Negative-feedback biasing#

Changes in the DC operating point are caused by supply changes, device tolerances, biasing errors and temperature changes. These changes are usually slow and have only frequency components at very low frequencies. If those frequencies do not occur in the signal, a distinction between biasing and signal components can be made in the frequency domain.

Fig. 311 shows a possible concept of negative feedback biasing for the voltage amplifier from Fig. 311A.

The integrating transadmittance amplifier G1 is the bias loop controller. It acts as a first-order low-pass filter with an infinite DC gain. Hence, for DC, it can be replaced with a nullor. As a consequence, the DC output voltage of the amplifier will equal $V_{ref}$. Since both the input impedance and the output impedance are infinite, the source-to-load transfer, the noise and the power efficiency will not be affected by the bias control loop. This, of course, is under the assumption that the integrator gain $g_B$ has been properly designed. In section High-pass cut-off and DC loop gain, we will study the design of the high-pass cut-off frequency and in Chapter Frequency compensation, we will discuss the design of the high-pass response and see the way in which the stability of negative feedback biasing can be assured. In the following example, we will derive symbolic expressions for the DC output voltage and the voltage transfer from the source to the output.

Example

\label{example-feedbackBiasing}

The netlist of the amplifier from Fig. 311A is shown below:

 1VampFeedbackBiasTotal
 2* file: VampFeedbackBiasTotal.cir
 3* SLiCAP netlist file
 4V1 1    0 V value = {V_s}
 5V2 6    0 V value = 0 dc={V_ref}
 6R1 1    2 {R_s}
 7R2 4    5 r value={R} dcvar={(R*sigma_r)^2}
 8R3 out  4 r value={19*R} dcvar={(19*R*sigma_r)^2}
 9C1 2    3 {C_a}
10C2 5    0 {C_b}
11X1 3 4 out  0 O_dcvar ; amplifier controller 
12+ sib={I_b*sigma_Ib} 
13+ sio={i_off} 
14+ svo={v_off} 
15+ iib={I_b}
16G1 3 0 out 6 {g_B/s}  ; bias loop controller
17.end  

Below, the SLICAP script for evaluation of the DC output voltage, the source-to-load transfer and the high-frequency approximation of the source-to-load transfer.

 1#!/usr/bin/env python3
 2# -*- coding: utf-8 -*-
 3# File VampFeedbackBiasTotal.py
 4
 5from SLiCAP import *
 6
 7fileName = 'VampFeedbackBiasTotal'
 8i1 = instruction()               # Creates instance of instruction object
 9i1.setCircuit(fileName + '.cir') # Checks, defines the local circuit object,
10                                 # and sets the index page to the circuit 
11                                 # index page
12i1.setSimType('symbolic')
13i1.setGainType('vi')
14i1.setDataType('dc')
15i1.setSource('V1')
16i1.setDetector('V_out')
17result = i1.execute()
18
19htmlPage('Feedback biasing')
20text2html('The DC output voltage $V_{outDC}$ is:')
21eqn2html('V_outDC', result.dc)
22
23# Laplace transfer function with feedback biasing
24
25i1.setGainType('gain')
26i1.setDataType('laplace')
27result = i1.execute()
28text2html('The voltage transfer $A_v$ from source to load is:')
29eqn2html('A_v', normalizeRational(result.laplace))
30hf = sp.limit(result.laplace, ini.Laplace, 'oo')
31text2html('For high frequencies this can be written as:')
32eqn2html('A_v', hf)

The html page with the results is shown in Fig. 312.

The results comply with our expectations:

1. The DC output voltage equals that of the reference voltage source.

2. The transfer has three poles and three zeros.

3. The high-frequency transfer equals the ideal gain of $20$.

Fig. 311B shows a possible implementation of this concept. The transadmittance gain $g_B$ is defined by the gain of the transimpedance integrator and the values of R4 and R5:

Auto-zero biasing#

If, during a short time interval, the load signal is not of interest, this time interval may be used to correct biasing errors with a feedback control signal. The corrected biasing should be maintained during the time of interest of the signal, which requires a memory element. If the load signal is not allowed to change due to this auto-zero process, a second memory element is required for storing the signal and passing it to the load during the auto-zero time interval. This principle is applied in so-called auto-zero operational amplifiers.