In nodal analysis, we use KCL at each node to determine the node voltages. This is why we need to determine the branch currents, which can be a little difficult to read if a voltage source exists between two nodes. In such cases, we use the supernode method to solve the system parameters. In circuit theory, a supernode is a theoretical construct that can be used to solve a circuit. To do this, we simply replace the voltage source with a short circuit, causing two nodes to act as a single one.
Following example will help to understand the concept. Find the node voltages and branch currents of the following given electric circuit using nodal analysis:
First, we need to determine all the nodes of the circuit and select the reference node, which must be located to ground. Then, we denote all the unknown node voltages and all the branch currents. The modified circuit will look like this, with six nodes in the circuit.
Here we have six nodes in the circuit. The reference node is represented by the green node, and the two pink color nodes represent the known voltage. The remaining three unknown nodes are represented by red color. The nodes A and B contain a voltage source, so we sort it and make these two nodes act as a single node, referred to as a supernode. The supernode construct is only required between two non-reference nodes.
The trick is that the number of equations needed to solve the problem is equal to the number of unknown nodes minus one. In this circuit, we have three unknown node voltages, so we need only two equations to solve this problem. To get the equations, we apply KCL at the supernode and one to any remaining node.
By using supernode analysis, we can simplify the circuit and reduce the number of equations needed to solve it. This technique is particularly useful in complex circuits with multiple voltage sources and non-reference nodes.
Summary:
To apply the supernode method, we first need to identify all the nodes in the circuit and select a reference node. Then we denote all the unknown node voltages and branch currents. If the circuit contains a voltage source between two non-reference nodes, we combine those two nodes into a single supernode.
The key trick in supernode analysis is to recognize that the number of equations needed to solve the problem is equal to the number of unknown node voltages minus one. For example, if a circuit has three unknown node voltages, only two equations are needed to solve for them.
To obtain the equations for supernode analysis, we apply KCL at the supernode and one additional node. This allows us to express the branch currents in terms of the node voltages and solve for the unknown
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