When we need to analyze a fairly complex circuit and we’re thrown back to our own resources because we don’t have access to SPICE or any other software tool, a problem arises that could be called the “oops factor”.
In more conventional terms, we want to make our error checking as simple as possible.
We have the choice to use loop analysis or node analysis. I personally prefer the nodes because with node analysis it seems so much easier to do the “accounting” as in the next sketch in Figure 1.
Figure 1 An example of node analysis with algebraic equations for each dependent node (E1, E2, E3, E4, E5, E6, E7). Each color indicates which equation belongs to which node. Source: John Dunn
My preference for node analysis stems from the fact that I find it very difficult to count how many loops has a circuit, but it’s easy to count how many nodes has a circuit. We need to write one algebraic equation for each loop or write one algebraic equation for each dependent node.
Count how many equations I need write versus how many equations I actually wrote is easy. If I have fewer equations than dependent nodes, I’ve omitted something. If I have more equations than dependent nodes then I’ve done something redundant or have a wrong equation somewhere.
On a side note, my preference for using node analysis as opposed to using loop analysis came from Dr. John Kelly, one of my professors at New York University in 1970. I took five of my MSEE classes with him and it was on purpose.
Coming back to the topic; there are seven dependent nodes in Figure 1 (E1 through E7) and seven node equations. The colors indicate which equation belongs to which node. With proper algebraic manipulations, an equation for each of the dependent nodes can be easily derived.
We will now use node analysis to examine a well-known circuit, the Wein Bridge (Figure 2). We turn our attention to the right side of the bridge. (Analyzing the voltage divider with two resistors on the left side of the bridge is just too easy.)
Figure 2 The Wein bridge circuit. The right side of this circuit is analyzed with node analysis. Source: John Dunn
We’ll look at the transfer function from an input connected at the top, to the top corner of the bridge, to an output on the right. Since we are dealing with only one dependent node, we only need one node equation to proceed as follows figure 3.
figure 3 Node analysis of the right side of the Wein bridge circuit. Source: John Dunn
This is absolutely correct, which is all well and good, but what if I had made an algebraic mistake somewhere, especially when setting up the original node equation. Any mistake made at that point could quite easily have gone unnoticed and led me to an incorrect result.
A mathematically equivalent approach based on inspection is useful as error checking (Figure 4).
Figure 4 A useful method to perform an error check of our node analysis is to identify impedance and admittance. Source: John Dunn
We divide the circuit into two “pairs” and identify each of them as the sum of two component expressions. We can reverse the admission expression of pair 2 to obtain an impedance expression and then construct a voltage divider equation as shown in Figure 5.
Figure 5 Inverting the pair 2 admission expression (green) to obtain a voltage divider equation. Source: John Dunn
Filling in the terms pair 1 and pair 2 gives us the exact same expression we had from the node analysis, but with visually comfortable traceability that gives us some reassurance that no mistake has been made along the way (Figure 6).
Figure 6 Achieving the desired transfer function from the second method (double checking our original node analysis for the Wein bridge). Source: John Dunn
From this point we clear fractions to get a useful result (Figure 7).
Figure 7 The final form of the transfer function of the Wein bridge circuit. Source: John Dunn
We happily note that where R1 = R2 and C1 = C2, this transfer function peaks at a value of 1/3, which is the classic Wein bridge result.
John Dunn is an electronics consultant and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and New York University (MSEE).