Introducing Supernodes!!! Nodal Analysis Introducing Supernodes!!!

Nodal Analysis Nodal Analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables. Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously.

Nodal Analysis In nodal analysis, we are interested in finding the node voltages. The number of non-reference nodes is equal to the number of independent equations that will be derived. Given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following steps:

Nodal Analysis 1. Select a reference node. Assign voltages v1, v2, … , vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node. 2. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s Law to express the branch currents in terms of node voltages. 3. Solve the resulting simultaneous equations to obtain the unknown voltages. The number of non-reference nodes is equal to the number of independent equations that will be derived.

1. Select a reference node: ground Examples 1. Select a reference node: ground 2. Label your non-reference nodes

Examples 3. Apply KCL to your non-reference nodes 4. Use Ohm’s Law to write your KCL equations in terms the node voltages. 5. Solve the resulting equations for the voltages.

Nodal Analysis with Voltage Sources Case 1: If a voltage source is connected between the reference node, simply set the voltage at the non-reference node equal to the voltage of the voltage source.

Nodal Analysis with Voltage Sources (cont.) Case 2: If a voltage source (dependent or independent ) is connected between two non-reference nodes, the two nodes form a supernode. Apply both KCL and KVL to determine the node voltages.

Supernodes We treat supernodes differently because nodal analysis uses KCL, and there is no way of knowing the current through a voltage source in advance.

Supernodes (cont.) Note the properties of a supernode: The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages A supernode has no voltage of its own A supernode requires the application of both KCL and KVL

Examples Case 1: If you have a voltage source connected between a reference node and a non-reference node, set the voltage at the non-reference node to the value of the voltage source. V1 = 10 V

Examples Case 2: If you have a voltage source connected in between two non-reference nodes, you have to apply both KCL and KVL to determine the node voltages.

Examples Find the node voltages in the circuit.

Examples Nodes 1 and 2 form a supernode; so do nodes 3 and 4. Apply KCL to the two supernodes. At supernode 1-2: 10 = i1 + i2 + i3 Rewriting in terms of the node voltages, the equation becomes: And simplifying yields: At supernode 3-4:

Examples Next, apply KVL to the loops containing the voltage sources. Since we already have 2 of the 4 needed equations, you only need to choose 2 of the 3 loops for your final equations. For loop 1, For loop 2, But, since the equation becomes For loop 3, But since - and it becomes The 5th equation here (loop 3) is redundant, but it can be used to check your answers.