In my previous articles, we discussed op-amp frequency compensation and one compensation method via shunt capacitance.

The frequency compensation technique in widest use today is called Miller frequency compensation, which we will explore in this article.

### What Is Miller Compensation?

Miller compensation is a technique for stabilizing op-amps by means of a capacitance *C _{ƒ}* connected in

*negative-feedback fashion*across one of the

*internal gain stages*, typically the second stage.

### Utilizing Miller Compensation

Using the PSpice circuit of Figure 1, which was introduced in the previous article on frequency compensation, we obtain the magnitude/phase plots of Figure 2, showing that the presence of *C _{ƒ}* causes the pole frequencies to

*split apart*. Specifically, the higher the value of

*C*is, the further apart the pole frequencies and, hence, the wider the intermediate frequency region of phase shifts close to –90°.

_{ƒ}

**Figure 1.** PSpice circuit to plot the open-loop gain magnitude and phase for different amounts of Miller compensation.

**Figure 1.**PSpice circuit to plot the open-loop gain magnitude and phase for different amounts of Miller compensation.

Figure 3 provides an expanded view of the region of the crossover frequencies to facilitate a visual estimation of phase margins. Given a magnitude curve, (1) we identify the position of its crossover frequency *ƒ _{x}* on the 0-dB axis, (2) then we turn to the corresponding phase curve below, and (3) finally we read the phase shift

*φ*at the right. Then the phase margin is

_{x}*φ*= 180° +

_{m}*φ*. For instance, for

_{x}*C*= 8 pF, corresponding to the 4

_{ƒ}^{th}curve after the

*C*= 0 curve, we estimate

_{ƒ}*φ*≈ –120°, so

_{x}*φ*≈ 60°.

_{m}Conversely, we can visually come up with a rough estimation for the value of *C _{ƒ}* required for a given

*φ*, and then refine

_{m}*C*using the trial-and-error approach via PSpice. As an example, for

_{ƒ}*φ*≈ 65.5°, which marks the onset of AC peaking, the above procedure yields

_{m}*C*= 9.90 pF. The corresponding pole frequencies are measured to be 63.4 Hz and 12.2 MHz.

_{ƒ}

**Figure 2.** Magnitude/phase plots of the circuit of Figure 1 for different values of the compensating capacitance C_{ƒ}: 0, 1 pF, 2 pF, 4 pF, 8 pF, 16 pF, and 32 pF.

**Figure 2.**Magnitude/phase plots of the circuit of Figure 1 for different values of the compensating capacitance C

_{ƒ}: 0, 1 pF, 2 pF, 4 pF, 8 pF, 16 pF, and 32 pF.

**Figure 3.** Expanded view of the area of the crossover frequencies of Figure 2.

**Figure 3.**Expanded view of the area of the crossover frequencies of Figure 2.

Using the PSpice circuit of Figure 4 with *C _{ƒ}* = 9.90 pF to provide Miller compensation, we get the plots of Figure 5, all of which are exempt from peaking!

**Figure 4**. PSpice circuit to plot closed-loop gains in 20-dB steps determined by R_{4}.

**Figure 4**. PSpice circuit to plot closed-loop gains in 20-dB steps determined by R

_{4}.

**Figure 5. ** Stepped responses of the PSpice circuit of Figure 4 after Miller compensation with C_{ƒ} = 9.90 pF.

**Figure 5.**Stepped responses of the PSpice circuit of Figure 4 after Miller compensation with C

_{ƒ}= 9.90 pF.

### The Miller Effect for Capacitance

In the previous article on frequency compensation, we found that making the first pole dominant required a shunt capacitance of* tens of* *nanofarads*. Miller compensation, on the other hand, requires only *picofarads*.

How come? The answer is provided by the Miller effect.

The Miller effect refers to the increase in equivalent capacitance that occurs when a capacitor is connected from the input to the output of an amplifier with large negative gain.

This concept is illustrated in Figure 6 for the capacitance case.

*(a) (b)*

**Figure 6**. Illustrating the Miller effect for capacitance.

**Figure 6**. Illustrating the Miller effect for capacitance.

In response to an applied voltage *v*, as in Figure *6(a)*, a capacitor *C* responds with the current $$i = C\frac{dv}{dt}$$. If we now connect the same capacitor *C* in feedback fashion across an *inverting voltage amplifier* with gain –*a _{v}*, as depicted in Figure

*6(b)*, then the current becomes

#### $$i=C\frac{d[v-(-a_{v}v)]}{dt}=C\frac{dv(1+a_{v})}{dt}=C_{M}\frac{dv}{dt}$$

**Equation 1**

**Equation 1**

### Miller Capacitance

The quantity *C _{M }*in Equation 1 is referred to as the

*Miller capacitance*and is calculated as follows

#### $$C_{M}=(1+a_{v})C$$

**Equation 2. **The Miller capacitance

**Equation 2.**The Miller capacitance

In words, the feedback capacitance *C* reflected to the input, gets multiplied by 1 + *a _{v}*. This makes it possible to synthesize large capacitances with relatively small physical capacitors.

With reference to the PSpice circuit of Figure 4, we have

*C _{M}* = (1 +

*G*)

_{m2}R_{2}*C*= (1 + 250)9.90 pF = 2.485 nF

_{ƒ}

The total capacitance seen by *R _{1}* is

*C*=

_{total}*C*+

_{M}*C*= 2.51 nF, so the dominant pole frequency is 1/(

_{1}*2πR*

_{1}C_{total}) = 63.4 Hz, in agreement with the value measured above via PSpice.

A systematic analysis of pole splitting, which is beyond the scope of this article, indicates that with Miller compensation, the new pole frequencies are approximately related to the originals *ƒ _{1}* and

*ƒ*as

_{2}

#### $$f_{1(new)}≈f_{1}\frac{1}{G_{m2}R_{2}}\times \frac{C_{1}}{C_{f}}$$

#### $$f_{2(new)}≈f_{2}\times G_{m2}R_{2}\times \frac{C_{f}}{C_{1}+C_{f}(1+\frac{C_{1}}{C_{2}})}$$

**Equation 3**

**Equation 3**

where *ƒ _{1}* = 1/(

*2πR*) and

_{1}C_{1}*ƒ*= 1/(

_{2}*2πR*). Since

_{2}C_{2}*ƒ*is

_{2(new)}*directly*proportional and

*ƒ*is

_{1(new)}*inversely*proportional to the second-stage gain

*G*, it is apparent that the larger this gain is, the wider the pole separation for a given

_{m2}R_{2}*C*. This is highly desirable because with a sufficiently high gain, the required

_{ƒ}*C*for a given phase margin can be kept small enough (not more than a few tens of picofarads) so it can be fabricated on-chip. Moreover, the smaller

_{ƒ}*C*is, the faster the op-amp dynamics, since the open-loop bandwidth, slew rate, and full power bandwidth are all inversely proportional to

_{ƒ}*C*.

_{ƒ}

### A Bit of History

The first integrated circuit (IC) op-amp to incorporate full compensation was the venerable µA741 op-amp (Fairchild Semiconductor, 1968), which used a 30-pF on-chip capacitor for Miller compensation. The open-loop gain characteristics of the µA741 macro model available in PSpice are shown in Figure 7.

**Figure 7**. Plotting the open-loop gain a of the µA741 op-amp.

**Figure 7**. Plotting the open-loop gain a of the µA741 op-amp.

The magnitude curve crosses the 0-dB axis at *ƒ _{x}* = 888.2 kHz, where Ph[

*a*] = –117°, for a phase margin of

*φ*≈ 63°. Things go

_{m}*as if*the µA741 had a second pole at

*ƒ*= tan(

_{2}*φ*)

_{m}*ƒ*= 1.743 MHz.

_{x}Before the advent of the µA741, all IC op-amps had to be compensated *externally* by the user. A popular uncompensated contemporary of the µA741was the LM301 (National Semiconductor), which offered the user three compensation options to meet different objectives: the *single-pole* compensation, the *double-pole* compensations, and the *feedforward* compensation.

Even though the µA741 offered far less compensation flexibility than the LM301, the mA741 took off meteorically, most likely because many users were relieved of the oft-unpleasant task of providing external compensation without having a thorough understanding of its inner workings.

In the next article, we'll look at decompensated operational amplifiers.

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