Basic Concepts

        You have learned how to use conductance matrix analysis in simple circuits.  However, the discussion in the first lesson did not give a formal algorithm for solution of these circuits.  Here are our goals for this lesson.

• Given an electrical circuit composed of resistors, voltage sources and current sources,
  • Be able to write the conductance matrix representation for the circuit,
  • Be able to solve the conductance matrix-source voltage equation.

        In this section we will consider how circuit equations are generated and how different elements affect those circuit equations.  We will examine how node equations are written.

        We start by looking at a typical node - extracted from some larger circuit.  That node is shown below.

• The node shown - Node "n" - is part of a larger circuit. The resistors connected to node "n" are connected to other nodes through other resistors - shown by dotted lines in the figure.
• We define currents flowing out of node "n" as positive when flowing away from the node. Obviously they can't all be positive, but using this convention helps in developing a formal methodology for setting up the node equations.

• Define a voltage at every node in the circuit.  A typical voltage can be denoted V_n.
  • That voltage symbol,V_n, denotes the voltage at node "n" measured with respect to ground.
• Note, that at this point we have to realize that every node voltage might not be an unknown in the system of equations that we are developing.  For example, in the circuit below, the voltage at the green node is known to be V_s, whereas the voltage,V_x, is not known a priori.
  • When you are setting up a group of equations to solve, it is important to know how many unknowns you have (because the level of difficulty of solution is related to the number of unknowns) and that information is important.

• Having defined a voltage at every node, it is now possible to write KCL at nodes where the voltage   is not known.  The KCL equations have a form that we can discover from the general node above.

• Then we write KCL at the node where V_x appears.
• Finally, we solve whatever equation results from writing KCL.

        We will look at these two steps separately, and we will work first on the problem of writing KCL at a node.

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Writing Node Equations

        Here is that node extracted from a larger circuit.

        Examining the sample node - node "n" - it is relatively easy to write the KCL equations for the node.  Using KCL for the sample node, we would have: I_x + I_y + I_z = 0

And, each of the currents can be expressed in terms of the voltage at the node and the voltage at one of the neighboring nodes.  For example:

I_x = (V_n -V_x)/R_x

Express every current in that manner, and put those expressions into the KCL equation:

(V_n -V_x)/R_x + (V_n -V_y)/R_y + (V_n -V_z)/R_z = 0

        Notice how this equation plays out.  By defining currents as positive when they are flowing away from the node (i.e., the current arrows defining current polarity point away from the node.), every term that involves V_n turns out to have a plus sign in the resultant equation.  That gives you a way to know when you have written the equation correctly.  There is a payoff to being systematic and consistent when you write the KCL equations.  You will find almost every textbook that discusses this topic writes the equations this way!  You should adopt that practice as well, and you should never go wrong.

        The case of nodes with only resistors connected is simple enough, but there are special situations that you will need to take care of.  Here are those special cases.

• One of the adjacent nodes might be connected to a voltage source.  Actually, there are two special cases here because the other end of the voltage source might be connected go ground - the first case -  or it might be connected to still another node - the second case.
• There might be a current source connected to the node at which you write KCL.

        Let's consider the first case.  Here is a picture of the situation with an adjacent node connected to a grounded voltage source.

In this situation, KCL is written exactly as it was before.  However, the voltage source serves to fix a node voltage and that node voltage is no longer a variable with an unknown value.  Instead, the voltage at node 'x' in the figure is known to be V_s.  After you have written KCL at the node, when you go to solve the KCL equations simultaneously, that known value will appear in a term on the right hand side (the knowns) of the equation.  Thus:

• If there is a grounded source connected to node "n", then:
  • No KCL equation is written at node "n".
  • We have V_n = V_s.

        There is another case in which there is a voltage source connected between two nodes - neither of which is ground.  In this case the voltage source establishes a voltage difference between two nodes, and the way this situation is handled is slightly different.

• If there is an ungrounded source connected between node "n" and node "z", then
  • V_n = V_z + V_s.
• If V_z can be found, thenV_z is known from this equation.
• One less KCL equation will be written for this circuit.
• Then, the approach would be to write a KCL equation at either node "n" or node "z", and use the equationV_n = V_z + V_s to find the voltage at the other node.
• The implication is that one less KCL equation will be written.