Design of feedback configurations#
The procedure for fixing a tranmission-1 parameter with the aid of negative feedback is as follows:
Sense the load quantity that needs to be related to a source quantity.
\label{N-converting}Multiply the sense result with the value of the transmission-1 matrix coefficient that needs to be fixed; this should yield a copy of the source quantity.
Nullify the error between the source quantity and the derived copy of it with the aid of a high-gain error amplifier or controller.
This amplifier controls the output signal in such a way that the feedback signal is the best possible copy of the source signal.
At this stage of the design of negative feedback amplifiers, we will use nullors as error amplifiers. Nullors are two-ports that have a nullator as the input port and a norator as the output port. They have infinite available power gain. Nullors are introduced in Chapter MNA stamps.
Implement the nullor with a practical error amplifier, define its performance requirements and design or select it. Chapter Amplifier performance and controller requirements, discusses in which way, and to what extend the amplifier’s performance depends on properties of its controller.
Step-by-step design of the feedback configuration#
The step-by-step design of a negative feedback voltage amplifier will be illustrated in the following example. The design procedure of other types of negative feedback amplifiers is similar to the one presented.
Example
Let us consider that we want to design a negative feedback voltage amplifier with a voltage gain \(A_{v}\). Such an amplifier should have an infinite input impedance and zero output impedance. The values of the transmission-1 parameters of this amplifier should be:
Fig. 215A shows the situation at the start: we have a signal voltage source with open-circuit voltage \(V_{s}\) and source impedance \(Z_{s}\) and a load impedance \(Z_{\ell}\). With the above values of the transmission-1 matrix parameters, the voltage across \(Z_{\ell}\) will equal \(A_{v}V_{s}\) and will not depend on \(Z_{s}\) and \(Z_{\ell}\).
We need to sense the load voltage \(V_{\ell}\) (see Fig. 215B).
Then, we create a copy of the source voltage by multiplying the sensed voltage by \(A=\frac{1}{A_{v}}\). To this end, we may use a voltage-controlled voltage source as shown in Fig. 215C. The gain of this so-called feedback element should equal the reciprocal value of the desired source-to-load transfer \(A_{v}\).
We then determine the difference between the source voltage and this copy by connecting these two voltages anti-series: see Fig. 215D.
Finally, we nullify this difference between the source voltage and the copy by placing a nullator between the anti-series connected voltage sources, and by controlling the load voltage through placement of a norator in parallel with the load (see Fig. 215E).
The nullor is a network abstraction that cannot be realized. After we have designed the negative feedback configuration with the aid of a nullor, we need to design the error amplifier that replaces it. This error amplifier must have a large gain so that the error signal will be negligibly small and the signal at the load is a sufficiently accurate copy of the source signal.
Fig. 215 A until F: step-by-step development of the circuit concept for a negative feedback voltage amplifier with gain \(\frac{V_{\ell}}{V_{s} }=A_{v}\). A: The voltage \(V_{s} \) of a grounded source must evoke a voltage \(V_{\ell} \) across the grounded load. Assume the voltage at the load equals its desired value \(V_{\ell}\). B: The load voltage is sensed. Sensing of a voltage requires the sense element to be placed in parallel with the voltage to be sensed. C: The sensed voltage is converted into a copy of the source voltage. Hereto, it is multiplied by the reciprocal value of the desired source-to-load transfer.#
D: The difference between the source voltage and its copy is obtained through anti-series connection (series connection with opposite polartities). E: This difference should be zero. The nullator sets the zero voltage and zero current condition for this {error signal} : the difference between the source signal and its intended copy. The norator drives the load and the feedback network in such a way that the condition set by the nullator will be satisfied. In this way, the load voltage \(V_{\ell }\) is generated.
F: The nullor is replaced with a two-port with a large available power gain, in such a way that negative feedback is obtained.
G: The controlled source in the feedback path is replaced with a nonenergic voltage attenuator (a transformer).
H: The controlled source in the feedback path is replaced with a passive voltage attenuator.
Fig. 215F shows the circuit after the nullor has been replaced with an error amplifier. The ports of the error amplifier are connected in such a way, that a unintentional decrease of the load voltage results in an increase of the error signal that establishes a correction of the load signal. This technique is called corrective, degenerative or negative feedback. If the opposite is the case, we have positive feedback, which may result in unstable behavior.
In Fig. 215C we used a voltage-controlled voltage source to generate a copy of the load voltage. This voltage-controlled voltage source is an active network element. We applied it to elucidate the synthesis of this negative feedback voltage amplifier. Amplifiers with active feedback elements are called active feedback amplifiers. Fig. 215G shows the amplifier with a transformer as the feedback element. An ideal transformer is a nonenergic network element, and therefore this amplifier is called a nonenergic feedback amplifier. Fig. 215H shows an arrangement in which the feedback network consists of a passive voltage divider. This technique is called passive feedback.
As already mentioned, the four transmission parameters of an amplifier can be independently fixed using negative feedback. Here is the procedure:
In order to fix the parameters \(A\) and \(C\) to a nonzero value, we need to sense the load voltage. Sensing of voltages means parallel connection of the sense network and the load impedance. This is called load voltage sensing, parallel sensing, output voltage feedback, output parallel feedback or output shunt feedback
In order to fix the parameters \(B\) and \(D\) to a nonzero value, we need to sense the load current. Sensing of a current means series connection of the sense network and the load impedance. This is called load current sensing or series sensing, output current feedback or output series feedback.
In order to fix the parameters \(A\) and \(B\) to a nonzero value, we need to compare the feedback voltage with the source voltage. Subtracting or comparison of voltages requires anti-series connection of the feedback voltage and the source voltage. This is called source voltage comparison, series comparison, input voltage feedback or input series feedback.
In order to fix the parameters \(C\) and \(D\) to a nonzero value, we need to compare the feedback current with the source current. Subtracting or comparing currents requires anti-parallel connection of the feedback current and the source current. This is called source current comparison, parallel comparison, input current feedback, input parallel feedback or input shunt feedback.
Fig. 216 shows a feedback amplifier in which all four transmission-1 parameters have been independently fixed through application of negative or corrective feedback. The amplifier has its input port and its output port both floating with respect to the ground. Each feedback loop fixes one parameter only. This orthogonal design of the transmission-1 parameters is possible due to the ideal sense and drive conditions of the feedback elements:
No voltage drop across current-sensing elements
No current through voltage-sensing elements
Zero output impedance of the feedback voltage sources
Infinite output impedance of the feedback current sources
With passive feedback and nonenergic feedback amplifiers elements, this is generally not the case. This will be discussed later.
Table 18 gives an overview of the way the transmission parameters of the amplifiers can be fixed with the aid of negative feedback.
Param. |
Feedback type at input port |
Feedback type at output port |
Transfer |
\(A\) |
V - comparison (series feedback) |
V - sensing (parallel feedback) |
[V/V] |
\(B\) |
V - comparison (series feedback) |
I - sensing (series feedback) |
[I/V] |
\(C\) |
I - comparison (parallel feedback) |
V - sensing (parallel feedback) |
[V/I] |
\(D\) |
I - comparison (parallel feedback) |
I - sensing (series feedback) |
[I/I] |
Table 19 gives an overview of the 16 amplifier types (including the nullor) that can be realized by leaving out one or more feedback elements from the circuit from Fig. 216.
Amplifier type |
\(A\) |
\(B\) |
\(C\) |
\(D\) |
Nullor |
\(0\) |
\(0\) |
\(0\) |
\(0\) |
Voltage amplifier (unilateral) |
\(\frac{1}{\mu}\) |
\(0\) |
\(0\) |
\(0\) |
Transadmittance amplifier (unilateral) |
\(0\) |
\(\frac{1}{\gamma}\) |
\(0\) |
\(0\) |
Transimpedance amplifier (unilateral) |
\(0\) |
\(0\) |
\(\frac{1}{\zeta}\) |
\(0\) |
Current amplifier (unilateral) |
\(0\) |
\(0\) |
\(0\) |
\(\frac{1}{\alpha} \) |
\(Z_{i}=\frac{\alpha}{\gamma};\){\small \ }\(Z_{o}=\infty\)%\ (unilateral) |
\(0\) |
\(\frac{1}{\gamma}\) |
\(0\) |
\(\frac{1}{\alpha} \) |
\(Z_{i}=\frac{\zeta}{\mu};\){\small \ }\(Z_{o}=0\)\ (unilateral) |
\(\frac{1}{\mu}\) |
\(0\) |
\(\frac{1}{\zeta}\) |
\(0\) |
\(Z_{i}=0;\) \(Z_{o}=\frac{\zeta}{\alpha}\)\ (unilateral) |
\(0\) |
\(0\) |
\(\frac{1}{\zeta}\) |
\(\frac{1}{\alpha}\) |
\(Z_{i}=\infty;\) \(Z_{o}=\frac{\mu}{\gamma}\)\ (unilateral) |
\(\frac{1}{\mu}\) |
\(\frac{1}{\gamma}\) |
\(0\) |
\(0\) |
Transformer-like amplifier (non-unilateral) |
\(\frac{1}{\mu}\) |
\(0\) |
\(0\) |
\(\frac{1}{\alpha}\) |
Gyrator-like amplifier (non-unilateral) |
\(0\) |
\(\frac{1}{\gamma}\) |
\(\frac{1}{\zeta}\) |
\(0\) |
Triple loop 1 (non-unilateral) |
\(\frac{1}{\mu}\) |
\(\frac{1}{\gamma}\) |
\(\frac{1}{\zeta}\) |
\(0\) |
Triple loop 2 (non-unilateral) |
\(\frac{1}{\mu}\) |
\(\frac{1}{\gamma}\) |
\(0\) |
\(\frac{1}{\alpha}\) |
Triple loop 3 (non-unilateral) |
\(\frac{1}{\mu}\) |
\(0\) |
\(\frac{1}{\zeta}\) |
\(\frac{1}{\alpha}\) |
Triple loop 4 (non-unilateral) |
\(0\) |
\(\frac{1}{\gamma}\) |
\(\frac{1}{\zeta}\) |
\(\frac{1}{\alpha}\) |
Quadruple loop |
\(\frac{1}{\mu}\) |
\(\frac{1}{\gamma}\) |
\(\frac{1}{\zeta}\) |
\(\frac{1}{\alpha}\) |
Direct sensing and comparison techniques#
Until now we discussed the conceptual design of so-called direct feedback amplifiers. In direct feedback amplifiers, the load quantity is sensed and the source quantity is compared with the feedback quantity. With this technique, high-quality amplifiers can be designed. The opposite to direct feedback is indirect feedback. We will discuss it at a later stage.
We will now discuss direct sensing and comparison techniques in more detail.
Direct voltage sensing techniques#
If we need to fix the transmission parameters \(A\) or \(C\) of an amplifier to nonzero values, we have to sense the voltage across the load. The input voltage of the feedback network will equal the load voltage if the input terminals of the feedback network are connected in parallel with the load. This is shown in Fig. 217. Hence, if the load must be floating with respect to the ground, the input of the feedback network should also be floating with respect to ground. If one terminal of the load is connected to the ground, the input port of the feedback network will also be grounded.
Direct voltage comparison techniques#
If we need to fix the transmission parameter \(A\) or \(B\) of an amplifier to a nonzero value, we have to subtract the output voltage of the feedback network from the source voltage. Methods for direct voltage comparison are shown in Fig. 218.
Fig. 218 Direct voltage comparison, or input voltage feedback. A: The controller input port, the source and the output port of the feedback network, are floating with respect to the ground. B: Grounded input port of the controller and grounded load. The output port of the feedback network has to be floating with respect to the ground. C: The input port of the controller and the output port of the feedback network are grounded. the source has to be floating with respect to the ground, D: The source and the output port of the feedback network are grounded. The input port of the controller has to be floating with respect to the ground.#
The principle of voltage comparison is to connect the source and the feedback network anti-series, with the controller input closing the loop. This is shown in Fig. 218A.
Ideally, the differential-mode input voltage, the differential-mode input current and the common-mode input current of the controller are zero (controller is a natural two-port with \(A,B,C\) and \(D\) equal to zero).
Fig. 218B though Fig. 218D show different grounding concepts for this loop. Similar as with direct current sensing, parasitic impedances in parallel with the source, or in parallel with the output of the feedback network, may affect the ideal gain of the amplifier (see section Ideal gain of a feedback amplifier). This is of particular interest if the input port of the controller needs to be floating with respect to the ground (see Fig. 218D).
Direct current sensing techniques#
Fig. 219 Direct current sensing, or output series series feedback. A: The controller output port, the load and the input port of the feedback network, are floating with respect to the ground. B: Grounded output port of the controller and grounded load. The input port of the feedback network has to be floating with respect to the ground. C: The output port of the controller and the input port of the feedback network are grounded. the load has to be floating with respect to the ground, D: The load and input port of the feedback network are grounded. The output port of the controller has to be floating with respect to the ground.#
If we need to fix the transmission parameters \(B\) and \(D\) of an amplifier to nonzero values, we have to sense the current through the load. The input current of the feedback network equals the load current if the input port of the feedback network is placed in series with the load.
Methods for current sensing are shown in Fig. 219. The principle of direct sensing of the load current is that the sensing element, the load and the output port of the controller are part of a current loop. This is shown in Fig. 219A. Fig. 219B though Fig. 219D show different grounding concepts for this loop.
If the load and the output port of the controller are grounded, the input port of the sensing element has to be floating with respect to the ground. This is shown in Fig. 219B. Fig. 219C illustrates a situation in which the output port of the controller and the input port of the sensing element are connected to the ground. In such a situation, the load cannot share one of its nodes with the ground. Fig. 219D shows a situation in which the load and the input port of the sensing element are connected to the ground. In this case, the output port of the controller has to be floating with respect to the ground.
In all cases, parasitic impedances in parallel with the load and in parallel with the input port of the sensing element have to be avoided.
Such impedances create parasitic current paths resulting in a difference between the load current and the input current of the sensing element. A parasitic impedance in parallel with the load results in a transfer that depends on the load impedance, while an impedance in parallel with the input port of the sensing element introduces an error in transfer of the feedback network, resulting in an error of the ideal gain of the feedback amplifier.
In practice, this complicates the design of a controller that requires a floating output port (see Fig. 219D). Parasitic impedances between the output terminals of the controller and the ground should be large enough to keep those errors within acceptable limits.
Direct current comparison techniques#
If we need to fix the transmission parameters \(C\) or \(D\) of an amplifier, we have to subtract the output current of the feedback network from the source current. To do so, the output port of the feedback network needs to be connected anti-parallel with the source. This is shown in Fig. 220.
If the source must be floating with respect to the ground, the output port of the feedback network should also be floating with respect to the ground. If one terminal of the source is grounded, the output port of the feedback network should also be grounded.
Direct current comparison is dual to direct voltage sensing.
Indirect sensing and comparison techniques#
Sometimes it is not possible to sense the load quantity or to compare the feedback quantity with the source quantity directly. In those cases, indirect feedback or model-based feedback can be applied. Basic indirect feedback techniques have been illustrated in Fig. 221. Indirect feedback techniques are often used in IC design. A more detailed discussion of indirect feedback will be presented in section Indirect feedback.
Fig. 221 Indirect feedback amplifiers use indirect sensing of the load signal and/or indirect comparison with the source signal.#