Asymptotic gain model

Asymptotic gain model#

In this section, we will discuss the asymptotic gain model. With this model, the source-to-load transfer of a feedback amplifier can be written in a way that supports the two-step design method, while it provides exactly the same results as network theory. The asymptotic gain model will be derived from the superposition model.

Superposition model#

The superposition model gives a formal approach to the analysis of negative feedback amplifiers. The model is based upon the application of the superposition theorem that holds for linear networks. In order to apply the superposition theorem, the feedback amplifier is modeled as a network that has at least one controlled source \(E_{c}\) that is controlled by a controlling quantity \(E_{i}\). The reference variable \(A,\) as shown in Fig. 325, describes the relation between an arbitrarily selected controlled quantity \(E_{c}\) and its controlling quantity \(E_{i}:E_{c}=AE_{i}\).

../_images/superposition-controlled.svg

Fig. 325 The superposition model is based upon the representation of a linear network with one arbitrarily selected controlled source.#

Fig. 325 shows a simplified schematic representation of an electrical network according to the superposition model. The source quantity \(E_{s}\), the load quantity \(E_{\ell}\), the controlled quantity \(E_{c}\) and the controlling quantity \(E_{i}\) can be a current or a voltage.

In the superposition model, we will mathematically break the loop that exists from the input of the controlled source to its output by replacing the controlled source \(E_{c}\) with an independent source \(E_{c}\), as shown in Fig. 326.

With the aid of the superposition theorem, each current and voltage of the feedback amplifier can now be written as a linear combination of the two independent sources in the circuit: the current or voltage of the signal source (the source quantity \(E_{s}\)) and the current or voltage of the independent source \(E_{c}.\) In this way, we write both the load quantity \(E_{\ell}\)\ and the controlling quantity \(E_{i}\) as a linear combination of the two sources \(E_{s}\) and \(E_{c}\):

(100)#\[\begin{split}\left( \begin{array} [c]{c} E_{\ell}\\ E_{i} \end{array} \right) =\left( \begin{array} [c]{cc} \rho & \lambda\\ \kappa & \lambda\beta\kappa \end{array} \right) \left( \begin{array} [c]{c} E_{s}\\ E_{c} \end{array} \right) . \label{ex-matrixSuperposition}\end{split}\]

This is the superposition model and its model parameters are defined as:

  • \(\rho=\left. \frac{E_{\ell}}{E_{s}}\right\vert _{E_{c}=0}:\) Direct transfer from the source to the load;

  • \(\lambda=\left. \frac{E_{\ell}}{E_{c}}\right\vert _{E_{s}=0}:\) Transfer from \(E_{c}\) to the load;

  • \(\kappa=\left. \frac{E_{i}}{E_{s}}\right\vert _{E_{c}=0}:\) Transfer from the source to the driving quantity \(E_{i}\);

  • \(\lambda\beta\kappa=\left. \frac{E_{i}}{E_{c}}\right\vert _{E_{s}=0}:\) Transfer from \(E_{c}\) to \(E_{i}\); a nonzero value of \(\lambda\beta\kappa\) indicates the existence of feedback.

../_images/superposition.svg

Fig. 326 The superposition model. The gain of an arbitrarily selected controlled source is defined as ‘reference variable’ \(A\), and the controlled source (diamond) is replaced by the independent source (circle) \(E_{c}\).#

The feedback factor \(\beta\) is thus defined as

\[\beta=\frac{1}{\lambda\kappa}\left( \left. \frac{E_{i}}{E_{c}}\right\vert _{E_{s}=0}\right) .\]

Please note that all parameters depend on the selection of the loop gain reference variable \(A\):

\[A=\frac{E_{c}}{E_{i}}\]

At a later stage we show that the loop gain reference \(A\) can be chosen in such a way that \(-\beta\) equals the transfer of the feedback network. In such cases this feedback model provides useful design information.

Source-to-load transfer#

We will now derive an expression for the transfer from source-to-load \(A_{f}\) of a feedback amplifier. The model equations are:

(101)#\[\begin{split}E_{\ell} & =\rho E_{s}+\lambda E_{c},\label{supmodel1}\\ E_{i} & =\kappa E_{s}+\lambda\beta\kappa E_{c},\label{supmodel2}\\ E_{c} & =AE_{i}. \label{supmodel3}\end{split}\]

Substitution of (supmodel3) in (supmodel1) and in (supmodel2 ) yields:

(102)#\[\begin{split}E_{\ell} & =\rho E_{s}+\lambda AE_{i},\label{supmodel5}\\ E_{i} & =\kappa E_{s}+\lambda\beta\kappa AE_{i}. \label{supmodel6}\end{split}\]

We now solve \(E_{i}\) from (supmodel6) and substitute the result in (supmodel5). We obtain:

\[E_{i}=\frac{\kappa}{1-\lambda\beta\kappa A}E_{s}.\]

Substitution of this result in (supmodel5) yields:

\[E_{\ell}=E_{s}\left( \rho+\frac{\lambda A\kappa}{1-\lambda\beta\kappa A}\right) .\]

We can now express the source-to-load transfer in terms of the parameters of the superposition model:

(103)#\[A_{f}=\frac{E_{\ell}}{E_{s}}=\rho-\frac{1}{\beta}\frac{-L}{1-L}, \label{eq-superposition}\]

where \(L=\lambda\beta\kappa A\) is called the loop gain with respect to the reference variable \(A\). Negative feedback gives a negative value for \(L\). The factor \(1-L\) is called the return difference with respect to the reference variable \(A\). Expression (103) gives an accurate description of the transfer from source-to-load for any selection of the reference variable. A different selection of the reference variable \(A\), however, yields different values for the other model parameters \(\rho\) and \(L\).

The superposition model does not support the two-step design approach as described in the introduction: if we have no circuit, we cannot select a reference variable and we do not know about \(\rho,\) \(\kappa\) and \(\lambda\). The asymptotic gain model that will be described in the next section is suited for this purpose.

Asymptotic gain model#

When the loop gain \(L\) approaches infinity, the source-to-load transfer obtains its so-called asymptotic gain value \(A_{f\infty}\):

(104)#\[A_{f\infty}=\lim_{L\rightarrow-\infty}\frac{E_{\ell}}{E_{s}}=\rho-\frac{1}{\beta}. \label{eq-asymptotic}\]

With the aid of ((104)), we are able to express the source-to-load transfer \(\frac{E_{\ell}}{E_{s}}\) from ((103)) in the asymptotic gain \(A_{f\infty}\), the loop gain \(L\) and the direct transfer \(\rho\):

\[A_{f}=\frac{E_{\ell}}{E_{s}}=\rho\frac{-L}{1-L}+\rho\frac{1}{1-L}-\frac{1}{\beta}\frac{-L}{1-L},\]
(105)#\[A_{f}=A_{f\infty}\frac{-L}{1-L}+\frac{\rho}{1-L} \label{eq-asymp1}\]

This expression resembles the expression obtained by Black’s model. Before we draw any further conclusions, we will first study the influence of the direct transfer \(\rho\).

Influence of the direct transfer#

In most practical situations, the direct transfer can be neglected. This can be seen as follows. Assume the intended gain of the amplifier equals the ideal gain \(A_{i}\). Also assume the loop gain reference variable \(A\) will be selected in such a way that the asymptotic gain equals the ideal gain.

We may then write ((105)) in the form

(106)#\[A_{f}=A_{i}\left( \frac{-L}{1-L}\right) \left( 1-\frac{1}{L}\frac{\rho }{A_{i}}\right) . \label{eq-asymp2}\]

The direct transfer \(\rho\) is usually much smaller than the ideal transfer \(A_{i}\), so in most practical situations, we have

\[\left\vert \frac{\rho}{A_{i}}\right\vert \ll1.\]

In addition, for an accurate match between the actual source-to-load transfer and the ideal transfer of the amplifier, we need \(\left\vert L\right\vert \gg1\), which justifies the conclusion:

\[\left\vert \frac{1}{L}\frac{\rho}{A_{i}}\right\vert \ll1.\]

So, with proper selection of the reference variable, we may write:

(107)#\[A_{f}=A_{i}\frac{-L}{1-L}. \label{eq-asymptotic-gain}\]

Hence, if the asymptotic gain equals the ideal gain, the source-to-load transfer of a negative feedback amplifier can be written as the product of the ideal gain and the servo function \(S\):

(108)#\[S=\frac{-L}{1-L}. \label{eq-servoDef}\]

This servo function is uniquely defined by the loop gain.

Now, we have our two-step design approach: first, we design the ideal gain, and second, we design the controller such that the errors with respect to this ideal gain are sufficiently small.

Comparison with Black’s model#

Expression ((107)) has the same appearance as ((98)), but it is more arrurate and it does not require the assumptions of Black’s feedback model.

However, only if the reference variable has been selected such that the ideal gain \(A_{i}\) equals the asymptotic gain \(A_{f\infty}\), the loop gain \(L\) is the only measure for the correspondence between the ideal gain and the gain. If so, the asymptotic gain model supports the desired two-step design approach.

Please notice that the servo function according to the asymptotic gain model ((108)) has \(-L\) in the numerator and in the denominator, while the servo function according to Black’s feedback model ((99)) has \(+L\) in the numerator and the denominator. This is because the asymptotic gain model does not include a subtracter. In the asymptotic gain model, a negative value of the loop gain indicates negative feedback, while in Black’s feedback model, a positive value of the loop gain indicated negative feedback.

Selection of the loop gain reference#

The two-step approach requires proper selection of the loop gain reference variable \(A\).

First of all, this reference variable needs to be selected such that the asymptotic gain \(A_{f\infty}\) equals the ideal gain \(A_{i}\). Hence, the condition \(L\rightarrow\infty\) for the selected reference variable should be equal by replacing the controller with a nullor. In other words, when the loop gain approaches infinity, the voltage across the input terminals of the controller and the current through the input port of the controller should both become zero:

Definition

The asymptotic gain \(A_{f\infty}\) of a direct feedback amplifier equals its ideal gain\ if the reference variable \(A\)\ is selected such that both the input voltage and the input current of the controller approach zero if \(A\rightarrow\infty\).

Secondly, the reference variable should be selected such that the contribution of the direct transfer \(\rho\) to the gain \(A_{f}\) should be minimized. This can usually easily be achieved over the frequency range of interest. Outside this range, at relatively high frequencies, the contribution of the direct transfer may become dominant.

If the above criteria have been met, expression ((107)) describes the source-to-load transfer and the design of the amplifier can be performed in two subsequent steps:

  1. The design of the ideal gain, as discussed in Chapter Design of feedback amplifier configurations

  2. The design of a controller that provides a sufficiently large loop gain over the operating range of interest.

In the following examples, we will show that the source-to-load relation of a feedback amplifier is always accurately described by ((105)), while it is only accurately described by ((107)) if the loop gain reference variable \(A\) has been properly selected.

Example

Fig. 327A shows a negative feedback voltage amplifier built with a current feedback operational amplifier as the controller. A simplified small-signal model of this operational amplifier is shown in Fig. 327B.

../_images/OpAmp-CF-Small-Signal-Simple.svg

Fig. 327 Left: Voltage amplifier with passive direct negative feedback. Right: Strongly simplified small-signal model of the current feedback operational amplifier that is used as active part for the voltage amplifier.#

The ideal source-to-load voltage transfer \(A_{v}\) of this amplifier is obtained by replacing the controller with a nullor, as shown in Fig. 328; we obtain:

\[A_{v}=\frac{V_{\ell}}{V_{s}}=\frac{R_{a}+R_{b}}{R_{b}}\]

Fig. 329 shows the small-signal equivalent model of the complete amplifier. We will evaluate the asymptotic gain, the direct transfer, the loop gain and the source-to-load transfer of this amplifier with the aid of the asymptotic gain feedback model. We will do this for two different selections of the loop gain reference variable \(A\). First, we will select the gain \(g_{m}\) of the voltage-controlled current source as the loop gain reference, and then we will select the gain \(A_{r}\) of the current-controlled voltage source.

The SLICAP netlist for the circuit from Fig. 329 is:

 1cfbVamp 
 2* file: cfbVamp.cir
 3* SLiCAP circuit file
 4V1 1 0 {V_s}
 5R1 1 2 {R_s}
 6R2 4 0 {R_ell}
 7R3 4 5 {R_a}
 8R4 5 0 {R_b}
 9G1 3 5 2 5 {g_m}
10H1 4 0 0 3 H value={A_r}
11.end
../_images/cfbVampLoopModel.svg

Fig. 329 Small signal equivalent model of the voltage amplifier from Fig. 327.#

The first part of the script calculates the gain:

 1#!/usr/bin/env python3
 2# -*- coding: utf-8 -*-
 3# File: cfbVamp.py
 4
 5from SLiCAP import *
 6
 7fileName = 'cfbVamp'
 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_4')
15i1.setSimType('symbolic')
16i1.setGainType('gain')
17i1.setDataType('laplace')
18result = i1.execute()
19
20gain = result.laplace

The second part of the script calculates the parameters of the asymptotic gain model with the voltage-controlled current source G1 selected as the loop gain reference and generates an html page displaying the results:

 1V_ell, V_s, A_f                = sp.symbols('V_ell, V_s, A_f')
 2L_G1, rho_G1, S_G1, A_infty_G1 = sp.symbols('L_G1, rho_G1, S_G1, A_oo_G1')
 3L_H1, rho_H1, S_H1, A_infty_H1 = sp.symbols('L_H1, rho_H1, S_H1, A_oo_H1')
 4
 5# Calculations with H1 as loop gain reference
 6
 7i1.setLGref('G1')
 8i1.setGainType('asymptotic')
 9result = i1.execute()
10AG1 = result.laplace
11
12i1.setGainType('loopgain')
13result = i1.execute()
14LG1 = result.laplace
15
16i1.setGainType('servo')
17result = i1.execute()
18SG1 = result.laplace
19
20i1.setGainType('direct')
21result = i1.execute()
22DG1 = result.laplace
23
24htmlPage('Asymptotic-gain model G1 ref')
25text2html('The gain of the circuit is obtained as:')
26eqn2html(V_ell/V_s, gain)
27
28text2html('The asymptotic-gain $A_{\\infty_G1}$ is found as:')
29eqn2html(A_infty_G1, AG1)
30
31text2html('The loop gain $L_{G1}$ is found as:')
32eqn2html(L_G1, LG1)
33
34text2html('The servo function $S_{G1}$ is found as:')
35eqn2html(S_G1, SG1)
36
37text2html('The direct transfer $\\rho_{G1}$ is found as:')
38eqn2html(rho_G1, DG1)
39
40text2html('The gain $A_f$ calculated from $A_{\\infty_{G1}}$, $S_{G1}$ ' +
41          'and  $\\rho_{G1}$ is obtained as:')
42eqn2html(A_f, sp.simplify(AG1*SG1 + DG1/(1-LG1)))

Fig. 330 shows the results of the gain calculation directly from the MNA matrix and the results according to the asymptotic gain model with the voltage-controlled current source G1 selected as the loop gain reference.

The results clearly show that the asymptotic gain does not equal the ideal gain, but the gain calculated according to the asymptotic gain model is the same as the one calculated directly from network analysis. This can be understood if we study the circuit from Fig. 329 in more detail. The asymptotic gain is calculated for the case in which \(g_{m}\) approaches infinity. If this is the case, the controlled source G1 can deliver any current while its controlling voltage equals zero. Hence, in this case, the input voltage of the controller equals zero. However, since the controlled source H1 has a finite gain, it requires a nonzero input current for any nonzero output voltage. This input current flows in the inverting input of the controller, thus the controller does not act as a nullor, and as a consequence, the asymptotic gain differs from the ideal gain.

../_images/cfbVampG.svg

Fig. 330 SLiCAP simulation results of the voltage amplifier with current-feedback operational amplifier. The voltage-controlled current source {\tt{G1}} has been selected as loop gain reference variable.#

The third part of the script calculates the parameters of the asymptotic gain model with the current-controlled voltage source H1 selected as the loop gain reference and generates an html page displaying the results:

 1# Calculations with H1 as loop gain reference
 2
 3i1.setLGref('H1')
 4i1.setGainType('asymptotic')
 5result = i1.execute()
 6AH1 = result.laplace
 7
 8i1.setGainType('loopgain')
 9result = i1.execute()
10LH1 = result.laplace
11
12i1.setGainType('servo')
13result = i1.execute()
14SH1 = result.laplace
15
16i1.setGainType('direct')
17result = i1.execute()
18DH1 = result.laplace
19
20htmlPage('Asymptotic-gain model H1 ref')
21text2html('The gain of the circuit is obtained as:')
22eqn2html(V_ell/V_s, gain)
23
24text2html('The asymptotic-gain $A_{\\infty_{H1}}$ is found as:')
25eqn2html(A_infty_H1, AH1)
26
27text2html('The loop gain $L_{H1}$ is found as:')
28eqn2html(L_H1, LH1)
29
30text2html('The servo function $S_{H1}$ is found as:')
31eqn2html(S_H1, SH1)
32
33text2html('The direct transfer $\\rho_{H1}$ is found as:')
34eqn2html(rho_H1, DH1)
35
36text2html('The gain $A_f$ calculated from $A_{\\infty_G1}$, $S_{G1}$ and ' +
37          '$\\rho_{G1}$ is obtained as:')
38eqn2html(A_f, sp.simplify(AH1*SH1 + DH1/(1-LH1)))
../_images/cfbVampH.svg

Fig. 331 SLiCAP simulation results of the voltage amplifier with a current-feedback operational amplifier. The voltage-controlled current source {\tt{H1}} has been selected as the loop gain reference variable.#

Fig. 331 shows the results of the gain calculation directly from the MNA matrix and the results according to the asymptotic gain model with the current-controlled voltage source H1 selected as the loop gain reference. In this case, the reference variable has been selected properly: the asymptotic gain equals the ideal gain. Since we also have no direct transfer (\(\rho=0\)), the servo function completely describes the non ideal behavior of the controller. Hence, the design of this amplifier can be performed in two subsequent steps: the design of the ideal gain and the design of the controller.

../_images/colorCode.svg

In the following example, we will use a more elaborate model for the operational amplifier and illustrate the use of the asymptotic gain model in more detail.

Example

Fig. 332 shows the circuit of the passive feedback voltage amplifier with a current feedback operational amplifier as the controller.

The SLICAP netlist of this circuit has been listed below. SLICAP has a built-in model for a current feedback operational amplifier. Please see the SLICAP help file for detailed information on this model.

 1cfbVampExtended
 2* file: cfbVampExtended.cir
 3* SLiCAP circuit file
 4V1 1 0 {V_s}
 5R1 1 2 {R_s}
 6O1 2 4 3 0 mycfb
 7R2 3 0 {R_ell}
 8R3 3 4 {R_a}
 9R4 4 0 {R_b}
10* Model definition for the operational amplifier 'mycfb'
11.model mycfb OC cp={C_i} gp={1/R_i} cpn={C_d} gpn={1/R_d} gm={g_m} 
12+               zt={R_t/(1+s*tau)} zo={R_o/(1+s*tau)}
13* parameter values for numeric simulation
14.param C_i=5p R_i=1M g_m=20m R_t=1M R_o=2k tau=50u R_a=2k R_b=500 R_s=100
15+             R_ell=500 V_s=1 C_d=2p R_d=10k
16.end

SLICAP expands this netlist to the circuit depicted in Fig. 332. The element H_O1 is a current-controlled voltage source with series impedance. It has two associated values: the first one for the transimpedance factor and the second one for the output impedance. The data of all circuit elements after expansion of the netlist can be displayed by SLICAP. The result is shown in Fig. 333. The script below shows how this should be done:

 1#!/usr/bin/env python2
 2# -*- coding: utf-8 -*-
 3"""
 4Created on Thu Jul  2 14:59:59 2020
 5
 6@author: anton
 7"""
 8from SLiCAP import *
 9
10fileName = 'cfbVampExtended'
11prj = initProject(fileName)    # Creates the SLiCAP libraries and the
12                               # project HTML index page
13i1 = instruction()             # Creates an instance of an instruction object
14i1.setCircuit(fileName+'.cir') # Checks and defines the local circuit object,
15                               # and sets the index page to the project index                          
16htmlPage('Circuit data')
17elementData2html(i1.circuit)
../_images/cfbVampExpanded.svg

Fig. 333 Netlist of the amplifier from Fig. 332 , expanded by SLiCAP.#

The second part of the script shows how to combine the magnitude characteristics and the phase of the transfers of the asymptotic gain model in one magnitude and one phase plot. Fig. 334 shows the Bode plots generated by this script. The part of the script for generating the Bode plots is shown below:

 1i1.setSource('V1')
 2i1.setDetector('V_3')
 3i1.setLGref('H_O1')
 4i1.setSimType('numeric')
 5i1.setDataType('laplace')
 6i1.setGainType('gain')
 7G = i1.execute()
 8i1.setGainType('asymptotic')
 9A = i1.execute()
10i1.setGainType('loopgain')
11L = i1.execute()
12i1.setGainType('servo')
13S = i1.execute()
14i1.setGainType('direct')
15D = i1.execute()
16figdBmag = plotSweep('cfbVampdBmag.svg', 'dB magnitude plots asymptotic-' +
17                     'gain model', [G,A,L,S,D], 1e4, 1e9, 200, 
18                     funcType = 'dBmag', show=True)
19figPhase = plotSweep('cfbVampPhase.svg', 'Phase plots asymptotic-gain model', 
20                     [G,A,L,S,D], 1e4, 1e9, 200, funcType = 'phase', show=True)
21htmlPage('Bode plots')
22fig2html(figdBmag, 800)
23fig2html(figPhase, 800)
../_images/colorCode.svg
../_images/cfbVampExtendedBodePlots.svg

Fig. 334 Bode plots of the transfers according to the asymptotic gain model of the voltage amplifier from Fig. 332.#

Mismatch between ideal gain and asymptotic gain#

In the above example, the asymptotic gain approximates the ideal gain over a wide frequency range. At low frequencies, the nonzero resistance of the source and the nonzero conductance between the noninverting controller input and the ground cause a small error: together, they constitute a voltage divider. At high frequencies, the asymptotic gain drops due to the nonzero values of the source resistance and the input capacitance of the controller. In general, a finite impedance between an input terminal of the controller and the ground will cause inaccuracy and nonlinearity in the transfer of amplifiers that have their controller inputs floating with respect to the ground. Such impedances can almost never be avoided and should be kept as large as possible because their influence cannot be reduced by increasing the loop gain (see direct voltage comparison in Fig. 218).

Below the part of the script for evaluation of the asymptotic gain.

1htmlPage('Symbolic asymptotic-gain')
2i1.setSimType('symbolic')
3i1.setGainType('asymptotic')
4result = i1.execute()
5A = result.laplace
6text2html('The asymptotic-gain is found as:')
7eqn2html('A_f_oo', A)

The html page generated by this script has been shown in Fig. 335.

../_images/cfbVampExtendedAsymptotic.svg

Fig. 335 Results of the symbolic evaluation of the asymptotic gain of the circuit from Fig. 332#

This result can be written as:

\[A_{f\infty}=\left( \frac{R_{a}+R_{b}}{R_{b}}\right) \left( \frac{R_{i} }{R_{s}+R_{i}}\right) \left( \frac{1}{1+sC_{i}\frac{R_{i}R_{s}}{R_{s}+R_{i} }}\right) ,\]

which clearly shows the influence of \(R_{i}\) and \(C_{i}\) on the asymptotic gain.

Two-step design approach#

The servo function approximates unity at frequencies at which the magnitude of the loop gain is much larger than unity, while it approximates the loop gain at frequencies for which the magnitude of the loop gain is much smaller than unity. In the above example, the magnitude of the direct transfer is much smaller than that of the asymptotic gain. Hence, a two-step design approach steps, seems to be possible.

Hand calculations of the loop gain#

Although symbolic analysis programs such as SLICAP provide an easy way to derive expressions for the different transfer functions of the asymptotic gain model, they do not always present these expressions in a design-friendly way. In other words, it is not always easy to derive design conclusions from them. In the following examples, we will perform hand calculations based upon the calculation of current division factors. This method is known as design-oriented analysis, resulting in so-called low-entropy expressions, as described by R.D. Middlebrook.\cite[-1cm]%{Middlebrook1991}

Ideal gain and asymptotic gain#

In the following example, we will evaluate the ideal gain and the asymptotic gain of a passive feedback voltage amplifier.

Example

../_images/simpleVampDCidealOpamp.svg

Fig. 336 Left: Voltage amplifier with a nullor as the controller for evaluation of the ideal gain. Right: Voltage amplifier with an operational amplifier for evaluation of the asymptotic-gain, the loop gain and the direct transfer.#

Fig. 336 shows two amplifier circuits: one with a nullor as the controller and another with an operational amplifier.

We will evaluate the ideal transfer from the configuration with the nullor. Since both the input voltage and the input current of the nullor equal zero, we find:

\[\frac{V_{\ell}}{V_{s}}=\frac{R_{1}+R_{2}}{R_{2}}.\]

The asymptotic gain will be evaluated from the configuration with the operational amplifier. The small-signal model of the operational amplifier that is used for the controller is shown in Fig. 337. It is a single-pole operational amplifier with a gain-bandwidth product of \(GB\) [Hz], a DC voltage gain \(A_{0}\), an input resistance \(R_{i}\) and an output resistance \(R_{o}\).

The small-signal model of the voltage amplifier with the operational amplifier is shown in Fig. 338. We will determine the asymptotic gain, the loop gain and the direct transfer. To this end, we need to select the loop gain reference variable. A proper selection is one in which the controller behaves as a nullor if the gain of this loop gain reference becomes infinity.

../_images/simpleVampDCsmallSignal.svg

Fig. 338 Small-signal equivalent circuit of the passive feedback voltage amplifier.#

This is the case if we select the voltage transfer of the operational amplifier

as the loop gain reference variable:

\[A=\frac{A_{0}}{1+s\frac{A_{0}}{2\pi GB}}\]

If the voltage gain of this voltage-controlled voltage source approaches infinity, the voltage across the input port of the controller approaches zero for any finite output voltage. Under this condition the current through the input port also approaches zero.

Hence, the asymptotic gain equals the ideal gain of the amplifier:

\[A_{f\infty}=\left. \frac{V_{\ell}}{V_{s}}\right\vert _{A\rightarrow\infty }=\frac{R_{1}+R_{2}}{R_{2}}.\]
../_images/colorCode.svg

Loop gain#

In the following example, we will determine the loop gain of the amplifier.

Example

\label{example-simpleVampLoopGain}

../_images/simpleVampDCloopGain.svg

Fig. 339 Small-signal equivalent circuits for calculation of the loop gain. The voltage gain of the operational amplifier has been selected as the loop gain reference. Left: Amplifier circuit in which the controlled source has been replaced with an independent source. Right: The circuit redrawn to easy hand calculations of the loop gain.#

For the evaluation of the loop gain, we need to replace the controlled source of the reference variable by an independent source \(V_{c}\). The controlling voltage is the dependent variable \(V_{i}\). According to its definition, the loop gain \(L=A\beta\) should be obtained as:

\[L=A\left. \frac{V_{i}}{V_{c}}\right\vert _{V_{s}=0}.\]

Fig. 339 shows the equivalent circuits for the above steps. The circuit on the left shows the small-signal model with the controlled source replaced with an independent source and the controlling voltage \(V_{i}\). To simplify the hand calculations, the circuit has been redrawn on the right side.

../_images/simpleVampDCloopGainModified.svg

Fig. 340 Simplified small-signal equivalent circuit for calculation of the loop gain.#

A method, that is very well suited for the analysis of so-called ladder networks is based upon current division analysis: the voltage across \(R_{i}\) is the product of the current through \(R_{i}\) and the resistance \(R_{i}\). The current \(I_{R_{i}}\) that flows through \(R_{i}\) can easily be obtained from network inspection. Fig. 340 shows the modified network in which the series connection of \(V_{c}\) and \(R_{o}\) has been replaced with its Norton equivalent. With the aid of this schematic, we write:

\[\begin{split}V_{i} & =-I_{R_{1}}\frac{R_{2}}{R_{2}+R_{i}+R_{s}}R_{i},\\ I_{R_{1}} & =\frac{V_{c}}{R_{o}}\frac{R_{\ell}^{\prime}}{R_{\ell}^{\prime }+R_{1}+\left( R_{2}{||}\left( R_{1}+R_{s}\right) \right) },\\ R_{\ell}^{\prime} & =\left( R_{o}{||}R_{\ell}\right) ,\end{split}\]

where the short notation \(\left( R_{a}{||}R_{b}\right) \) represents the equivalent resistance of the parallel connection of \(R_{a}\) and \(R_{b}\):

\[\left( R_{a}{||}R_{b}\right) =\frac{R_{a}R_{b}}{R_{a}+R_{b}},\]

from which we obtain:

\[V_{i}=\frac{V_{c}}{R_{o}}\frac{\left( R_{o}{||}R_{\ell}\right) }{\left( R_{o}{||}R_{\ell}\right) +R_{1}+\left( R_{2}{||}\left( R_{i}+R_{s}\right) \right) }\frac{-R_{2}R_{i}}{R_{2}+R_{1}+R_{s}}.\]

For design purposes, it is often convenient to leave the short notations for parallel connections. From this, we find

\[\lambda\beta\kappa=\left. \frac{V_{i}}{V_{c}}\right\vert _{V_{s}=0} =\frac{R_{\ell}}{R_{o}{||}R_{\ell}}\frac{R_{2}}{\left( R_{o}{||}R_{\ell}\right) +R_{1}+\left( R_{2}{||}\left( R_{i} +R_{s}\right) \right) }\frac{-R_{i}}{R_{2}+R_{i}+R_{s}}.\]

The loop gain \(L=\lambda\beta\kappa A\) is now found as:

(109)#\[L=-A_{0}\frac{1}{1+s\frac{A_{0}}{2\pi GB}}\frac{R_{\ell}}{R_{o}+R_{\ell}} \frac{R_{2}}{\left( R_{o}{||}R_{\ell}\right) +R_{1}+\left( R_{2}{||}\left( R_{i}+R_{s}\right) \right) }\frac{R_{i}}{R_{2} +R_{i}+R_{s}}. \label{ex-fiveLoopGain}\]

The five terms of the loop gain expression ((109)) give the following design information:

  1. The first term \(-A_{0}\) shows the contribution of the DC voltage gain of the operational amplifier to the loop gain.

  2. The second term \(\frac{1}{1+s\frac{A_{0}}{2\pi GB}}\) represents a unity gain low-pass transfer due to the pole of the operational amplifier.

  3. The third term \(\frac{R_{\ell}}{R_{o}+R_{\ell}}\) represents the DC attenuation due to the finite value of the load resistance \(R_{\ell}\) and the nonzero value of the output resistance \(R_{o}\) of the operational amplifier.

  4. The fourth term \(\frac{R_{2}}{\left( R_{o}{||}R_{\ell}\right) +R_{1}+\left( R_{2}{||}\left( R_{i}+R_{s}\right) \right) }\) represents the DC attenuation in the loop gain caused by the feedback network. Note that this attenuation also depends on \(R_{o}\), \(R_{i}\), \(R_{s}\) and \(R_{\ell}\). For ideal drive conditions of the feedback network: \(R_{o}=0\), \(R_{i}=\infty\), \(R_{s}=0\) and \(R_{\ell}=\infty\), this term simplifies to \(\frac{R_{2}}{R_{1}+R_{2}}\), which is the reciprocal value of the ideal gain of the voltage amplifier.

  5. The fifth term \(\frac{R_{i}}{R_{2}+R_{i}+R_{s}}\) shows the DC attenuation in the loop gain caused by the nonzero input resistance \(R_{i}\) of the operational amplifier.

Writing the loop gain as a product of static and dynamic transfers as above is very helpful for finding ways to implement frequency compensation. We will show this at a later stage.

../_images/colorCode.svg

Direct transfer#

In the following example, we will derive the direct transfer \(\rho\) of this passive feedback voltage amplifier. It has been defined as the source-to-load transfer with the reference variable set to zero.

Example

Fig. 341 shows the equivalent circuits for evaluation of the direct transfer. The circuit on the left side is the small-signal equivalent circuit with the reference variable set to zero. The circuit on the right has the components rearranged to facilitate hand calculations. It clearly shows zero direct transfer if \(R_{i}=\infty\) or if \(R_{o}=0\). With the aid of the current division method, we obtain:

\[\rho=\frac{R_{2}}{R_{s}+R_{i}+R_{2}}\frac{\left( R_{o}{||}R_{\ell }\right) }{R_{2}{||}\left( R_{s}+R_{i}\right) +R_{1}+\left( R_{o}{||}R_{\ell}\right) }.\]
../_images/colorCode.svg
../_images/simpleVampDCdirectTransfer.svg

Fig. 341 Small-signal equivalent circuit for calculation of the direct transfer, with the voltage gain of the operational amplifier selected as loop gain reference variable.#

Impedance model#

With the asymptotic gain model, we have a negative feedback model that relates one design parameter (the loop gain \(L\)) to the error of the source-to-load transfer of the feedback amplifier with respect to its ideal gain. This error includes:

  1. Imperfect sensing of the load source quantity due to a non-ideal input impedance of the amplifier

  2. Imperfect driving of the load due to a non-ideal output impedance of the amplifier

  3. Imperfect input-to-output transfer due to a lack of controller gain.

In many cases, we are only interested in the total error of the source-to-load transfer and it is not necessary to resolve this error in the above contributions. However, if accurate termination of the source and/or the load is required, we need to know the values obtained for the input and/or output impedance of the amplifier. We will show that the asymptotic gain model can also be used for this purpose.

../_images/impedance-model.svg

Fig. 342 Measurement of the port immitances of a negative feedback amplifier with ideal output sensing and input comparison conditions: A: Measurement of the input admittance \(Y_x=\frac{I_x}{V_x}\). The input shunt feedback is ineffective. The feedback current \(I_{comp} \) cannot change the input voltage or current of the controller, because it flows only in \(V_x\). B: Measurement of the output admittance \(Y_x=\frac{I_x}{V_x}\). The output shunt feedback is ineffective. The feedback voltage \(V_{sense} \) cannot be changed by the controller, because it equals \(V_x\). C: Measurement of the input impedance \(Z_x=\frac{V_x}{I_x}\). The input series feedback is ineffective. The feedback voltage \(V_{comp} \) cannot change the input voltage or current of the controller, because the driving impedance at the opposite terminal of the controller input is infinity. D: Measurement of the output impedance \(Z_x=\frac{V_x}{I_x}\). The output series feedback is ineffective. The feedback current \(I_{sense} \) cannot be changed by the controller, because it equals \(I_x\).#

Fig. 342 shows the setup for the determination of the port immittance

of a feedback amplifier. We will show that the immittance of the amplifier’s input or output port can be expressed in its asymptotic value and two functions that express the influence of the finite loop gain for two different termination conditions:

  1. The port is shorted, \(L=L_{sc}\)

    This is the case if we determine the input admittance \(Y_{xf}\) of the port by driving it from a voltage source \(V_{x}\) while measuring the port current \(I_{x},\) as shown in Fig. 342A for the input port of the amplifier and Fig. 342C for its output port

    \[Y_{xf}=\frac{I_{x}}{V_{x}}.\]

    Any shunt feedback to or from this port will become maximally ineffective while series feedback is maximally effective and the loop gain obtained is \(L_{sc}\).

  2. The amplifier port is left open, \(L=L_{o}\)

    This is the case if we determine the input impedance \(Z_{xf}\) of the port by driving it from a current source \(I_{x}\) while measuring the port voltage \(V_{x},\) as shown in Fig. 342C for the input port of the amplifier and Fig. 342D for its output port

    \[Z_{xf}=\frac{V_{x}}{I_{x}}.\]

    Any series feedback to or from this port will become maximally ineffective while shunt feedback is maximally effective and we obtain a loop gain \(L_{o}\).

We will now demonstrate the use of the superposition model and asymptotic gain model for evaluation of the port impedance of port \(x\) (input or output). For measurement of the port impedance, we drive the port with a current \(I_{x}\) and measure the response voltage \(V_{x}\) across the port.

For the current drive condition from Fig. 342B,\ we apply the superposition model with the following substitutions:

\[\begin{split}E_{s} & =I_{x},\qquad\text{current flow in the port}\\ E_{\ell} & =V_{x},\qquad\text{voltage across the port}\\ \frac{E_{\ell}}{E_{s}} & =Z_{xf},\qquad\text{port impedance}\end{split}\]

The model equations of the superposition model now become

(110)#\[\begin{split}\left( \begin{array} [c]{c} V_{x}\\ E_{i} \end{array} \right) =\left( \begin{array} [c]{cc} \rho & \lambda\\ \kappa & \lambda\beta_{o}\kappa \end{array} \right) \left( \begin{array} [c]{c} I_{x}\\ E_{c} \end{array} \right) . \label{eq.matrix.impedance}\end{split}\]

The direct transfer \(\rho\) is now defined as the port impedance when \(E_{c}\) has been set to zero:

\[\rho=\left. \frac{V_{x}}{I_{x}}\right\vert _{E_{c}=0}.\]

The parameter \(\lambda\beta_{o}\kappa\) is defined as the transfer from the controlled quantity \(E_{c}\) to the controlling quantity \(E_{i}\) at \(I_{x}=0\); in words, when the port has been left open

\[\lambda\beta_{o}\kappa=\left. \frac{E_{i}}{E_{c}}\right\vert _{I_{x}=0}.\]

We now have the following set of equations:

(111)#\[\begin{split}V_{x} & =\rho I_{x}+\lambda E_{c},\label{imp-sup-1}\\ E_{i} & =\kappa I_{x}+\lambda\beta_{o}\kappa E_{c},\label{imp-sup-2}\\ E_{c} & =AE_{i}. \label{imp-sup-3}\end{split}\]

Substitution of (imp-sup-3) in (imp-sup-1) and in (imp-sup-2 ) yields

(112)#\[\begin{split}V_{x} & =\rho I_{x}+\lambda AE_{i},\label{imp-sup-5}\\ E_{i} & =\kappa I_{x}+\lambda\beta_{o}\kappa AE_{i}. \label{imp-sup-4}\end{split}\]

We now solve \(E_{i}\) from (imp-sup-4) and obtain

\[E_{i}=\frac{\kappa I_{x}}{1-\lambda\beta_{o}\kappa A}.\]

We substitute this result in (imp-sup-5), which yields:

\[V_{x}=I_{x}\left( \rho+\frac{\lambda\kappa A}{1-\lambda\beta_{o}\kappa A}\right) .\]

We now obtain an expression for \(Z_{xf}\) in terms of the parameters of the superposition model:

(113)#\[Z_{xf}=\rho\left( 1+\frac{\lambda\kappa A}{\rho}\frac{1}{1-\lambda\beta _{o}\kappa A}\right) . \label{imp-sup-6}\]

We will now define a term \(\lambda\beta_{sc}\kappa\) as

(114)#\[\lambda\beta_{sc}\kappa=\left. \frac{E_{i}}{E_{c}}\right\vert _{V_{x}=0} \label{eq-betasc}\]

We can obtain this factor \(\lambda\beta_{sc}\kappa\) from (imp-sup-5) and (imp-sup-6) for \(V_{x}=0\):

\[\begin{split}0 & =\rho I_{x}+\lambda E_{c},\\ E_{i} & =\kappa I_{x}+\lambda\beta_{o}\kappa E_{c}.\end{split}\]

We then eliminate \(I_{x}\):

(115)#\[\begin{split}I_{x} & =-\frac{\lambda E_{c}}{\rho},\\ E_{i} & =E_{c}\left( \lambda\beta_{o}\kappa-\frac{\lambda\kappa}{\rho }\right) . \label{eq-eibetasc}\end{split}\]

From ((115)) we obtain:

\[\left. \frac{E_{i}}{E_{c}}\right\vert _{V_{x}=0}=\lambda\beta_{sc} \kappa=\lambda\beta_{o}\kappa-\frac{\lambda\kappa}{\rho}.\]

From which we obtain

(116)#\[\beta_{sc}=\beta_{o}-\frac{1}{\rho}. \label{eq.beta.sc}\]

The direct transfer \(\rho\) is the port impedance with the loop gain reference set to zero. It is found as:

\[\rho=\frac{1}{\beta_{o}-\beta_{sc}}.\]

We may then rewrite (imp-sup-6) as:

(117)#\[Z_{xf}=\rho\left( \frac{1-\lambda\beta_{sc}\kappa A}{1-\lambda\beta_{o}\kappa A}\right) . \label{imp-sup-7}\]

If we substitute \(\lambda A\beta_{o}\kappa=L_{o}\) and \(\lambda A\beta _{sc}\kappa=L_{sc}\) we obtain:

(118)#\[Z_{xf}=\rho\frac{1-L_{sc}}{1-L_{o}}. \label{ex-Blackmann}\]

In the following sections, we will discuss this result for both single-loop feedback amplifiers and multiple-loop feedback amplifiers.

Port impedance of single-loop feedback amplifiers#

In single-loop feedback amplifiers, we have either parallel feedback or series feedback at a port. In a case of solely parallel feedback, the ideal value of the port impedance is zero. In case of solely series feedback at a port, the ideal value of the port impedance is infinite. We have already concluded this in Chapter Design of feedback amplifier configurations. These conclusions can also be derived from ((118)).

Parallel feedback at a port#

In a case of parallel feedback, the loop gain will be zero if we short the port. Hence, we have \(L_{sc}=0,\) and the port impedance can be written as

(119)#\[Z_{xf}=\rho\frac{1}{1-L_{o}}. \label{ex-Blackman-1}\]

The asymptotic value of the port impedance then equals zero:

\[Z_{f\infty}=\lim_{A\rightarrow\infty}Z_{xf},\]

from which we obtain

\[Z_{f\infty}=\lim_{-L_{o}\rightarrow\infty}\rho\frac{1}{1-L_{o}}=0.\]

This is, of course, as expected: both forms of parallel feedback, voltage sensing at the output port and current comparison at the input port, were used to create zero port impedance.

Series feedback at a port#

In a case of series feedback, the loop gain will be zero if we leave the port open. Hence, we have \(L_{o}=0,\) and the port impedance can be written as

(120)#\[Z_{xf}=\rho\left( 1-L_{sc}\right) . \label{ex-Blackman-2}\]

The asymptotic value of the port impedance then equals infinity:

\[Z_{f\infty}=\lim_{-L_{sc}\rightarrow\infty}\rho\left( 1-L_{sc}\right) =\infty.\]

This is also as expected: both forms of series feedback, current sensing at the output port and voltage comparison at the input port, were used to create an infinite port impedance.

Determination of the output impedance of an operational amplifier#

In the following example, we will evaluate the output impedance of a negative feedback voltage amplifier. As stated earlier, the asymptotic gain model accurately describes the source-to-load transfer of negative feedback amplifiers, including error contributions resulting from the non-ideal input and output impedance of the amplifier. For this reason, there is no need to evaluate the port impedances of single-loop negative feedback amplifiers. The ideal port impedances of these amplifiers equal either zero or infinite. However, many data sheets of operational amplifiers specify the so-called ‘closed loop’ output impedance. This ‘closed loop’ output impedance is a property of a negative feedback amplifier equipped with the operational amplifier, rather than a property of the operational amplifier itself. The following example shows the way in which the output impedance of the operational amplifier itself

relates to the output impedance of the negative feedback amplifier.

Example

Let us consider the unity gain voltage amplifier from Fig. 343A. It comprises an operational amplifier the model of which is shown in Fig. 343B. For the sake of simplicity, the impedance of the signal source has been taken as zero. In practice, it would, of course, be useless to cascade an ideal voltage source with a voltage follower!

../_images/vAmpZout.svg

Fig. 343 A: Unity gain negative feedback voltage amplifier B: Small-signal equivalent model of the voltage follower from (A).#

The circuit for the determination of the output impedance is shown in Fig. 344A. A current is driven into the output port and the voltage across the output port is measured. The output impedance \(Z_{f}\) is defined as

\[Z_{f}=\frac{V_{o}}{I_{o}}.\]
../_images/vAmpZoutModel.svg

Fig. 344 A: Circuit for determination of the output impedance of the voltage follower. B: Small-signal equivalent circuit of (A).#

Fig. 344B shows the model for evaluation of the output impedance with the aid of the negative feedback model. The voltage gain \(A_{v}\) of the operational amplifier will be selected as the loop gain reference:

\[A=\frac{A_{0}}{1+s\frac{A_{0}}{2\pi G_{B}}}.\]

The controlled source will be replaced with an independent source \(V_{c}\).

../_images/vAmpZoutRhoBeta.svg

Fig. 345 A: Circuit for determination of \(\rho\). B: Circuit for determination of \(\lambda\beta_o\kappa\).#

Fig. 345A shows the circuit for evaluation of \(\rho\) and Fig. 345B shows the circuit for evaluation of \(\lambda\beta_{o}\kappa\):

\[\rho=\left. \frac{V_{o}}{I_{o}}\right\vert _{V_{c}=0},~\lambda\beta_{o} \kappa=\left. \frac{V_{i}}{V_{c}}\right\vert _{I_{o}=0}.\]

Please notice that the output impedance of the amplifier is defined with no load connected; the output port is left open.

The direct transfer \(\rho\) can easily be found from network inspection of the circuit from Fig. 345A\(:\)

\[\rho=\frac{R_{o}}{1+sR_{o}(C_{d}+\frac{C_{c}}{2})}.\]

The loop gain \(L_{o}\) is found from the circuit from Fig. 345B:

\[L_{o}=A\lambda\beta_{o}\kappa=\frac{A_{0}}{1+s\frac{A_{0}}{2\pi G_{B}}} \frac{-1}{1+sR_{o}(C_{d}+\frac{C_{c}}{2})}.\]

The output impedance \(Z_{f}\) can now be obtained with the aid of ((119)) as

\[Z_{f}=\frac{\frac{R_{o}}{1+sR_{o}(C_{d}+\frac{C_{c}}{2})}}{1+\frac{1}{1+sR_{o}(C_{d}+\frac{C_{c}}{2})}\frac{A_{0}}{1+s\frac{A_{0}}{2\pi G_{B}}}}.\]

With \(A_{0}\gg1,\) this expression can be simplified to

\[Z_{f}=\frac{R_{o}}{A_{0}}\frac{1+s\frac{A_{0}}{2\pi G_{B}}}{1+s\left( \frac{R_{o}\left( \frac{1}{2}C_{c}+C_{d}\right) }{A_{0}}+\frac{1}{2\pi G_{B}}\right) +s^{2}\frac{R_{o}\left( \frac{1}{2}C_{c}+C_{d}\right) }{2\pi G_{B}}}.\]

From this expression, we see that the low frequency value of the output impedance of the voltage follower is \(A_{0}\) times smaller than the output resistance of the operational amplifier. The output impedance has a zero at the pole of the voltage gain of the amplifier. In most cases, this zero will be dominant. At higher frequencies, we find two poles. Hence, at frequencies above the frequency of the zero, the output impedance of the voltage follower will be inductive, while at the highest frequencies, it will be capacitive. This can be seen by letting \(s\) approach infinity:

\[\left( Z_{f}\right) _{s\rightarrow\infty}=\frac{1}{s\left( \frac{1}{2} C_{c}+C_{d}\right) }.\]
../_images/colorCode.svg

Port impedance of multi-loop feedback amplifiers#

In this section, we will show that the port impedance of a multi-loop feedback amplifier can be expressed in its asymptotic-value and the product of two servo functions. These two servo functions are defined by the loop gain in the situation in which the port is shorted, and in the situation in which the port is left open. If the loop gain reference has been properly selected, the asymptotic value of the port impedance is equal to its ideal value, just as we found for the source-to-load transfer.

The asymptotic value of the port impedance is defined as the value of the port impedance (imp-sup-7) for \(A\rightarrow\infty\):

\[Z_{f\infty}=\lim_{A\rightarrow\infty}Z_{xf}=\rho\frac{\beta_{sc}}{\beta_{o}}.\]

From this expression, we obtain:

(121)#\[\rho=Z_{f\infty}\frac{\beta_{o}}{\beta_{sc}}. \label{imp-sup-8}\]

After we substitute (imp-sup-8) in (imp-sup-7) we obtain an expression for the port impedance in terms of its asymptotic value \(Z_{f\infty}\) and two servo functions. These servo functions describe the influence of the finite loop gains \(L_{o}\) and \(L_{sc}\):

(122)#\[Z_{xf}=Z_{f\infty}\frac{L_{o}}{1-L_{o}}\frac{1-L_{sc}}{L_{sc}}. \label{eq-asymptotic-gain-imp}\]

Just as in the case of the source-to-load transfer, if the reference variable is selected properly, the asymptotic value of the port impedance is equal to the ideal value of the port impedance. The ideal value is obtained assuming nullor properties for the controller.

Expression (122) shows that idealized single loop negative feedback amplifiers have either zero or infinite port impedances. The realization of accurate finite nonzero port impedances requires the application of two feedback loops, one establishing series feedback and another establishing shunt feedback at that port.

Application of asymptotic gain model in balanced amplifiers#

In section Design of balanced amplifiers, we have discussed the design of balanced amplifiers. We obtained balanced amplifiers through anti-series connection of unbalanced amplifiers. We have seen that the differential-mode transmission-1 parameters of balanced amplifiers show a simple relationship with the transmission-1 matrix parameters of their unbalanced version. The design and analysis of the differential mode behavior of a balanced amplifier is therefore usually performed by designing their unbalanced version and then connecting two of those unbalanced amplifiers anti-series. Cross-coupling of feedback networks can be used to change the sign of the a (differential-mode) transmission parameter of a balanced amplifier with multiple feedback loops (see section Dual-loop balanced passive feedback amplifiers).

We have seen that in truly balanced amplifiers, the common-mode to differential-mode conversion and the differential-mode to common-mode conversion are both zero (see section Common-mode behavior of balanced amplifiers for decomposition of balanced circuits into common-mode and differential-mode equivalent circuits.). In such cases we speak of orthogonality between common-mode and differential-mode quantities.

Although the information processing is governed by the differential-mode behavior, the common-mode behavior of balanced negative feedback amplifiers also needs to have our interest. Even in the absence of common-mode to differential-mode conversion, the common-mode signal excursions should be within the linear operating range of the devices and common-mode stability must be ensured. If not, the quality of the differential-mode transfer will be adversely affected by common-mode signals.

The asymptotic gain model can be for the design of both the common-mode behavior and the differential-mode behavior of balanced amplifiers. To this end, we need to decompose the balanced amplifier into a differential-mode and a common-mode equivalent circuit, as described in section Dual-loop balanced passive feedback amplifiers. The differential-mode behavior can then be design using the differential-mode equivalent circuit and the common-mode behavior can be designed with the aid of the common-mode equivalent circuit, both under application of the asymptotic gain model. In this way we find differential-mode and common-mode values for the asymptotic gain, the loop gain, the servo function and the direct transfer. It should be clear that the differential-mode ideal gain and the common-mode ideal gain are obtained after replacing the controller in their respective equivalent circuits with a nullor.

Asymptotic gain model and network analysis#

In this section we will discuss the way in which the ideal gain, the asymptotic gain, the loop gain, the servo function, the direct transfer and the gain can be calculated from the network equations, that are obtained from modified nodal analysis. We will illustrate this using an example of a transimpedance amplifier that uses a voltage-controlled current source as a controller. The small-signal equivalent circuit of this amplifier is shown in Fig. 346.

Ideal gain#

The ideal gain of a feedback amplifier is obtained as the source-to-load transfer of the circuit in which the controller has been replaced with a nullor. This will be demonstrated in the following example.

Example

Fig. 347 shows the transimpedance amplifier from Fig. 346, in which the controller has been replaced with a nullor.

The MNA matrix equation of the circuit is:

\[\left( \begin{array} [c]{ccc} I_{s} & 0 & 0 \end{array} \right) ^{T}=\mathbf{M}\left( \begin{array} [c]{ccc} V_{i} & V_{\ell} & I_{N} \end{array} \right) ^{T},\]

where

\[\begin{split}\mathbf{M}=\left( \begin{array} [c]{ccc} \frac{1}{Z_{s}}+\frac{1}{Z_{f}} & -\frac{1}{Z_{f}} & 0\\ g_{m}-\frac{1}{Z_{\ell}}-\frac{1}{Z_{f}} & \frac{1}{Z_{\ell}}+\frac{1}{Z_{f}} & 1\\ 1 & 0 & 0 \end{array} \right) .\end{split}\]

The source-to-load transfer \(\frac{V_{\ell}}{I_{s}}\) can be obtained as

\[\frac{V_{\ell}}{I_{s}}=\frac{\mathcal{C}_{1,2}}{\det\mathbf{M}}~,\]

where \(\mathcal{C}_{2,1}\) is an element of the cofactor matrix of \(\mathbf{M}\) (see Chapter Network Theory (selected topics)).

In this way we obtain

\[\begin{split}\frac{V_{\ell}}{I_{s}} & =\frac{\left( -1\right) ^{1+2}\det\left( \begin{array} [c]{cc} g_{m}-\frac{1}{Z_{\ell}}-\frac{1}{Z_{f}} & 1\\ 1 & 0 \end{array} \right) }{\det\mathbf{M}}~,\\ \frac{V_{\ell}}{I_{s}} & =-Z_{f}.\end{split}\]
../_images/colorCode.svg

Asymptotic-gain#

The asymptotic gain of a feedback amplifier is obtained as the source-to-load transfer of the circuit in which the loop gain reference has been replaced with a nullor.

Example

For calculating the asymptotic gain of the circuit from Fig. 346, we need to replace the selected reference variable with a nullor. In this simple circuit, the voltage-controlled current source is the controller and the loop gain reference variable. Hence, replacement of the reference variable with a nullor yields the circuit from Fig. 347 and the asymptotic gain \(A_{f\infty}\) equals the ideal gain:

(123)#\[A_{f\infty}=-Z_{f}. \label{eq-trimpAsymptotic}\]
../_images/colorCode.svg

Loop gain#

In 1945, Bode [46] published ‘Network Analysis and Feedback Amplifier Design’. In this fundamental work, Bode stated that a useful distinction between a forward transfer and a feedback transfer as it was made in Black’s feedback model, cannot be made in real physical circuits. This is because in such circuits, parasitic feedback paths almost always exist. Therefore, in practice, the loop gain as the product of the forward gain \(H\), and the feedback factor \(k\), is not always a measure for the quality improvement of the signal transfer.

For this reason Bode introduced the terms return difference and return ratio that have a meaning for a specific selection of a controlled source. The relation between the return difference \(F\) and the return ratio \(T\) is

\[F=\left( 1+T\right) .\]

The asymptotic gain model is based upon the work of Bode. The loop gain as defined in the asymptotic gain model can be obtained from the return ratio as

\[L=-T.\]

Bode showed, that for a given selection of the reference variable, the return difference can be calculated as

(124)#\[F=\frac{\det\mathbf{M}}{\det\mathbf{M}^{0}}~, \label{eq-BodeReturnDifference}\]

where \(\mathbf{M}\) is the MNA matrix of the network and \(\mathbf{M}^{0}\) is obtained from \(\mathbf{M}\), after replacing the gain of the selected reference variable with zero.

Example

Let us now calculate the loop gain for the circuit from Fig. 346. We will first calculate the return difference \(F\) with \(g_{m}\) as reference. The MNA matrix equation of the network can be formulated as

\[\begin{split}\left( \begin{array} [c]{c} I_{s}\\ 0 \end{array} \right) =\mathbf{M}\left( \begin{array} [c]{c} V_{i}\\ V_{\ell} \end{array} \right) ,\end{split}\]

where

\[\begin{split}\mathbf{M}=\left( \begin{array} [c]{cc} \frac{1}{Z_{s}}+\frac{1}{Z_{f}} & -\frac{1}{Z_{f}}\\ g_{m}-\frac{1}{Z_{f}} & \frac{1}{Z_{\ell}}+\frac{1}{Z_{f}} \end{array} \right) .\end{split}\]

The matrix \(\mathbf{M}^{0}\) is obtained after substitution of \(g_{m}=0\) in \(\mathbf{M}\):

\[\begin{split}\mathbf{M}^{0}=\left( \begin{array} [c]{cc} \frac{1}{Z_{s}}+\frac{1}{Z_{f}} & -\frac{1}{Z_{f}}\\ -\frac{1}{Z_{f}} & \frac{1}{Z_{\ell}}+\frac{1}{Z_{f}} \end{array} \right) .\end{split}\]

With the aid of ((124)) the return difference can be obtained as

(125)#\[F=1+\frac{g_{m}Z_{s}Z_{\ell}}{Z_{f}+Z_{s}+Z_{\ell}}. \label{eq-returnDiffExample}\]
../_images/colorCode.svg

The loop gain \(L\), as it has been defined in the asymptotic gain model, can be obtained from the return difference as

(126)#\[L=1-F. \label{eq-loopgainReturnDifference}\]

Example

Let us now calculate the loop gain with \(g_{m}\) as reference variable for the circuit from Fig. 346. After substitution of ((125)) in ((126)), we obtain

(127)#\[L=-\frac{g_{m}Z_{s}Z_{\ell}}{Z_{f}+Z_{s}+Z_{\ell}}. \label{eq-trimpLoopGain}\]
../_images/colorCode.svg

Servo function#

The servo function can be obtained from the return difference as

(128)#\[S=\frac{1-F}{F}. \label{eq-servoReturnDifference}\]

Example

We will evaluate the servo function for the circuit from Fig. 346 using \(g_{m}\) as loop gain reference. After substitution of ((124)) in ((128)), we obtain

\[S=\frac{\det\mathbf{M}^{0}}{\det\mathbf{M}}-1,\]

which yields

(129)#\[S=\frac{-Z_{s}g_{m}Z_{\ell}}{Z_{f}+Z_{s}+Z_{\ell}+Z_{s}g_{m}Z_{\ell}}. \label{eq-trimpServo}\]
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Direct transfer#

The direct transfer is defined as the source-to-load transfer with the gain of the loop gain reference set to zero. Hence, it can be calculated from the matrix \(\mathbf{M}^{0}\).

Example

The direct source-to-load transfer for the circuit from Fig. 346, with \(g_{m}\) selected as loop gain reference variable can be obtained as

\[\rho=\left. \frac{V_{\ell}}{I_{s}}\right\vert _{g_{m}=0}=\frac{\left( -1\right) ^{3}\left( -\frac{1}{Z_{f}}\right) }{\det\mathbf{M}^{0}},\]

which yields

(130)#\[\rho=\frac{Z_{s}Z_{\ell}}{Z_{f}+Z_{s}+Z_{\ell}}. \label{eq-trimpDirect}\]
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Gain#

An important property of the asymptotic-gain model is that the source-to-load transfer can be calculated from the asymptotic gain, the loop gain and the direct transfer, or directly from the matrix \(\mathbf{M}\). Both calculation methods yield the same result.

Example

\[\begin{split}\frac{V_{\ell}}{I_{s}}=\frac{\left( -1\right) ^{3}\left( g_{m}-\frac{1}{Z_{f}}\right) }{\det\left( \begin{array} [c]{cc} \frac{1}{Z_{s}}+\frac{1}{Z_{f}} & -\frac{1}{Z_{f}}\\ g_{m}-\frac{1}{Z_{f}} & \frac{1}{Z_{\ell}}+\frac{1}{Z_{f}} \end{array} \right) },\end{split}\]

which yields

(131)#\[\frac{V_{\ell}}{I_{s}}=\frac{-Z_{s}Z_{\ell}\left( Z_{f}g_{m}-1\right) }{Z_{f}+Z_{s}+Z_{\ell}+Z_{s}g_{m}Z_{\ell}}. \label{eq-trimpGainMNA}\]

This expression could also have been obtained through application of the asymptotic gain model:

(132)#\[\frac{V_{\ell}}{I_{s}}=A_{f_{\infty}}S+\frac{\rho}{1-L}. \label{eq-trimpSimpleAsGmodel}\]

after substitution of ((123)), ((129)), ((130)) and ((127)) into ((132)), we also obtain ((131)).

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Conclusions#

We have seen that the asymptotic gain model is very well suited for a two-step design of negative feedback amplifiers. The only important condition is that the loop gain reference variable is chosen properly.

The ideal gain as it has been fixed with the feedback elements, matches the asymptotic gain, if the controller behaves as a nullor when the controlled source that has been selected as the loop gain reference variable is replaced with a nullor.

This will not be the case if:

  1. The reference variable is selected in a local feedback loop in the controller.

    With operational amplifiers as controllers, this situation can almost always be avoided.

  2. Parasitic impedances at one or more port terminals:

  • Introduce an attenuation in the coupling between the source and the amplifier, and/or the amplifier and the load

  • Introduce an attenuation in the feedback network(s).

In such cases, the asymptotic gain model can still be applied. Those impedances can easily be extracted from the controller and assume to be part of the feedback network, the source or the load. In doing so, the ideal gain is modified and the asymptotic gain will equal this modified ideal gain.

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