Component failure analysis

There is a lot of truth to that quote from Dirac. With a little modification, I can extend his wisdom to electric circuits by saying, “I consider that I understand a circuit when I can predict the approximate effects of various changes made to it without actually performing any calculations.”


Also shown at the end of the series and parallel circuits chapter was how the table method works just as well for aiding failure analysis as it does for the analysis of healthy circuits. We may take this technique one step further and adapt it for total qualitative analysis. By “qualitative” I mean working with symbols representing “increase,” “decrease,” and “same” instead of precise numerical figures. We can still use the principles of series and parallel circuits, and the concepts of Ohm’s Law, we’ll just use symbolic qualities instead of numerical quantities. By doing this, we can gain more of an intuitive “feel” for how circuits work rather than leaning on abstract equations, attaining Dirac’s definition of “understanding.”

Enough talk. Let’s try this technique on a real circuit example and see how it works:

This is the first “convoluted” circuit we straightened out for analysis in the last section. Since you already know how this particular circuit reduces to series and parallel sections, I’ll skip the process and go straight to the final form:

R3 and R4 are in parallel with each other; so are R1 andR2. The parallel equivalents of R3//R4 and R1//R2 are in series with each other. Expressed in symbolic form, the total resistance for this circuit is as follows:

RTotal = (R1//R2)–(R3//R4)


Next, we need a failure scenario. Let’s suppose that resistor R2 were to fail shorted. We will assume that all other components maintain their original values. Because we’ll be analyzing this circuit qualitatively rather than quantitatively, we won’t be inserting any real numbers into the table. For any quantity unchanged after the component failure, we’ll use the word “same” to represent “no change from before.” For any quantity that has changed as a result of the failure, we’ll use a down arrow for “decrease” and an up arrow for “increase.” As usual, we start by filling in the spaces of the table for individual resistances and total voltage, our “given” values:

The only “given” value different from the normal state of the circuit is R2, which we said was failed shorted (abnormally low resistance). All other initial values are the same as they were before, as represented by the “same” entries. All we have to do now is work through the familiar Ohm’s Law and series-parallel principles to determine what will happen to all the other circuit values.




In case you’re not familiar with the qualitative assessment of an equation, it works like this. First, we write the equation as solved for the unknown quantity. In this case, we’re trying to solve for current, given voltage and resistance:

Now that our equation is in the proper form, we assess what change (if any) will be experienced by “I,” given the change(s) to “E” and “R”:


Therefore, Ohm’s Law (I=E/R) tells us that the current (I) will increase. We’ll mark this conclusion in our table with an “up” arrow:


Fundamentally, what we’re doing here with a qualitative usage of Ohm’s Law and the rules of series and parallel circuits is no different from what we’ve done before with numerical figures. In fact, its a lot easier because you don’t have to worry about making an arithmetic or calculator keystroke error in a calculation. Instead, you’re just focusing on the principles behind the equations. From our table above, we can see that Ohm’s Law should be applicable to the R1//R2 and R3//R4 columns. For R3//R4, we figure what happens to the voltage, given an increase in current and no change in resistance. Intuitively, we can see that this must result in an increase in voltage across the parallel combination of R3//R4:

But how do we apply the same Ohm’s Law formula (E=IR) to the R1//R2 column, where we have resistance decreasing and current increasing? It’s easy to determine if only one variable is changing, as it was with R3//R4, but with two variables moving around and no definite numbers to work with, Ohm’s Law isn’t going to be much help. However, there is another rule we can apply horizontally to determine what happens to the voltage across R1//R2: the rule for voltage in series circuits. If the voltages across R1//R2 and R3//R4 add up to equal the total (battery) voltage and we know that the R3//R4 voltage has increased while total voltage has stayed the same, then the voltage across R1//R2 must have decreased with the change of R2‘s resistance value:



Applying Ohm’s Law vertically to those columns with unchanged (“same”) resistance values, we can tell what the current will do through those components. Increased voltage across an unchanged resistance leads to increased current. Conversely, decreased voltage across an unchanged resistance leads to decreased current:

Once again we find ourselves in a position where Ohm’s Law can’t help us: for R2, both voltage and resistance have decreased, but without knowing how much each one has changed, we can’t use the I=E/R formula to qualitatively determine the resulting change in current. However, we can still apply the rules of series and parallel circuits horizontally. We know that the current through the R1//R2 parallel combination has increased, and we also know that the current through R1 has decreased. One of the rules of parallel circuits is that total current is equal to th sum of the individual branch currents. In this case, the current through R1//R2 is equal to the current through R1 added to the current through R2. If current through R1//R2 has increased while current through R1 has decreased, current through R2 must have increased:



Related Posts: