Black’s feedback model

Black’s feedback model#

In 1927, Black built the first negative-feedback amplifier [15]. He applied negative feedback to obtain linear and stable-gain repeater amplifiers for long-distance telephone systems. Black’s patent was awarded in 1937 [42]. The theoretical understanding of the high-frequency stability of negative feedback amplifiers was developed by Black in cooperation with Nyquist [45] and Bode [46].

The use of Black’s feedback model for negative feedback systems is wide - spread. The model is shown in Fig. 320. The model uses three basic building blocks: a feedback network, a subtracter and a loop amplifier or controller.

Model description#

We will now derive the input-output relation of a negative feedback system according to Black’s feedback model.

Let \(E_{i}\) be the input signal and \(E_{o}\) be the output signal of a feedback system that consists of a feedback element with a transfer \(k\), a controller with transfer \(H\) and a subtracter, as depicted in Fig. 320. From Chapter Design of feedback amplifier configurations, we know that the transfer \(k\) of the feedback element has been designed as the reciprocal gain of the desired input-to-output transfer of the feedback system. The output signal of the feedback element should be an accurate copy of the input signal of the system. Hence, the input signal of the controller, also called the error signal \(\varepsilon\), should approach zero. The error signal \(\varepsilon\) is the difference between the output signal of the feedback network and the input signal \(E_{i}\). According to the block diagram from Fig. 320 the error signal should satisfy two equations:

\[\begin{split}\varepsilon & =E_{i}-kE_{o},\\ \varepsilon & =\frac{1}{H}E_{o}.\end{split}\]

Hence, we may write:

\[\frac{1}{H}E_{o}=E_{i}-kE_{o},\]

or:

\[E_{o}\left( \frac{1}{H}+k\right) =E_{i},\]

from which we obtain the expression for the input-output relation of the feedback system:

(98)#\[\frac{E_{o}}{E_{i}}=\frac{H}{1+Hk}. \label{ex-Black}\]

If we consider \(H\) the gain of an amplifier, then we may say that negative feedback reduces the gain by a factor \((1+Hk)\). The product of the gain of the controller and that of the feedback network\ is called the loop gain \(Hk\). It is the gain enclosed in the loop that consists of the controller, the feedback element and the subtracter.

From a design point of view, we better consider \(H\) as the gain of a controller, that has to minimize the error between the \(E_{i}\) and the signal at the output of the feedback network. If the controller gain is infinite, the loop gain is infinite and the input-to-output transfer becomes the reciprocal transfer of the feedback network: the ideal gain of the negative feedback amplifier. This complies with the results from Chapter Design of feedback amplifier configurations :

\[\lim_{H\rightarrow\infty}\left( \frac{E_{o}}{E_{i}}\right) =\frac{1}{k}.\]

Hence, a design-friendly notation that explicitly writes ((98)) as the product of the intended gain, and a factor that represents the effect of a limited controller gain, is:

\[\frac{E_{o}}{E_{i}}=\frac{1}{k}\left( \frac{Hk}{1+Hk}\right) .\]

Application of the model#

Expression (98) describes the transfer from the input to the output of the system, rather than the transfer from the signal source to the load. This is a consequence of the modeling technique. Currents and voltages cannot easily be distinguished in these kinds of block diagrams.

The model also supposes ideal subtraction of the feedback signal from the input signal and no interaction between unilateral building blocks.

For a negative feedback voltage amplifier, for example, these conditions imply that the controller should have an infinite \(CMRR,\) infinite differential-mode and common-mode input impedances and zero output impedance. These, indeed, are prerequisites for voltage operational amplifiers, but they seriously complicate their design, because no electronic components with such properties exist.

In the following examples, we will show the application of Black’s feedback model for electronic circuits.

Example

.

Fig. 321 shows a passive-feedback voltage amplifier in which the controller has been modeled as a simple opamp: a voltage-controlled voltage source with voltage gain \(A_{v}\). The SLICAP netlist of this circuit is:

 1"Voltage amplifier with VCVS controller"
 2* file: vAmpBlack.cir
 3* SLiCAP circuit file
 4V1 1 0 {V_s}
 5R1 1 2 {R_s}
 6R2 3 0 {R_ell} 
 7E1 3 0 2 4 {A_v}
 8R3 3 4 {R_a}
 9R4 4 0 {R_b}
10.end

The following script can be used to obtain a symbolic expression for the transfer from V1 to the voltage across R2:

 1#!/usr/bin/env python3
 2# -*- coding: utf-8 -*-
 3# File: vAmpBlack.py
 4
 5from SLiCAP import *
 6
 7fileName = 'vAmpBlack'
 8prj = initProject(fileName)    # Creates the SLiCAP libraries and the
 9                               # project HTML index page
10i1 = instruction()             # Creates an instance of an instruction object
11i1.setCircuit(fileName+'.cir') # Checks and defines the local circuit object,
12                               # and sets the index page to the project index 
13i1.setSource('V1')
14i1.setDetector('V_3')
15i1.setSimType('symbolic')
16i1.setGainType('gain')
17i1.setDataType('laplace')
18result = i1.execute()
19
20htmlPage('Voltage amplifier with VCVS controller')
21text2html('The gain of the system is obtained as:')
22V_ell, V_s = sp.symbols('V_ell, V_s')
23eqn2html(V_ell/V_s, result.laplace)

The html page with the result of this run is shown in Fig. 322.

../_images/SLiCAP-vAmpBlack.svg

Fig. 322 SLiCAP simulation results for the negative feedback voltage amplifier with VCVS controller.#

The result can be written as

\[\frac{V_{\ell}}{V_{s}}=\frac{R_{a}+R_{b}}{R_{b}}\left( \frac{A_{v}\frac{R_{b}}{R_{a}+R_{b}}}{1+A_{v}\frac{R_{b}}{R_{a}+R_{b}}}\right) .\]

In this notation, we clearly see that the loop gain \(L\) can be written as the product of the voltage gain of the controller and the voltage gain of the feedback network:

\[L=A_{v}\frac{R_{b}}{R_{a}+R_{b}}.\]

The ideal gain is the gain of the amplifier in which the controller is replaced with a nullor. It equals the reciprocal value of the gain of the feedback network.

../_images/colorCode.svg

The above example shows the application of Black’s feedback model for a simple situation. In this particular case, it appears as if the model supports a two-step design approach: the source-to-load transfer can be written as the product of the ideal gain \(\frac{R_{a}+R_{b}}{R_{b}}\), and the servo function \(S\), which is uniquely defined by the loop gain:

(99)#\[S=\frac{L}{1+L}. \label{eq-servoBlack}\]

This servo function is a measure for the error between the ideal gain, and the actual gain in which the controller a high-gain amplifier, rather than a nullor.

It appears as if the model of Black supports a two-step design approach:

The first step is the design of the ideal transfer using a nullor as controller, while the second step comprises the design of a controller that provides a sufficiently large loop gain.

However, the above is only true because the controller in this voltage amplifier is an ideal voltage-controlled voltage source: it has an infinite input impedance and zero output impedance. In this case, this has the following consequences:

  1. The current through the source resistor \(R_{1}\) equals zero, thus the input voltage of the amplifier equals the source voltage.

  2. Current through the feedback network and current through load impedance do not affect the output voltage of the controller.

  3. There is no direct transfer from the source to the load through the feedback network.

In fact, for negative-feedback voltage amplifiers, one would like to use an ideal voltage-controlled voltage source as controller. This would make the loop gain independent from the source and load impedance. However, from a design point of view it is not at all necessary. The design goal is to design servo function transfer close to unity.

This makes Black’s feedback model not the best model for the design of negative feedback amplifiers. It does not support the two-step design approach.

We will elucidate this in the next example in which we select a current-controlled current source instead of a voltage-controlled voltage source as controller in the voltage amplifier.

Example

Fig. 323 shows the passive feedback voltage amplifier from Fig. 321 but with the controller replaced with a current-controlled current source. The SLiCAP netlist file for this model is:

 1"Voltage amplifier with CCCS controller"
 2* file: vAmpBlackF.cir
 3* SLiCAP circuit file
 4V1 1 0 {V_s}
 5R1 1 2 {R_s}
 6R2 3 0 {R_ell} 
 7F1 0 3 2 4 {A_i}  ;The SLiCAP syntax for a CCCS differs from SPICE syntax
 8R3 3 4 {R_a}
 9R4 4 0 {R_b}
10.end

The following script can be used to obtain a symbolic expression for the transfer from V1 to the voltage across R2:

 1#!/usr/bin/env python3
 2# -*- coding: utf-8 -*-
 3# File: vAmpBlackF.py
 4
 5from SLiCAP import *
 6
 7fileName = 'vAmpBlackF'
 8prj = initProject(fileName)    # Creates the SLiCAP libraries and the
 9                               # project HTML index page
10i1 = instruction()             # Creates an instance of an instruction object
11i1.setCircuit(fileName+'.cir') # Checks and defines the local circuit object,
12                               # and sets the index page to the project index 
13i1.setSource('V1')
14i1.setDetector('V_3')
15i1.setSimType('symbolic')
16i1.setGainType('gain')
17i1.setDataType('laplace')
18result = i1.execute()
19
20htmlPage('Voltage amplifier with CCCS controller')
21text2html('The gain of the system is obtained as:')
22V_ell, V_s = sp.symbols('V_ell, V_s')
23eqn2html(V_ell/V_s, result.laplace)

The result of the newly generated html page is shown in Fig. 324.

../_images/SLiCAP-vAmpBlackF.svg

Fig. 324 SLiCAP simulation results for the negative feedback voltage amplifier with CCCS controller.#

It is clear that the transfer now depends on the source impedance, the load impedance and the loop gain, while the loop gain cannot simply be obtained as the product of the gain of the controller and the transfer of the feedback network.

../_images/colorCode.svg

The above example shows that changing the controller type does not only change the expression of the source-to-load transfer, but also complicates the interpretation of the results that are obtained with Black’s feedback model.

In fact, there are three causes for the modification of the source-to-load transfer when we change the controller in the negative feedback voltage amplifier from a VCVS to a CCCS:

  1. With the CCCS, the source current is no longer zero, hence the input voltage of the circuit does not equal the source voltage.

  2. With the CCCS, the load voltage is not uniquely defined by the output quantity of the controller.

  3. With the CCCS, there exist two current paths from the source to the load with a nonzero transfer: one through the controller and one direct transfer from the source to the load through the feedback network. The latter one is not accounted for in Black’s feedback model, which assumes unidirectional transfer only.

In general, any practical controller circuit:

  1. will have nonzero finite port impedances

  2. will have bidirectional transfer

  3. does not show natural two-port behavior (see Chapter Two-port conditions for two-port conditions and natural two-port behavior).

Conclusions#

At this stage, we may conclude that Black’s feedback model is very well suited to system level analysis with building blocks that show no interaction and have unilateral behavior, but it is not very suited to the design and the analysis of negative feedback amplifiers.

For the analysis of feedback circuits to match those obtained with network analysis, we need a model that does not pose additional requirements to circuit parts that can be identified as controller or as feedback network. In the next section, we will show that the asymptotic gain model satisfies such requirements while also supporting the two-step design method in the best possible way.

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