Minus signs in algebraic derivations are a pain. It’s really easy to make mistakes. And if you don’t think about what your results mean, you may not realize you have errors. Phase angles can be even harder to get right.
For Design-Oriented Analysis, arithmetic, algebra, and even circuit analysis techniques like nodal or loop analysis are just tools that allow us to mindlessly, systematically and automatically convert circuits into equations or numbers.
For students and beginning engineers, this mindlessness can lead to equations or numbers that don’t mean anything. In contrast, the main purpose of Design-Oriented Analysis is to understand how circuits work.
So what do minus signs mean? Fortunately, when applied to transfer functions, amplifier or filter gains, etc., signs (or even phase angles) usually aren’t very important. They tell you the delay or phase shift between input and output. But we usually don’t care about that delay. (For example, we don’t care how many cycles we missed while the signal was being transmitted to our cell phone.)
On the other hand, if signals that go through two paths are going to be compared, added or subtracted, their signs and phases must be right. Getting the sign of a feedback signal wrong can turn a bias circuit into a latch. Swapping a sign can turn a band-pass fliter into a band-reject filter.
So what to do? My suggestion is to use the meaning of the circuit to make sure the signs are right. The common-source and common-emitter configurations are the only ones that invert phase. The others, common-gate, common-base, source-followers and emitter-followers, do not invert. Current mirrors invert phase because they have common-source or common-emitter paths. Resistive paths do not invert phase. Diode-connected transistors are equivalent to resistors at DC, and so they do not invert phase either. Only common-source or common-emitters invert.
So, when you’re doing a feedback loop-gain analysis, and you want to know whether the feedback is positive or negative, don’t bother to keep track of signs. Just count the number of common-source/emitter stages. If the feedback loop includes an even number of inversions, then the feedback is positive. If there is an odd number of inverting stages (probably one or three) then the feedback is negative.
This works great for getting the signs right in feedback loops. Now, unfortunately, there are times where you just have to be careful and systematic. But even in those cases, always remember at the end of your algebraic analysis to turn your mind back on and think about what your results mean. If your results don’t make sense physically, then they’re probably wrong and you need to go back and find out why. Sign confusion is the most common cause of errors in algebraic derivations.
