Pole-splitting

If phantom zero compensation is not possible, frequency compensation may also be accomplished by changing the initial positions of the poles of the loop gain. A compensation technique based on this principle is the application of local capacitive feedback around one or more gain stages in the controller. With this technique, the distance between two poles can be increased while their product remains virtually unaffected. This technique is called pole-splitting. Pole-splitting reduces the controller gain in the frequency range between the two poles. In fact, it trades controller gain with local loop gain in the gain stages enclosed in the capacitive feedback loop. Reduction of the controller gain may adversely affect the nonlinearity of the feedback amplifier. Since phantom zero compensation increases the controller gain, it is preferred over pole-splitting. Pole-splitting should only be applied if phantom zero compensation is not feasible.

Pole-splitting in operational amplifier circuits

Fig. 416A shows a gain stage that is part of a controller. This gain stage consists of a single-pole operational amplifier driven from an \(RC\) network.

Fig. 416 A: Controller gain stage with a single-pole operational amplifier B: Gain stage with pole-splitting.

The two poles of this circuit can be estimated from network inspection: \(p_{1}=-\frac{1}{R_{a}C_{a}}\) and \(p_{2}=-\frac{1}{\tau}\). These poles are the solutions of the characteristic equation.

The determinant of the MNA matrix.

The characteristic equation can be calculated symbolically by SLICAP. The netlist file of the circuit from Fig. 416A has is below:

"Pole splitting with opamps"
* file: poleSplitOpamp.cir
* SLiCAP netlist file
V1 1 0 0
R1 1 2 {R_a}
C1 2 0 {C_a}
E1 0 3 2 0 {A_0/(1+s*tau)}
C2 2 3 {C_c}
.param C_c=0
.end

The script for calculating the poles (with \(C_{c}=0\)) is:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# File: poleSplitOpAmp.py

from SLiCAP import *

fileName = 'poleSplitOpamp'
prj = initProject(fileName)
i1 = instruction()
i1.setCircuit(fileName + '.cir')
i1.setSimType('numeric') # This substitutes C_c=0 in the element expressions
i1.setDataType('poles')
i1.setGainType('gain')
i1.setDetector('V_3')
poles = i1.execute().poles
print("Poles:",poles)

The result is shown in the PYTHON console window:

Poles: [-1/tau, -1/(C_a*R_a)]

This corresponds with the estimation above.

After adding \(C_{c}\) to the circuit as shown in Fig. 416B, the characteristic equation of the network changes to:

(174)\[1+s\left( R_{a}\left( C_{a}+C_{c}\left( 1+A_{0}\right) \right) +\tau\right) +R_{a}\left( C_{a}+C_{c}\right) \tau s^{2}=0. \label{eq-poleSplitOpamp}\]

Insertion of \(C_{c}\) did not change the order of the characteristic equation. This is because C2 (with capacitance \(C_{c}\)) is part of a loop of capacitors and voltage sources. Hence, it does not introduce a new independent capacitor voltage.

Expression ((174)) shows if \(C_{c}\ll C_{a}\) the product of the poles \(\frac{1}{\left( C_{a}+C_{c}\right) \tau}\) does not change with \(C_{a}\). However, if \(C_{c}\left( 1+A_{0}\right) \gg C_{a}+\tau\), the magnitude of the sum of the poles increases considerably with \(C_{c}\). This means that one of the poles must shift to a higher frequency while the other one moves to a lower frequency. This will be illustrated with a numeric example.

Example

Fig. 417 Pole-splitting in the gain stage from Fig. 416 . The pole with the highest frequency moves away from the origin while increasing \(C_c\)

We will plot the pole positions of the circuit from Fig. 416B as a function of \(C_{c}\). In this numerical example, we use: \(C_{a}=100\)pF, \(R_{a}=10\)k\(\Omega\), \(\tau=1\)ms and \(A_{0}=10^{5}\). Fig. 417 shows the pole positions as a function of \(C_{c}\). It can clearly be seen that the pole with the highest frequency moves to a higher frequency.

Fig. 419 shows the low-frequency pole moving towards the origin for increasing \(C_{c}\).

Pole-splitting can also clearly be observed in the voltage transfer of the network. Fig. 418 shows the magnitude plots of the voltage transfer with \(C_{c}\) stepping from \(0\cdots2\)pF. At very high frequencies, all curves approach the same asymptote. This indicates that the product of the poles is not affected by \(C_{c}\).

Fig. 418 Pole splitting in the circuit from Fig. 416B.

Fig. 419 Pole splitting in the gain stage from Fig. 416 . The pole with the lowest frequency moves towards the origin while increasing \(C_c\).

Undesired pole-splitting may occur due to parasitic capacitances that establish negative feedback around gain stages in a feedback amplifier. As a consequence, poles that move towards higher frequencies may no longer belong to the dominant group. This results in reduction of the achievable MFM bandwidth of the amplifier.

Miller effect

The earliest description of pole-splitting in amplifier stages dates from 1920.[41] In the honour of its discoverer, this phenomenon is often referred to as the Miller effect.

Fig. 420 shows the small-signal equivalent circuit of a current-driven basic amplification stage driving an RC load (\(R_{\ell} || C_{\ell}\)). In a case of vacuum tubes and MOS transistors, the resistance \(r_{i}\) is extremely large and is usually left out of the model. If the reverse capacitance \(c_{r}\) equals zero, the poles of the transimpedance gain can easily be found from network inspection: \(p_{1}=-\frac{1}{r_{i}c_{i}}\) and \(p_{2}=-\frac{1}{R_{\ell}C_{\ell}}\).

Fig. 420 Pole-splitting in a basic amplifier stage caused by the reverse transfer capacitance \(c_r\).

If \(c_{r}\ll c_{i}\) and \(c_{r}\ll C_{\ell},\) the\(\ \)reverse capacitance does not affect the product of the poles. If, in addition, \(g_{m}R_{\ell}\gg1\) and \(g_{m}R_{\ell}c_{r}\gg c_{i}\), the sum of the poles increases considerably with \(c_{r}\) and pole-splitting occurs. This can be seen by solving the characteristic equation of the circuit.

A similar circuit is analysed in example example-MillerNetworkTheory.

Interaction with other performance aspects

We have seen that pole-splitting reduces the controller gain in the frequency range between the split poles. This may increase the nonlinearity of the feedback amplifier. This may be of particular interest if an amplifier stage with a current-limiting character drives a stage in which pole-splitting has been implemented. This is because an amplifier stage in which pole splitting has been implemented by means of parallel capacitive feedback obtains an integrating transimpedance character. If such a stage is driven from a current-limiting stage, the rate of change of the output voltage is limited to the quotient of the drive current and the feedback capacitance. In fact, pole-splitting by means of local capacitive feedback is the common cause of the voltage slew rate limitation as it occurs in operational amplifiers.

The deterioration of the signal-to-noise ratio and of the power efficiency that is associated with this technique can usually be neglected. This is because the feedback capacitance that provides the pole splitting is usually small with respect to existing capacitances in the amplifier stage.