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Nodal Analysis Steps
1) Identify all nodes in the circuit. Call the number of nodes N.
2) Select a reference node. Label it with reference (ground) symbol. As a general rule, the reference node is usually chosen to be
a node with largest number of elements connected to it, or
a node which is connected to the maximum number of voltage sources, or
a node of symmetry.
3) Assign a variable for each node whose voltage is unknown. If a voltage source is connected between a node and the reference node, the voltage is already known and it is not necessary to assign a variable. If there is a voltage source between two nodes, the difference between the node voltages equals to the voltage of the source. In this case, to reduce the number of unknowns assign a variable for one of the nodes and express the voltage of the other one with respect to the assigned variable.
4) If there are dependent sources in the circuit, write down equations that express their values in terms of node voltages.
5) Write down a KCL equation for each node by setting the total current flowing out of the node to zero. Recall that the KCL states that the algebraic sum of all currents entering and exiting a node is equal to zero. It is always a good idea to rearrange these equations into the form where and are some constants. If there are voltage sources between two unknown voltages, join the two nodes as a supernode. Note that you should have only one unknown variable for a supernode because the voltage of one the nodes can be expressed with respect to the voltage of the other one. For a supernode, the currents of the two nodes are combined in a single equation, and a new equation for the voltages is formed. For a circuit with N nodes and M voltage sources N – M – 1 simultaneous linearly independent equations can be written.
Here are some solved problems posted in solved-problems.com:
Solving a Circuit with Three Nodes by the Nodal Analysis
Solving by Nodal Analysis – Circuit with Four Nodes
The nodal analysis method is generally straightforward to apply, but becomes rather difficult in the following cases.
Non-grounded Voltage Sources
Since the current of a voltage source is independent of the voltage, it cannot be used in writing KCL equations. If one node of a voltage source is connected to the reference node, we do not need to know the current passing through the voltage source. The reason is that the voltage of the node can be easily determined by the voltage of the voltage source and there is no need to write KCL equation for the node.
Complicated cases are the ones where a voltage source is located between two non-reference nodes. In these cases, a supernode method should be used. A simple supernode is consist of a source and its nodes. In general, supernodes can have more than one voltage sources. After identifying a supernode, we need to define only one voltage variable for one of the nodes of the supernode and find the voltage of other node(s) with respect to that voltage variable. This equation relates node voltages of the supernode to each other. Then, we should treat a supernode as a node and write a KCL equation for all currents entering and leaving the super node. Now we have one equation and two unknowns (the node voltages). This equation should be added to the set of equations derived for other nodes and the new set of equations should be solved to determine all node voltages.
Check out this solved problem:
Nodal Analysis – Supernode
Dependent Current Source
When there is a dependent current source in the circuit, it should be treated as an independent current source but the variable which the current source depends on should be expressed in terms of node voltages. For example, if it is current of a resistor, Ohm’s law should be used to state the variable in term of the node voltages of the resistor.
Here is a solved problem with a dependent current source:
Nodal Analysis – Dependent Current Source
Dependent Voltage Sources
A dependent voltage source can make the solution a bit challenging. The solution follows the same steps mentioned for dependent source with an extra step. After writing super-node KCL equation, the variable that the dependent source depends on should be written in terms of the node voltages.
The circuit of the following solved problem has a dependent voltage source: