# Integrator Limitations: Magnitude and Phase Errors

## In this article, we will examine the deviation of the actual transfer function H from the ideal \(H_{ideal}\) in finer detail.

In our first article on integrators, we investigated the limitations stemming from using a constant gain-bandwidth-product (constant GBP) op-amp, and we found that the best frequency range of operation is between the op-amp’s pole frequency* f _{b}* and the integrator’s unity-gain frequency

*f*. This is the range over which the

_{0}*loop gain*T is maximized and, hence, the range over which the deviation of the

*actual*transfer function H from the

*ideal*H

_{ideal}is minimized.

We now wish to examine this deviation in finer detail. Before proceeding, we recall from our second article on integrator limitations that *H* is affected also by the op-amp’s output impedance z_{o}, because z_{o }allows for signal feed-through around the op-amp.

In the following, we assume |z_{o}| to be so much smaller than the integrator’s resistance *R*, that feed-through can be ignored. Under these conditions, we recycle our findings from previous articles and write, for the circuit of Figure 1(a),

\[H(j f )= \frac {V_o}{V_i} = H_{ideal}(j f) \frac{1}{1+1/T(j f)} \: \: \: \: \: \: \: \: \: \: \: \:(1)\]

where

\[H_{ideal}(j f) = \frac{-1}{jf/ f_o} \: \: \: \: \: \: \: \: \: \: \: \:(2)\]

\[f_o = \frac {1}{2 \pi RC} \: \: \: \: \: \: \: \: \: \: \: \:(3)\]

Moreover, the loop gain |*T*| is visualized as the difference between the open-loop gain |*a*| and the noise-gain |1/*β*| (see Figure 1(b), top), where

\[\frac {1}{\beta} = H_{ideal} +1 \: \: \: \: \: \: \: \: \: \: \: \:(4)\]

**Figure 1. **(a) Integrator, and (b) linearized plots of magnitude |H| (top) and phase Ph[H] (bottom).

**Figure 1.**(a) Integrator, and (b) linearized plots of magnitude |H| (top) and phase Ph[H] (bottom).

### Magnitude Error

Using Equations (1) and (2), we write

\[\left | H \right | = \left | H_{ideal} \right | \times \left | \frac {1}{1+1/T} \right | = \frac {1}{f/f_o} \times \frac {1}{\left |1+1/T \right |} = \frac {1}{f/(f_o/\left | 1+1/T \right |)}\]

indicating that magnitude-wise, the circuit still acts as an integrator, but with a unity-gain frequency of

\[f_o ' = \frac {f_o}{\left | 1+1/T \right |} \: \: \: \: \: \: \: \: \: \: (5)\]

With reference to Figure 1(b), top, we estimate |*T(jf _{0})*| by exploiting the constancy of the GBP: |

*T(jf*|×f

_{0})_{0}= 1×f

_{t}, which gives |

*T(jf*)| = f

_{0}_{t}/f

_{0}. Substituting into Equation (5) gives

\[f_o ' \cong \frac {f_o}{1+f_o/f_t} \: \: \: \: \: \: \: \: \: \: (6)\]

The downshift in the unity-gain frequency provides a convenient means for visualizing the departure of |*H*| from |*H _{ideal}*|. This departure is referred to as

*magnitude error*.

### Phase Error

By Equation (2), the phase angle of the integrator is, ideally, Ph[*H _{ideal}*] = Ph[–1] – Ph[

*j*] = 180 – 90 = 90°. However, because of the integrator’s pole pair, Ph[

*H*] will depart from 90° both at the low and at the high ends of its useful frequency range (see Fig, 1

*b*, bottom).

We will see in the next article that the phase error in the vicinity of *f _{0}* is of particular concern in integrator-based filters such as the state-variable and the biquad filter types. This error is ϵ

_{ϕ}= –tan

^{-1}(

*f/f*). A well-designed integrator has f

_{t}_{0}<< f

_{t}, so in the vicinity of f

_{0}we approximate as

\[\epsilon _\phi \cong -f/f_t \: \: \: \: \: \: \: \: (7)\]

**Figure 2.** (a) PSpice integrator with f_{0} = 100 kHz and f_{t} = 1 MHz. (b) Magnitude and phase plots.

**Figure 2.**(a) PSpice integrator with f

_{0}= 100 kHz and f

_{t}= 1 MHz. (b) Magnitude and phase plots.

As a practical example, consider the PSpice integrator of Figure 2(a), which uses a Laplace block to simulate an op-amp with GBP = 1 MHz. By Equation (3), *f _{0}* = 100 kHz.

Equations (6) and (7) provide the estimates *\(f_o ' \cong \)* 90.9 * *kHz and ϵ_{ϕ}\((f_o ') \cong \) -5.71°, in fair agreement with the measured values of 90.53 kHz and –4.64°.

We observe that the magnitude error is not necessarily bad because we can always compensate for it by suitably *predistorting* the value of the *RC* product. For instance, lowering C from 159.155 pF to 142.473 pF in Figure 1, while leaving *R* unchanged, will raise f_{0} by the amount necessary to ensure \(f_o ' = 100.0\) kHz.

Alternatively, we can leave *C* unchanged and lower *R* from 10.0 kΩ to 8.952 kΩ.

### Active Phase-Error Compensation

While the magnitude error can readily be compensated for via component-value predistortion, phase-error compensation requires that we introduce a suitable amount of *phase lead* to combat the phase *lag* due to the pole frequency *f _{t}*, which is responsible for said error.

In Figure 3, this lead is provided by *OA*_{2}, a matched twin of *OA _{1}*.

*OA*operates as a unity-gain voltage follower, so its closed-loop gain has a pole frequency of f

_{2}_{t}.

**Figure 3.** Active phase-error compensation circuit giving \(\epsilon _\phi \cong -(f/f_t)^3\).

**Figure 3.**Active phase-error compensation circuit giving \(\epsilon _\phi \cong -(f/f_t)^3\).

A detailed analysis indicates that near \(f_o '\) we now have

\[\epsilon _\phi \cong - (f/f_t)^3 \: \: \: \: \: \: \: \: \: \: (8)\]

which, for *f* << *f _{t}*, is clearly much better than the error of Equation (7).

**Figure 4. **(a) PSpice integrator with active compensation. (b) Magnitude and phase plots.

**Figure 4.**(a) PSpice integrator with active compensation. (b) Magnitude and phase plots.

As a practical example, consider the PSpice integrator of Figure 4(a), designed for *f _{0}* = 100 kHz and using matched op-amps with

*f*= 1 MHz.

_{t}Without compensation, we measure \(f_o' = 90.60\) kHz and \(\epsilon _\phi (f_o') = -4.71 \)°.

With compensation, we measure \(f_o' = 91.60\) kHz and \(\epsilon _\phi (f_o') = -0.04 \)°.

Figure 5 shows an alternative circuit for phase-error manipulation.

**Figure 5.** An alternative phase-error compensation circuit, giving \(\epsilon _\phi \cong + f/f_t\).

**Figure 5.**An alternative phase-error compensation circuit, giving \(\epsilon _\phi \cong + f/f_t\).

*OA*2 is now operated as a unity-gain inverting amplifier, so its closed-loop gain has a pole frequency of *f _{t }*/2. Due to the inversion provided by

*OA*

_{2}, the inputs of

*OA*

_{1}must be swapped, so the resulting transfer function

*H = V*is that of a

_{o}/V_{i}*non-inverting*integrator.

To compare its phase error to the non-compensated case, we need to plot Ph[*H*] + 180°, as depicted in Figure 6.

**Figure 6.** (a) PSpice integrator with an alternative phase compensation. (b) Magnitude and phase plots.

**Figure 6.**(a) PSpice integrator with an alternative phase compensation. (b) Magnitude and phase plots.

It can be shown that the phase error is now *positive*,

\[\epsilon _\phi \cong + f/f_t \: \: \: \: \: \: (9)\]

a property that comes in very handy when a compensated *non-inverting* integrator is connected in cascade with an *uncompensated* inverting type, so their opposing phase errors *cancel* each other out. In fact, cursor measurements in Figure 6 give \(f_o'\) = 93.48 kHz and ϵ_{ϕ} (\(f_o'\)) = +4.67°, the latter being almost exactly the opposite of the uncompensated value of \(\epsilon _\phi (f_o') = -4.71\)° found in connection with Figure 4.

This concludes the third article on integrator limitations. Please share questions and comments regarding the information here in the comments below.

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