# Can a controller have a negative gain?

I'm working on a problem that involves a Ćuk converter. The question asks "find k_c (a gain for a controller) such that the system is marginally stable".

So I got the properly space-state model with matrices A and B. Then, I set the real part of the eigenvalues of A equal to zero and try to solve for the unknown k_c. After that I got that kc must be equal to -1.8 approximately. That is, a negative gain.

1. Is it physically feasible to implement in real life?
2. Does it make a difference to use a positive gain or a negative gain?
• Generally, the negative gain means only that the voltage is inverted. – Marko Buršič Sep 8 '17 at 17:45
• If it an inverting converter, it must have negative gain. – Sunnyskyguy EE75 Sep 8 '17 at 17:58

1) Sure, this inverting amplifier circuit has a negative gain. Gain = output/input, so a negative gain simply means that the polarity of the signal is inverted between input and output.

2) OH YES in a system with negative feedback, like this: The sign of the feedback signal (running through (H) ) cannot be changed sign-wise or the feedback would not longer be negative feedback but positive. That results in a completely different system behavior.

Imagine cruise-control in your car, when going up a hill the cruise-control compensates to keep the speed constant. That's negative feedback, the speed becomes less (we're going up a hill) power is increased. In a positive feedback system, power would be decreased so you'd slow down ! That's not what you want.

If it an inverting converter, it must have negative gain.

But if not then the negative feedback would cause positive loop gain and if loop gain =>1 @ 360 deg then it is unstable.

Plot the voltage conversion ratio as a function of the control variable (in your case this is the duty cycle). You will see that the slope (DC gain of your small signal model) is always negative. You must therefore have another negative gain in the forward path (controller as you suggested). The same occurs with an LLC converter operating in the inductive region.