Essential Nodes And Essential Branches (Worked Examples)

Contents

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Key learnings:

Essential Node Definition: An essential node is defined as a point where three or more circuit elements connect, making it crucial for circuit analysis.
Essential Branch Definition: An essential branch connects two essential nodes without passing through another essential node.
Nodal Analysis: Essential nodes reduce the complexity of circuit analysis by minimizing the number of equations needed.
Reference Node Selection: In nodal analysis, the essential node connected to the most branches is chosen as the reference node for easier calculations.
Practical Example: Using nodal and mesh analysis with essential nodes and branches simplifies solving for voltages and currents in a circuit.

What is an Essential Node?

A node is a connected set of ideal conductors whose element terminals share one voltage. An essential node has three or more connected circuit elements.

The first circuit has seven nodes. Four are essential and marked green; the other three join only two elements and are marked red.

What is an Essential Branch?

A branch is a path between two nodes that contains a circuit element. An essential branch connects two essential nodes without passing through a third essential node.

An essential branch can pass through nodes that join only two elements. The next diagram shows how this rule changes the branch count.

The circuit contains seven essential branches, labelled B1 to B7.

B3 is one essential branch even though it passes through non-essential node 4. The preceding diagram shows the node labels.

B4 and B5 are separate essential branches. There is no single essential branch from the top node, node 2, to the bottom node, node 7, because the path passes through essential node 3.

Node 3 therefore divides that path into two essential branches.

Essential Node Example

Essential nodes help organise nodal analysis. In the circuit below, the nodes inside series paths can be handled within their essential branches.

The worked example uses Kirchhoff’s current law to solve for the unknown essential-node voltages.

First, identify the nodes, choose a reference and write the branch currents in terms of node voltages.

essential node example — Essential Node Example

The circuit has five total nodes, including the three essential nodes V1 to V3. The other two nodes lie within branches and do not require separate KCL equations in this setup.

Choose one node as the zero-volt reference. The choice is arbitrary, although a grounded node or a node connected to many branches often makes the equations shorter.

For this example, V3 is the reference node. V1 and V2 are measured relative to V3.

Here, n is the number of essential nodes in the reduced circuit.

With one reference node, this circuit has n-1 = 2 unknown essential-node voltages. Circuits with voltage sources between unknown nodes may also need supernode constraints.

Apply KCL at node V1:

$\frac{V1-10}{4} + \frac{V1}{2} + \frac{V1-V2}{4} = 0$

Apply KCL at node V2:

$\frac{V2-V1}{4} + \frac{V2}{2} -10 = 0$

Solving the two equations gives node voltages V1 = 6.363 V and V2 = 15.454 V relative to V3.

$V1 = 6.363$

$V2 = 15.454$

Essential Branch Example

For a connected planar circuit, mesh analysis needs one equation for each independent mesh. The same count is b-n+1 when branches and nodes are counted consistently.

For this diagram:

There are 7 individual element branches.
There are 5 essential branches, B1 to B5.
There are 3 essential nodes, V1 to V3.

Using the reduced essential graph, b = 5 and n = 3, so b-(n-1) = 3 independent mesh equations. The unreduced count gives the same result: 7-5+1 = 3.

The circuit therefore requires three independent equations.

Apply KVL to the three independent meshes with the current directions shown.

Loop 1

$10 = 10I_1 - 2I_2$

Loop 2

$0 = 10I_2 - 2I_1 - 4I_3$

Loop 3

$-5 = 10I_3 - 4I_2$

Solving this system gives the following mesh currents. A negative result means the actual current is opposite to its assumed direction.

$I_1 = 1A$

$I_2 = 0A$

$I_3 = -0.5A$

Use these mesh currents to calculate power supplied or absorbed by a voltage source, branch current and the voltage across each element. Apply the passive sign convention when assigning power.