Given a set of equations arising from applying the node voltage method, there are several solution options.  So far we can see the following.

• With one unknown node voltage, one equation results and it can be solved easily.
• With two unknown node voltages, two equations result, and they can be solved with the method of back substitution.
• With three unknown node voltages, three equations would result.  We could use back substitution to eliminate one of the variables, getting two equations, then use back substitution again to get to one variable and solve.  It looks and sounds messy, but we could do it.

• As long as we have linear resistors  - and no diodes, transistors, etc. - we will always get a set of simultaneous linear equations when we write the node equations.
• Back substitution is a well-known way of solving sets of linear equations, but other methods based on determinants and matrices are widely used.
• Many mathematical analysis packages - like Mathcad and Matlab, for example - have built-in routines for solving sets of simultaneous linear equations.   If we want to use methods based on determinants and matrices, and especiallly if we want to use the packages available, we will need to be able to represent our node equations in a way that is different.
• We need to be able to represent our equations in a vector-matrix format.

• V_x(1/R_a+ 1/R_b + 1/R_c) -V_y/R_c = V_s/R_a
• -V_x/R_c+ V_y(1/R_c+ 1/R_d)  = 0

        A vector-matrix representation that contains the same information is given below.

The vector-matrix formulation has several important characteristics.

• The information about the circuit connections are contained in a matrix that has numerous reciprocal resistance values.  Actually, they are conductances, and this matrix is called theconductance matrix.
• The unknown node voltages are represented as components in a voltage vector.
• The voltage source gets embedded in a source vector on the right hand side of the vector-matrix equation for the circuit.

• The matrix, G, is the conductance matrix.  It has units of ohms^-1.
• The vector, v, is the node voltage vector.  It has units of volts.
• The vector, I, is the source vector.  It has units of current.

Getting it is another story.  How many ways do you know to take the inverse of a matrix?  How do you invert a matrix?  Let us count the ways.

• The inverse of a matrix,G, is given by:
  • Adj(G)T/|G| so, you can compute an inverse by hand.
• Mathcad and Matlab have matrix inversion functions.  Use them.

        We wanted to show you this first because we wanted to build a case that it may not be a good idea to compute this inverse symbolically by hand.  This is not a large circuit, and the analytic inverse looks fairly horrendous.

• Even a circuit as simple as this points to the fact that numerical solutions to circuits are very important.  In turn, that implies the you need to be familiar with numerical ways to solve the kinds of equations that come out of linear circuits.  That meansyou need to understand the algorithms and you need to be able to use those algorithms in various incarnations especially as they appear in analysis applications like Mathcad and Matlab.

• R_a, R_b  andR_c are all attached to the first node, whereV_x appears.
  • In the conductance matrix, the reciprocals of all those resistances are added together to give element (1,1).
• Similarly, R_d are attached to the second node, and their reciprocals are added to give element (2,2) in the conductance matrix.
• Finally, note that R_c is shared between the two nodes, and the negative reciprocal of R_c appears in elements (1,2) and (2,1) in the conductance matrix.

        It should be clear that there's a pretty slick way to fill in a conductance matrix in the rules above.  Review that material to be sure you understand it. You can always write down a conductance matrix from a circuit by following these two rules.

• If you have a resistance between two nodes, node m and node n,
  • Then that resistance contributes a term that is the nevative reciprocal of the resistance to the (n,m) entry in the conductance matrix, and to the (m,n) entry in the conductance matrix.
  • All such contributions are added together.
• If you have a resistance attached to node n,
  • Then that resistance makes a positive contrbution to the (n,n) entry.

        That's it.  At this point, you are invited to check those results for this circuit.

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Example

E1   Here is a small ladder network.

And, here is the conductance matrix equation for the circuit above.  Notice how the conductance matrix is built up from the conductances attached to each node using the method we outlined above.

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        Determining the node voltages for this circuit involves more than just the conductance matrix.  The complete set of equations is given again.  We have an algorithm for determining the conductance matrix on the left hand side of this equation, but we also need a way to determine the right hand side.  We are lucky in this circuit because the one voltage source has one end connected to ground.  That leads to a simple expression on the right hand side of the set of equations.

        When we have a voltage source with one end connected to ground, then the right hand side of the circuit equation set takes on a particularly simple form.

• The voltage source appears in the equation for any node with which it shares a resistor.  In the example circuit,V_s sharesR_a with the first node -V_x.  The voltage source appears on the right hand side of the equation for this node, being multipled by the reciprocal of the shared resistance.