#### What is in this article?:

- Stability Criteria Of A Control System
- 3.2 Stability Criteria
- Stable or Unstable?

“Designing Control Loops for Linear and Switching Power Supplies” is the latest book from Christophe Basso, a past contributer to Power Electronics. This book focuses on what engineers really need to know to compensate or stabilize a given control system. This article contains excerpts from the section of the book covering stability criteria.

In the electronic field, an oscillator is a circuit capable of producing a self-sustained sinusoidal signal. In a lot of configurations, cranking up the oscillator involves the noise level inherent to the adopted electronic circuit. As the noise level grows at power-up, oscillations are started and self-sustained. This kind of circuit can be formed by assembling blocks such as those appearing in *Fig. 1*. As you can see, the configuration looks very similar to that of our control system arrangement.

In our example, the excitation input is not the noise but a voltage level, V_{in}, injected as the input variable to crank the oscillator. The direct path is made of the transfer function H(s), while the return path consists of the block G(s). To analyze the system, let us write its transfer function by expressing the output voltage versus the input variable:

If we expand this formula and factor V_{out}(s), we have

The transfer function of such a system is therefore

In this expression, the product G(s)H(s) is called the loop gain, also noted T(s).To transform our system into a self-sustained oscillator, an output signal must exist even if the input signal has disappeared. To satisfy such a goal, the following condition must be met:

To verify this equation in which V_{in} disappears, the quotient must go to infinity. The condition of the quotient to go to infinity is that its characteristic equation, D(s), equals zero:

1+ G(s)H(s) = 0 (5)

To meet this condition, the term G(s)H(s) must equal -1. Otherwise stated, the magnitude of the loop gain must be 1 and it sign should change to minus. A sign change with a sinusoidal signal is simply a 180° phase reversal. These two conditions can be mathematically noted as follows:

|G(s)H(s)| = 1 (6)

ArgG(s)H(s) = –180^{o} (7)

*Fig. 2*displays these two curves where conditions for oscillation are met. Should the system slightly deviate from these values (e.g., temperature drift, gain change), output oscillations would either exponentially decrease to zero or diverge in amplitude until the upper/lower power supply rail is reached. In an oscillator, the designer strives to reduce as much as possible the gain margin so the conditions for oscillations are satisfied for a wide range of operating conditions.