# Conductance Matrix

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We saw in lecture the other week that you can represent a system of equations for a circuit in the form:

where \textbf{G} is the conductance matrix, \textbf{e} is a column vector of node voltages and \textbf{S} is a column vector of combinations derived from the independent sources and other parameters.

Following the node method, develop a set of simultaneous equations for the network shown below that can be solved to determine the unknown node voltages e_1, e_2 and e_3. Express the set of equations in the form

You need not solve the set of equations for the node voltages.

Write your answers in terms of R, the coefficient for the dependent voltage source, G, the coefficient for the dependent current source, and the conductances G1, G2, G3, G4, G5 and instead of the resistances for the respective resistors (trust us it'll be cleaner this way).

Note that e_4 is not included in the analysis because it is directly sourced by the dependent voltage source from ground, and the source voltage can be expressed in terms of the first three node voltages. In this sense, it is treated like an independent voltage source.

State how e_4 can be determined from e_1, e_2 and e_3 once the latter node voltages are determined.