Home / Exercises 03 / Conductance Matrix

Conductance Matrix

The questions below are due on Wednesday February 26, 2020; 11:59:00 PM.
 
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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:

\textbf{G}\textbf{e} = \textbf{S}

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

\begin{bmatrix} G_{11} & G_{12} & G_{13} \\ G_{21} & G_{22} & G_{23} \\ G_{31} & G_{32} & G_{33} \end{bmatrix} \begin{bmatrix} e_1\\ e_2\\ e_3\\ \end{bmatrix} = \begin{bmatrix} -I\\G_1G_4RV\\G_4V\end{bmatrix}

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.

Give an expression for G_{11} where S_1 involves the independent sources contributing to current into or out of a node
Give an expression for G_{12} where S_1 involves the independent sources contributing to current into or out of a node
Give an expression for G_{13} where S_1 involves the independent sources contributing to current into or out of a node
Give an expression for G_{21} where S_2 involves the independent sources contributing to current into or out of a node.
Give an expression for G_{22} where S_2 involves the independent sources contributing to current into or out of a node.

Give an expression for G_{23} where S_2 involves the independent sources contributing to current into or out of a node.

Give an expression for G_{31} where S_3 involves the independent sources contributing to current into or out of a node.
Give an expression for G_{32} where S_3 involves the independent sources contributing to current into or out of a node.

Give an expression for G_{33} where S_3 involves the independent sources contributing to current into or out of a node.

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

Give an expression for the voltage e_4 in terms of node voltages (use e_1 written as e1) and component values: