Passive feedback#
In this section, we will discuss the design of amplifier types that have one or more transmission parameters fixed to a nonzero value with the aid of a passive feedback network. We will discuss the design of single loop feedback configurations in section Single-loop passive feedback configurations. We will study their influence on the power efficiency and on the noise performance in sections Noise behavior of passive feedback configurations and Power efficiency of passive feedback configurations, respectively. The design of multiple-loop passive feedback configurations will be treated in section Dual-loop passive feedback configurations.
Single-loop passive feedback configurations#
In the previous sections, we have seen that \(15\) types of negative feedback amplifiers can be realized using controlled sources as feedback elements. Application of nonenergic feedback elements resulted in the best possible noise performance and power efficiency of a feedback amplifier. In practice, nonenergic feedback cannot always be implemented. Gyrators can only be realized with the aid of active circuits, while transformers are expensive, show far from ideal behavior and cannot always be integrated in silicon technology. Practical application of nonenergic feedback is often restricted to followers in which short circuits and open circuits can be used as feedback elements.
In this section, we will focus on the design of the four basic single-loop configurations using passive feedback elements. These basic configurations are shown in Fig. 227. They can be designed according to the procedure explained in Fig. 215. Their ideal gain is the reciprocal transfer of the feedback network. It can easily be evaluated through network inspection, using the zero voltage and the zero current condition for the nullator.
Fig. 227 Single-loop passive feedback amplifiers for source and load referenced to ground.#
\item[A.] Voltage amplifier: \(\frac{V_{\ell}}{V_{s}}=\frac{Z_{1}+Z_{2}}{Z_{1} }\)
\item[B.] Transadmittance amplifier: \(\frac{I_{\ell}}{V_{s}}=-\frac{1}{Z_{1}}\)
\item[C.] Transimpedance amplifier: \(\frac{V_{\ell}}{I_{s}}=-Z_{1}\)
\item[D.] Current amplifier: \(\frac{I_{\ell}}{I_{s}}=\frac{Z_{1}+Z_{2}}{Z_{1} }\)
Both the voltage and the current amplifier have noninverting transfers, and the transadmittance and transimpedance amplifiers have inverting transfers. This is a consequence of the noninverting character of the transfer of the passive feedback elements. With natural two-ports as feedback elements (such as transformers) we can realize both inverting and noninverting transfers of the feedback network. Using non natural two-ports, we have to find other ways to realize both inverting and noninverting amplifiers. These techniques are:
Active feedback
Balanced feedback
Indirect feedback.
These techniques will be discussed in sections Active feedback, Design of balanced amplifiers and Indirect feedback, respectively.
Another restriction due to the application of non-natural two-ports as feedback elements, is their lack of port isolation. This is because passive feedback networks establish interconnections between the input and the output port. With the aid of active feedback, balanced feedback and indirect feedback we will be able to design amplifier configurations with floating ports.
Noise behavior of passive feedback configurations#
It is to be expected that the insertion of the passive feedback elements into the signal path causes a deterioration of the signal-to-noise ratio. It can be shown that such deterioration can be kept small when compared to the application of brute force techniques.
The influence of the feedback network on the noise performance of passive feedback amplifiers can be evaluated in several ways. Hand calculations can be done using source transformation techniques. Alternatively, one could use symbolic mathematical tools for this purpose. Network simulation programs (SPICE-like programs) can be used to obtain numeric results. We will demonstrate various methods in the following examples.
First, we will show the application of source transformation techniques. Then, we will show the combination of source transformation techniques and superposition. Finally, we will demonstrate the use of modified nodal analysis. The latter techniques are more suited for automated noise analysis as it has been implemented in SLICAP.
Example
In this example, we will illustrate the use of source transformation technique.
Fig. 228 Passive-feedback voltage amplifier with noise sources.#
Fig. 228\ shows a passive feedback voltage amplifier with its noise sources. In order to evaluate the deterioration of the signal-to-noise ratio due to passive feedback, these noise sources have to be transformed into one total equivalent input noise voltage source. The noise sources that have to be accounted for are:
The noise voltage \(V_{ns}\) associated with the signal source. The spectral density of this source is given by \(4kT\operatorname{Re}\left\{ Z_{s}\right\} \) [V\(^{2}\)/Hz]
The equivalent input noise voltage \(V_{eq}\) of the nullor. The spectral density of this source is given by \(S_{veq}\) [V\(^{2}\)/Hz]
The equivalent input noise current \(I_{eq}\) of the nullor. The spectral density of this source is given by \(S_{ieq}\) [A\(^{2}\)/Hz]
The thermal noise \(V_{1}\) associated with \(R_{1}\). The spectral density of this source is given by \(4kTR_{1}\) [V\(^{2}\)/Hz]
The thermal noise \(V_{2}\) associated with \(R_{2}\). The power spectral density of this source is given by \(4kTR_{2}\) [V\(^{2}\)/Hz].
The starting situation is shown in Fig. 228. The transformation of all the noise sources to one equivalent input voltage source will be illustrated in six steps.
In the first step, as shown in Fig. 229, the equivalent current source of the nullor is redirected via the ground. It is then represented by two equal sources: one in parallel with \(R_{1}\) and one in parallel with the source.
Fig. 229 Application of the current-split theorem to \(I_{eq}\).#
With the aid of the Th’{e}venin transformation, both current sources can now be transformed into voltage sources, as shown in Conversion of noise current sources into voltage sources.. This results in two voltage sources, one in series with \(R_{1}\) that has a value \(I_{eq}R_{1}\), and one with a value of \(I_{eq}Z_{s}\) in series with the source. Both sources are correlated, and therefore, we will keep track of the signs of these sources.\medskip
Fig. 230 Conversion of noise current sources into voltage sources.#
The next step, the result of which is depicted in Fig. 231, shows how the voltage sources in series with \(R_{1}\) are shifted into the branch with \(R_{2}\) and the branch with the input of the nullor.\medskip
The voltage sources in series with the input of the nullor can be shifted through the input port of the nullor. They now appear in series with the source. The voltage sources in series with \(R_{2}\) can be shifted into the branch connected to the output of the voltage amplifier and the branch connected to the output of the nullor. This is shown in Fig. 232.
The sources in series with the output port of the nullor can be transformed into equivalent input sources of the nullor by multiplying them by the corresponding transmission parameters. Since all of the transmission parameters of the nullor equal zero, these sources can be neglected.
Fig. 231 The voltage sources \(V_{1}\) and \(I_{eq}R_{1} \) are shifted through node (a).#
Fig. 232 The noise voltage sources at the input are shifted through the input port of the nullor and the sources in series with \(R_{2}\) are shifted through node (b).#
The sources in series with the output port of the voltage amplifier can be transformed into equivalent input sources of the voltage amplifier by multiplying them by the corresponding transmission parameters of the voltage amplifier. Since, for the voltage amplifier, only the parameter \(A\) is fixed at a nonzero value, these output voltage sources can be represented by equivalent input voltage sources that have a value \(A\) times larger, with \(A=R_{1}/(R_{1}+R_{2}).\) These steps are shown in Fig. 233.
Fig. 233 Transformation of all noise sources to the input.#
Fig. 234 shows the final result of the noise transformations. The power spectral density of the total equivalent input noise voltage source can be expressed as:
\[S_{V_{n,tot}}=4kT\operatorname{Re}\{Z_{s}\}+S_{veq}+4kT\frac{R_{1}R_{2}}{R_{1}+R_{2}}+S_{ieq}\left\vert Z_{s}+\frac{R_{1}R_{2}}{R_{1}+R_{2} }\right\vert ^{2}.\]
Fig. 234 Final result of the noise transformations.#
Thus, the influence of the feedback network on the noise performance of the passive feedback voltage amplifier can be calculated as if the parallel connection of \(R_{1}\) and \(R_{2}\) is in series with the signal source.
This confirms the equivalence of the noise models from Fig. 228 and Fig. 235.\medskip
Fig. 235 Noise model of the voltage amplifier, equal to that of Fig. 228.#
Conclusions with respect to the noise behavior of the other single loop passive feedback amplifier configurations can be derived in a similar way:
The total equivalent input noise of the voltage amplifier from Fig. 227A can be calculated from the individual noise contributions as if the parallel connection of the feedback elements is in series with the source.
The total equivalent input noise of the transadmittance amplifier from Fig. 227B can be calculated from the individual noise contributions as if the feedback element is in series with the source.
The total equivalent input noise of the transimpedance amplifier from Fig. 227C can be calculated from the individual noise contributions as if the feedback element is in parallel with the source.
The total equivalent input noise of the current amplifier from Fig. 227D can be calculated from the individual noise contributions as if the series connection of the feedback elements is in parallel with the source.
In the next example, we will demonstrate the use of source transformation techniques and the application of the superposition theorem. We start with the determination of the contribution of each individual noise source to the output noise of the amplifier. We then obtain the spectral density of the output noise by adding all of these contributions. The spectral density of the input noise is then obtained by dividing the output noise spectrum by the squared gain of the amplifier.
Example
We will evaluate the contributions of the individual noise sources to the total output noise spectrum \(S_{Vout}\).
The noise voltage \(V_{ns}\) associated with the signal source. The spectral density \(S_{vn}\) of this source equals: \(4kT\operatorname{Re}\left\{ Z_{s}\right\} \) [V\(^{2}\)/Hz].
The contribution \(S_{1}\) of this noise source to \(S_{Vout}\) can be calculated by multiplying \(S_{vn}\) by the squared gain (see Fig. 236):
\[S_{1}=4kT\operatorname{Re}\left\{ Z_{s}\right\} \left( \frac{R_{1}+R_{2} }{R_{1}}\right) ^{2}.\]The equivalent input noise voltage of the nullor \(V_{eq}.\) The spectral density of this source is given by: \(S_{veq}\) [V\(^{2}\)/Hz].
The contribution \(S_{2}\) of this noise source to \(S_{Vout}\) can be calculated in a similar way:
\[S_{2}=S_{veq}\left( \frac{R_{1}+R_{2}}{R_{1}}\right) ^{2}.\]The equivalent input noise current \(I_{eq}\) of the nullor. The power spectral density of this source is given by: \(S_{ieq}\) [A\(^{2}\)/Hz].
Fig. 237 Evaluation of the contribution of \(I_{eq}\) to the output noise.#
We can split this source into two correlated sources as shown in Fig. 237. The current source in parallel with the source impedance causes an output voltage \(V_{1}\):
\[V_{1}=I_{eq}Z_{s}\frac{R_{1}+R_{2}}{R_{1}}.\]The current source in parallel with \(R_{1}\) causes an output voltage \(V_{2}\):
\[V_{2}=I_{eq}R_{2}.\]These are two correlated contributions, so we add these voltages and find their contribution \(S_{3}\) to the spectrum of the output noise voltage by multiplying the spectral density of \(I_{eq}\) by the squared magnitude of the transfer from \(I_{eq}\) to the output voltage. In this way, we obtain
\[S_{3}=S_{ieq}\left\vert Z_{s}\frac{R_{1}+R_{2}}{R_{1}}+R_{2}\right\vert ^{2}.\]This can be written as
\[S_{3}=S_{ieq}\left( \frac{R_{1}+R_{2}}{R_{1}}\right) ^{2}\left\vert Z_{s}+\frac{R_{1}R_{2}}{R_{1}+R_{2}}\right\vert ^{2}.\]The thermal noise \(V_{1}\) associated with the feedback element \(R_{1}\). The spectral density of this source is given by: \(4kTR_{1}\) [V\(^{2}\)/Hz].
The gain of this noise source to the output voltage equals \(-R_{2}/R_{1}\). The contribution \(S_{4}\) to the spectral density of the out[put noise voltage is thus obtained by multiplying \(4kTR_{1}\) with the squared magnitude of this transfer. We obtain:
\[S_{4}=4kT\frac{R_{2}^{2}}{R_{1}}\]
Fig. 238 Evaluation of the contribution of the thermal noise of \(R_2\) to the output noise.#
The thermal noise \(V_{2}\) associated with the feedback element \(R_{2}\). The spectral density of this source is given by \(4kTR_{2}\) [V\(^{2}\)/Hz].
This noise source can be converted into a current source \(I_{2}\) with a spectral density \(S_{2}=4kT/R_{2}\) [A\(^{2}\)/Hz]. It can be split into two correlated sources, as shown in Fig. 238. The source in parallel with the norator has no effect. The remaining source contributes \(S_{5}\) to the output spectrum:
\[S_{5}=4kTR_{2}.\]The spectral density of the output voltage noise is obtained by adding the contributions of \(S_{1}\) through \(S_{5}\):
\[\begin{split}S_{Vout} & =\left( \frac{R_{1}+R_{2}}{R_{1}}\right) ^{2}\left\{ 4kT\operatorname{Re}\left\{ Z_{s}\right\} +S_{veq}+S_{ieq}\left\vert Z_{s}+\frac{R_{1}R_{2}}{R_{1}+R_{2}}\right\vert ^{2}\right\} +\nonumber\\ & +4kT\left( \frac{R_{2}^{2}}{R_{1}}+R_{2}\right) ,\end{split}\]which can be written as:
\[\begin{split}S_{Vout} & =\left( \frac{R_{1}+R_{2}}{R_{1}}\right) ^{2}\times\nonumber\\ & \times\left\{ 4kT\operatorname{Re}\left\{ Z_{s}\right\} +S_{veq} +S_{ieq}\left\vert Z_{s}+\frac{R_{1}R_{2}}{R_{1}+R_{2}}\right\vert ^{2}+4kT\frac{R_{2}R_{1}}{R_{2}+R_{1}}\right\} .\end{split}\]The source-referred noise spectrum is obtained after multiplication of this result by the reciprocal value of the squared magnitude of the voltage gain:
\[S_{V_{n,tot}}=4kT\operatorname{Re}\{Z_{s}\}+S_{veq}+S_{ieq}\left\vert Z_{s}+\frac{R_{1}R_{2}}{R_{1}+R_{2}}\right\vert ^{2}+4kT\frac{R_{1}R_{2} }{R_{1}+R_{2}}.\]
Finally, we will evaluate the total equivalent input voltage noise of the passive-feedback voltage amplifier with the aid of the MNA method. This method has been implemented in SLICAP.
Example
In order to keep the matrices as small as possible, we will replace the voltage noise sources \(V_{1}\) and \(V_{2}\) with current sources \(I_{1}\) and \(I_{2}\) and combine the noise sources \(V_{ns}\) and \(V_{eq}\) in one voltage source. In addition, we will omit the signal source \(V_{s}.\) The simplified equivalent noise model is shown in Fig. 239. Using this model, we will calculate the total equivalent input noise from the total output noise. We will evaluate the total output noise and obtain the source-referred noise after multiplication of the output noise voltage by the reciprocal value of the voltage gain: \(\frac{R_{1}}{R_{1}+R_{2}}.\) The total output noise will be determined using MNA and Cramer’s rule.
Fig. 239 Equivalent noise model of the voltage amplifier for MNA.#
The circuit can be described with the following matrix equation:
where
The total equivalent input noise voltage \(V_{ntot}\) can be found from
The output voltage \(V_{out}\) equals \(V_{4}\). It can be found with the aid of Cramer’s rule:
where
With the aid of a symbolic math tool, this can quickly be simplified to
Hence, for the spectral density \(S_{V_{ntot}}\) of \(V_{ntot},\) we write:
which can again be written as:
In the following example, we will demonstrate the way in which SLICAP can be used to perform such noise calculations.
Example
Fig. 240 shows the SLICAP circuit diagram for the noise analysis of the passive feedback voltage amplifier. The spectral densities of all uncorrelated noise sources are given in [V\(^{\text{2}}\)/Hz] or in [A\(^{\text{2}}\)/Hz].
The SLICAP netlist for this circuit is shown below. The controller of the amplifier is the ‘noisy nullor’ x1. It is a SLiCAP library element (sub circuit N_noise) that consists of a nullor with two equivalent-input noise sources. The spectral densities of these noise sources are parameters of this sub circuit.
1VampNoise.cir
2X1 2 3 out 0 N_noise si={S_i} sv={S_v}
3R1 out 3 {R_b}
4R2 3 0 {R_a}
5I1 out 3 I noise={4*k*T/R_b}
6I2 3 0 I noise={4*k*T/R_a}
7R3 2 1 {R_s}
8V1 1 0 V noise={4*k*T*R_s}
9.end
The SLICAP script for evaluation of the source-referred noise and the detector-referred noise is shown below. The signal source is the voltage source Vn1. The detector is a voltage detector between node (5) and the ground.
1#!/usr/bin/env python3
2# -*- coding: utf-8 -*-
3"""
4Created on Mon Mar 29 20:31:51 2021
5
6@author: anton
7"""
8from SLiCAP import *
9
10fileName = 'VampNoise';
11prj = initProject(fileName)
12i1 = instruction()
13i1.setCircuit(fileName + '.cir')
14htmlPage('Circuit data')
15head2html('Circuit diagram: ' + fileName)
16img2html(fileName + '.svg', 600)
17netlist2html(fileName + '.cir')
18#
19i1.setGainType('vi')
20i1.setDataType('noise')
21#
22# Define the source and the detector
23i1.setSource('V1');
24i1.setDetector('V_out');
25#
26htmlPage('Symbolic noise analysis')
27i1.setSimType('symbolic')
28noiseResultSym = i1.execute()
29noise2html(noiseResultSym)
The HTML output, generated by SLiCAP is shown in Fig. 241.
Fig. 241 SLiCAP symbolic noise results.#
Example
In this example we will demonstrate the determination of show-stopper values for \(R_{a}\), \(R_{b}\), \(S_{v}\) and \(S_{i}\) for a given source resistance, noise figure and voltage gain. To do so, we assign values to \(R_{s}\) and the voltage gain \(A_{v}\) and establish a relation between \(R_{a}\) and \(R_{b}\).
and calculate the noise as a function of these parameters. This is done in lines 30-42 of the script:
1#
2# Let us find show-stopper values for R_a, S_v, and S_i for the case:
3# - source resistance: R_s = 600 Ohm
4# - voltage gain : A_v = 20
5# - noise figure : NF = 2 (3dB)
6#
7i1.defPar('R_s', 600)
8# Define R_b = (A_v-1)*R_a
9i1.defPar('R_b', '(A_v-1)*R_a')
10i1.defPar('A_v', 20)
11i1.setSimType('numeric')
12# Calculate the noise with the given parameters.
13noiseResultNum = i1.execute()
The remaining symbolic variables \(R_{a}\), \(S_{v}\) and \(S_{i}\) as well as their show-stopper values \(R_{a_{\max}}\), \(S_{v_{\max}}\) and \(S_{i_{\max}},\) and the noise factor \(NF\) are defined in the python environment in lines 43-47 of the script:
1#
2# Determine the noise figure NF: (the given procedure works with
3# frequency-independent noise spectra only)
4#
5R_a, S_v, S_i, NF, R_a_max, S_i_max, S_v_max = sp.symbols('R_a, S_v, S_i,' +
6 'NF, R_a_max, S_i_max, S_v_max')
The noise factor is calculated as the ratio of the total source-referred noise spectrum and the spectrum of the noise associated with the signal source. This is correct if the noise spectra do not depend on frequency (white noise); see lines 49-57 of the script:
1htmlPage('Show-stopper values')
2#
3text2html('Let us find show-stopper values for $R_a$, $S_v$, and $S_i$ for ' +
4 'the case in which the noise factor $NF$ equals 2 (3dB).')
5head2html('Noise factor NF')
6text2html('The noise factor NF [-] is obtained as:')
7NFact = sp.simplify(noiseResultNum.inoise/noiseResultNum.inoiseTerms['V1'])
8eqn2html(NF, NFact);
9#
The show stopper values of \(R_{a_{\max}}\), \(S_{v_{\max}}\) and \(S_{i_{\max}}\) are found by solving the equation of \(NF\) for one variable, while assuming the other noise contributions zero. Since zero value for the feedback resistances is not meaningful, the show stopper values \(S_{v_{\max}}\) and \(S_{i_{\max}}\) are calculated as a function of \(R_{a}\); see lines 49-57 of the script. Fig. 242 shows the results.
1# Show stopper (= maximum) value $R_{amax}$ for R_a with S_i=0 and S_v=0
2Ra_max = sp.N(sp.solve(NFact.subs([(S_v, 0), (S_i, 0)])-2, R_a)[0], 3)
3head2html('Show-stopper value $R_a$');
4text2html('The show stopper value $R_{amax}$ for $R_a$ with $NF=2$, ' +
5 '$S_v=0$ and $S_i=0$ is obained as:');
6eqn2html(R_a_max, Ra_max);
7#
8# Show stopper (= maximum) $S_{v,max}$ for S_v as a function of R_a and S_i=0
9Sv_max = sp.N(sp.solve(NFact.subs(S_i, 0)-2, S_v)[0], 3);
10head2html('Show-stopper value $S_v$');
11text2html('The show stopper value for $S_v$ with $NF=2$ and $S_i=0$ can be ' +
12 'obained a function of $R_a$ (setting $R_a$ to zero would be ' +
13 'meaningless):')
14eqn2html(S_v_max, Sv_max);
15#
16# Show stopper (= maximum) $S_{i,max}$ for S_i as a function of R_a and S_i=0
17Si_max = sp.N(sp.solve(NFact.subs(S_v, 0)-2, S_i)[0], 3);
18head2html('Show-stopper value $S_i$');
19text2html('The show stopper value for $S_i$ with $NF=2$ and $S_v=0$ can be ' +
20 'obained a function of $R_a$: (setting $R_a$ to zero would be ' +
21 'meaningless):');
22eqn2html(S_i_max, Si_max);
Fig. 242 Derivation and solution of the component design equations for the noise performance.#
Power efficiency of passive feedback configurations#
In this section we will summarize the influence of the feedback networks of the basic passive feedback amplifier configurations on their power efficiency. These conclusions simply follow from network inspection:
Definition
The power efficiency of the voltage amplifier from Fig. 227A is affected by the feedback elements as if the series connection of the feedback elements is in parallel with the load impedance.
The power efficiency of the transadmittance amplifier from Fig. 227B is affected by the feedback elements as if the feedback element is in series with the load impedance.
The power efficiency of the transimpedance amplifier from Fig. 227C is affected by the feedback elements as if the feedback element is in parallel with the load impedance.
The power efficiency of the current amplifier from Fig. 227D is affected by the feedback elements as if the parallel connection of the feedback elements is in series with the load impedance.
Dual-loop passive feedback configurations#
Fig. 243 shows the basic single-nullor dual-loop feedback configurations that can be realized with passive feedback alone. These are the gyrator-like configuration and the transformer-like configuration.
The gyrator-like configuration from Fig. 243A has some interesting properties. The impedance \(Z\) establishes parallel feedback at both the input and the output port (load voltage sensing and source current comparison). Hence, it fixes transmission parameter \(C\). The admittance \(Y\) establishes input and output series feedback (load current sensing and source voltage comparison), which fixes parameter \(B\). Due to voltage drop across the current sensing element \(Y\) and current though the voltage-sensing element \(Z\), all transmission parameters have been fixed by \(Z\) and \(Y\). However, only two of them can be fixed independently.
The transmission parameters of this configuration are:
This configuration is often applied in characteristic impedance systems. Impedance matching at both ports is obtained if: \(Z_{s}=Z_{\ell}=\sqrt{\frac{Z}{Y}}.\)
In the transformer-like configuration from Fig. 243B, a voltage attenuator\ (\(Z_{1},Z_{2}\)) and a current attenuator (\(Z_{3},Z_{4}\)) are used to fix transmission parameters \(A\) and \(D\), respectively. Due to current flow through the voltage attenuator and voltage drop across the current attenuator, parameter \(C\) is also fixed. The transmission parameters are:
An accurate input impedance can be obtained if the output port is left open. This is true in practice, if \(Z_{\ell}\gg Z_{1}+Z_{2}\).
The configuration has an accurate output impedance if it is driven from an ideal current source; in practice, if \(Z_{s}\gg Z_{3}+Z_{4}\).
Other passive single-nullor dual-loop configurations have opposite signs for their transmission parameters, which results in a negative input or output impedance. This will be demonstrated in the next example.
Example
Let us try to design a passive feedback amplifier that has a finite nonzero input impedance and zero output impedance. For such an amplifier, we need to fix the parameters \(A\) and \(C\) to a finite nonzero value, while having \(B=D=0\). Fixing \(A\) requires parallel sensing at the output port and series comparison at the input port. Fixing \(C\) requires parallel sensing at the output port and parallel comparison at the input port. With passive feedback, this would give the circuit shown in Fig. 244.
Circuit analysis shows that
which yields:
Hence, the input impedance is negative! In order to obtain a positive input impedance, we need to create a signal inversion in one of the feedback loops. Means to achieve this will be discussed in the following sections.