Feedback Equation and Stability

Figure 5–7 shows the canonical form of a feedback loop with control system and electronic system terms. The terms make no difference except that they have meaning to the system engineers, but the math does have meaning, and it is identical for both types of terms.

The electronic terms and negative feedback sign are used in this analysis, because subsequent chapters deal with electronic applications. The output equation is written in Equation 5–1.

Figure 5–7. Comparison of Control and Electronic Canonical Feedback Systems

The error equation is written in Equation 5–2.

Combining Equations 5–1 and 5–2 yields Equation 5–3.

Collecting terms yields Equation 5–4.

Rearranging terms yields the classic form of the feedback Equation 5–5.

When the quantity Aβ in Equation 5–5 becomes very large with respect to one, the one can be neglected, and Equation 5–5 reduces to Equation 5–6, which is the ideal feedback equation. Under the conditions that Aβ >>1, the system gain is determined by the feedback factor β. Stable passive circuit components are used to implement the feedback factor, thus in the ideal situation, the closed loop gain is predictable and stable because β is predictable and stable.

The quantity Aβ is so important that it has been given a special name: loop gain. In Figure 5–7, when the voltage inputs are grounded (current inputs are opened) and the loop is broken, the calculated gain is the loop gain, Aβ. Now, keep in mind that we are using complex numbers, which have magnitude and direction. When the loop gain approaches minus one, or to express it mathematically 1∠–180°, Equation 5–5 approaches 1/0 ⇒ ∝ .

The circuit output heads for infinity as fast as it can using the equation of a straight line. If the output were not energy limited, the circuit would explode the world, but happily, it is energy limited, so somewhere it comes up against the limit.

Active devices in electronic circuits exhibit nonlinear phenomena when their output approaches a power supply rail, and the nonlinearity reduces the gain to the point where the loop gain no longer equals 1∠–180°. Now the circuit can do two things: first it can become stable at the power supply limit, or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail.

The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed and reapplied. The second state where the circuit bounces between power supply limits is named oscillatory.

Remember, the loop gain, Aβ, is the sole factor determining stability of the circuit or system. Inputs are grounded or disconnected, so they have no bearing on stability.

Equations 5–1 and 5–2 are combined and rearranged to yield Equation 5–7, which is the system or circuit error equation.

First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. As the loop gain increases, the error decreases, thus large loop gains are attractive for minimizing errors.