 – G2*V2 = G1*Vs1

KCL at Node 2:
(V2 – V1) / R2 + (V2 – 0) / R3 = 0
G2(V2 – V1) + G3(V2) = 0
-G2*V1 + (G2 + G3)V2 = 0

Solving these two simultaneous linear equations for V1 and V2 yields:

V2 = (G1 * Vs1 * G2) / (G1*G2 + G1*G3 + G2*G3)

V1 = (G1 * Vs1 * (G2 + G3)) / (G1*G2 + G1*G3 + G2*G3)

Example Calculation

Let's use the following values to demonstrate the calculator's function:

• Voltage Source Vs1 = 10 Volts
• Resistor R1 = 100 Ohms
• Resistor R2 = 200 Ohms
• Resistor R3 = 300 Ohms

First, calculate the conductances:

• G1 = 1/100 = 0.01 Siemens
• G2 = 1/200 = 0.005 Siemens
• G3 = 1/300 ≈ 0.003333 Siemens

Now, apply the formulas:

Denominator = (0.01 * 0.005) + (0.01 * 0.003333) + (0.005 * 0.003333)
= 0.00005 + 0.00003333 + 0.000016665
≈ 0.0001

V2 = (0.01 * 10 * 0.005) / 0.0001
= 0.0005 / 0.0001
V2 ≈ 5.0000 Volts

V1 = (0.01 * 10 * (0.005 + 0.003333)) / 0.0001
= (0.1 * 0.008333) / 0.0001
= 0.0008333 / 0.0001
V1 ≈ 8.3330 Volts

These results match the calculator's output, demonstrating the application of node analysis for this circuit.

Limitations

This calculator is specifically designed for the two-node circuit configuration described above. For circuits with more nodes, different configurations, or containing other types of components (e.g., dependent sources, capacitors, inductors in AC analysis), a more advanced node analysis tool or manual calculation involving matrix methods would be required.