# Phase margin estimation using the closing rate

How can an engineer measure stability in an analog negative feedback circuit? For minimum phase circuits, you may consider using the closing rate.

In everyday analog design, engineers often need to quickly estimate the degree of stability (or lack thereof) of a negative feedback loop. A convenient tool that is applicable to minimum phase The circuits (so called because all their poles and zeros are in the left half of the complex plane) are the closing rate (ROC).

To prepare the background, consider the familiar block diagram in Figure 1st:

##### Figure 1. (a) Block diagram of a negative feedback circuit, and (b) display of the loop gain T.

This diagram consists of a error amplifier with profit a(jf), a feedback network with transfer function Î² (jf), and an adder block that generates the Smi error signal,

##### Equation (1)

Collect and solve for Smi gives

##### Equation (2)

where T = aÎ² it’s called the loop gain because any signal going into the amplifier and rotating clockwise around the loop will be amplified first a and then by Î², for a total gain of aÎ². Obviously bigger T leads smaller Smifor a given input Syo.

Rewriting as T = a / (1 / Î²), taking the logarithms, and multiplying by 20 to convert to decibels, gives

##### Equation (3)

which indicates that we can visualize the decibel graph of |T| As the difference between the decibel plots of |a| and |1 / Î²|. This is represented in the figure 1 B for the case of an op amp with constant bandwidth gain and independent frequency Î².

The frequency FX in which the two curves intersect, aptly called the crossover frequency, plays an important role in the stability of the circuit (or lack thereof). At this frequency, we have |T (jfX)| dB = 0, or |T (jfX) | = 1. If the phase ph

The[T (jfX)]ever reach â€“180 Â°, then we would have T (jfX) = â€“1, which, substituted in Eq. (2), indicates that Smi It would explode and lead to oscillation. (Note that even if you set Syo = 0, the intrinsic noise of the circuit would cause the accumulation of Smi.) To avoid oscillation, we must ensure that ph

The[T (jfX)]it is far from the dreaded value of â€“180 Â° for a sufficient quantity, appropriately called phase margin Ï†meter,

### Ï†meter = 180 Â° + ph[T(jf[T(jfX)]

##### Equation (4)

The phase margins of practical interest are Ï†meter = 90 Â°, Ï†meter = 65.5 Â° (which marks the beginning of in peak hours in the answer to c), Ï†meter = 76.3 Â° (which marks the start of buzz in the transient response), and Ï†meter = 45 Â° (which we will see to lend itself to easy geometric visualization, although it results in a 2.4 dB peak and a 23% excess timbre).