Feedback biasing frequency compensation

Until now, we have only paid attention to the design of the low-pass roll-off characteristic of the transfer of negative feedback amplifiers. Any transfer of a physical system will show a low-pass behavior due to the fundamental limitation of speed.

Amplifiers that exhibit AC coupling, may also show a high-pass character at low frequencies. AC coupling may be the result of the design of the desired transfer function or of the biasing. In the negative-feedback integrator, discussed in section High-pass cut-off and DC loop gain, AC coupling and the high-pass character of the servo function were a result of the design of the integration operation. In the voltage amplifier discussed in section Evaluation of biasing errors, the high-pass character of the transfer was implemented during biasing.

In this section, we will discuss the design of the high-pass characteristic when using negative feedback biasing.

Negative feedback biasing concepts

Negative feedback biasing, as it was introduced in section Evaluation of biasing errors, can only be applied if DC signal components are not of interest. In that case DC bias quantities can be separated from the signal. A DC control loop that keeps the bias quantities in their desired range can then be added to the amplifier without affecting the transmission coefficients of the amplifier in the frequency range of interest.

Fig. 436 shows two arrangements in which a bias loop controls the DC operating voltage of the output port of a negative feedback amplifier. Fig. 436A shows a circuit concept in which the bias controller is modeled as a voltage-controlled current source. Fig. 436B shows a circuit concept in which the bias controller is modeled as a voltage-controlled voltage source. Similar arrangements can be designed to control the DC operating current of the output port.

Fig. 436 Feedback biasing concepts A. Negative feedback biasing concept with VCCS bias controller B. Negative feedback biasing concept with VCVS bias controller

In the circuit from Fig. 436A, the bias circuit controls the DC operating voltage of the output port by adding a DC current to the input of the negative feedback amplifier. This feedback biasing scheme can be applied if the negative feedback amplifier has a nonzero DC transimpedance factor. AC coupling at the source or the load is not always necessary. It has to be applied if no DC current is allowed to flow through the source or the load and if the DC source impedance or the DC load impedance equals zero.

In the circuit from Fig. 436B, the bias circuit controls the DC operating voltage of the output port by adding a DC voltage to its input. This feedback biasing scheme can be applied if the negative feedback amplifier has a nonzero DC voltage gain factor. AC coupling at the source or the load is not always necessary. However, it has to be applied if no DC current is allowed to flow through the source or the load and if the DC load impedance equals zero. If the source has been AC coupled to the amplifier, a nonzero DC bypass has to be created for setting the DC voltage at the noninverting input of the feedback amplifier. In the circuit from Fig. 436B, \(Z_{p}\) performs this task.

Dynamic behavior with feedback biasing

In the frequency range of interest, the values of the transmission parameters of the amplifier should not be affected by the biasing. Hence, at those frequencies, the coupling capacitors should behave as short circuits and the gain in the bias control loop should approximate zero. In the concept from Fig. 436A, the transmission parameter \(C\) can be affected by the bias loop, while in the concept from Fig. 436A, the transmission parameter \(A\) could be changed. Aside from affecting the transmission parameters of the amplifier, too small values of the coupling capacitors also deteriorate the signal-to-noise ratio and the power efficiency.

The poles introduced by the coupling capacitors, as well as the dynamic behavior of the controller, should be designed such that a stable MFM high-pass character will be obtained. The general approach to the design of feedback biasing circuitry is as follows:

1. Use the idealized model for the negative feedback amplifier.

   This is allowed if the low-pass cut-off and the high-pass cut-off of the amplifier are well separated.

2. If necessary, apply AC coupling of the source and the load and define the lower limits of the coupling capacitors on grounds of:

   1. Their influence in the noise behavior

   2. The high-pass cut-off frequency

   3. The deterioration of the power efficiency of the amplifier

3. Design the desired transfer characteristic of the bias controller.

   Including the frequency compensation of the biasing.

   This is done by evaluating the source-load transfer of the amplifier, including its bias loop, and equating the coefficients of \(s\) of this transfer with those of the desired high-pass characteristic.

We will demonstrate this for the design of the biasing of the charge integrator from example example-IntegratorHighPass.

Example

In example example-IntegratorHighPass, we discussed the design of the high-pass behavior of a current integrator. We derived a requirement for the DC gain of the controller such that the high-pass cut-off of the integrator was below \(1\)kHz. However, we did not finalize the design and did not focus on the biasing. The design of the biasing will be discussed in this example. We will use the following additional requirements for the biasing of the integrator:

1. A bias current of maximally \(\pm1\mu\)A is allowed to flow through the source.

2. A DC current of maximally \(+2.5\)mA is allowed to flow through the load.

3. The DC voltage at the output of the amplifier should be \(2.5\pm0.01\)V.

4. A power supply of \(\pm5\)V is available.

Fig. 437 shows the current integrator in which an operational amplifier has been used for the controller and feedback biasing according to Fig. 436B has been applied. The aim of the biasing circuit is to keep the quiescent DC output voltage within specifications.

Fig. 437 Current integrator from example example-IntegratorHighPass with negative feedback biasing according to Fig. 436B.

During the design of the biasing, the transfer of the reference voltage \(V_{R}\) to the DC output voltage is of interest. The ideal value of this transfer should be unity. We will use the asymptotic gain model to evaluate the dynamic behavior of this biasing circuit and determine the requirements for the controller. To this end, we select \(A_{v}\) of the controller as the reference variable and evaluate the asymptotic gain, the loop gain and the servo function. With \(A_{v}\) as the loop gain reference variable, the asymptotic gain equals unity. Hence, it equals the ideal gain and the servo function will describe the deviation from this ideal transfer.

In order to find an expression for the servo function, we will determine the loop gain \(L_{B}(s)\) of the biasing loop. For the evaluation of the loop gain, we will approximate the dynamic behavior of the integrator by its low-frequency behavior.

Low-frequency behavior is the behavior below midband frequencies as described in section Filter design approach.

By doing so, we obtain

\[L_{B}(s)=-A_{v}(s)\frac{A_{0}}{1+sA_{0}R_{s}C_{i}}.\]

We can now evaluate the servo function of the bias loop:

\[\frac{-L_{B}(s)}{1-L_{B}(s)}=\frac{A_{v}(s)A_{0}}{1+sA_{0}R_{s}C_{i} +A_{v}(s)A_{0}}.\]

The biasing accuracy will be completely determined by the DC properties of the bias controller if it approximates nullor behavior. For this reason, we will design the bias controller as an ideal integrator:

\[A_{v}(s)=\frac{1}{s\tau}.\]

With this controller, the servo function becomes:

\[\frac{-L_{B}(s)}{1-L_{B}(s)}=\frac{1}{1+s\frac{\tau}{A_{0}}+s^{2}\tau R_{s}C_{i}}.\]

For a second order cut-off at \(1\)kHz, we need

\[\tau=\frac{1}{4\pi^{2}10^{6}R_{s}C_{i}}=0.1\text{s.}\]

The loop gain has one pole in the origin (\(s=0\)) and one pole very close to the origin (\(s=-\frac{1}{2\pi A_{0}R_{s}C_{i}}\)Hz). A smooth MFM high-pass cut-off can be achieved with phantom zero compensation. With the two poles of the loop gain in the origin, this zero should be located at: \(z=-\frac{1000}{\sqrt{2}}\)Hz.

In the next example, we will verify the results with SLICAP.

Example

Fig. 438 Current integrator from Fig. 437 with negative feedback biasing with phantom zero compensation

Fig. 438 shows the circuit of the biased charge amplifier. The phantom zero for frequency compensation of the biasing has been implemented with \(C_{c}\). The OPA627 has been selected as the operational amplifier for implementation of both controllers. The SLICAP netlist of the small-signal equivalent circuit is listed below:

QampBias
* file: QampBias.cir
* SLiCAP circuit file
I1 0 1 0
R1 1 0 {R_s}
R2 2 0 {R_ell}
R3 2 4 {R_B}
C1 1 2 {C_i}
C2 2 4 {C_c}
C3 3 4 {C_B}
O1 3 1 2 0 OPA627
O2 0 4 3 0 OPA627
.param R_s=50k R_ell=2k C_i=5p R_B=1M C_B=100n C_c=0
.end

With \(R_{B}=1\)M\(\Omega\) and \(C_{B}=100\)nF, we have \(\tau=0.1\)s, and with \(C_{c}=220\)pF, we have the value of the phantom zero as defined in the previous example. The SLICAP script for plotting the magnitude characteristics of the uncompensated and compensated circuit, as well as for plotting the pole positions while stepping \(C_{c},\) is listed below:

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

from SLiCAP import *

fileName='QampBias'
prj = initProject(fileName)
i1 = instruction()
i1.setCircuit(fileName+ '.cir')
htmlPage('Circuit data')
netlist2html(fileName+ '.cir')
i1.setSource('I1')
i1.setDetector('V_2')
i1.setSimType('numeric')
i1.setGainType('gain')
i1.setDataType('laplace')
GainUncomp = i1.execute()
GainUncomp.label = 'uncomp.' # Assign a plot label to this result
i1.defPar('C_c','220p')
GainComp = i1.execute()
GainComp.label = 'phz comp.' # Assign a plot label to this result
htmlPage("Bode plots charge amplifier with feedback biasing")
dBmag = plotSweep('dBmagQamp', 'Charge amplifier with feedback biasing', [GainUncomp, GainComp], 100, 10e6, 500, funcType='dBmag', show=True)
phase = plotSweep('phaseQamp', 'Charge amplifier with feedback biasing', [GainUncomp, GainComp], 100, 10e6, 500, funcType='phase', show=True)
fig2html(dBmag, 800)
fig2html(phase, 800)
i1.setDataType('poles')
i1.stepOn()
i1.setStepVar('C_c')
i1.setStepStart(0)
i1.setStepStop('500p')
i1.setStepNum(20)
i1.setStepMethod('lin')
pzPlot = plotPZ('RLqAmp', 'Biased charge amplifier', i1.execute(), xmin=-3, xmax=0, ymin=-1.5, ymax=1.5, xscale='k', yscale='k', show=True)

The Bode plots are shown in Fig. 439. They clearly show a second order high-pass character of the integrator with a high-pass cut-off frequency of \(1\)kHz. This is exactly as designed. The uncompensated amplifier shows a large peaking in the vicinity of the high-pass cut-off frequency.

Fig. 439 Magnitude plots of the uncompensated and the compensated charge amplifier with negative feedback biasing.

Fig. 440 Poles of the gain of the charge amplifier as a function of \(C_c\) (25pF per step).

Fig. 440 shows that the two poles of the uncompensated amplifier, that determine the high-pass behavior are almost located on the imaginary axis. This confirms the peaking in the magnitude characteristic and is according to the calculations from the previous example. Fig. 440 also shows that phantom zero compensation with \(225\)pF brings the poles into MFM positions. The magnitude characteristic of the compensated amplifier shows a smooth transition from a differentiator below \(1\)kHz to an integrator above \(1\)kHz.