Negative feedback amplifier

When a fraction of the output of an amplifier is combined with the input, feedback exists; if the feedback opposes the original signal, it is negative feedback and if it increases the signal it is positive feedback. A negative feedback amplifier, or more commonly simply a feedback amplifier, is an amplifier which uses negative feedback to improve performance (gain stability, linearity, frequency response, step response) and reduce sensitivity to parameter variations due to manufacturing or environmental uncertainties. A single feedback loop with unilateral blocks is shown in Figure 1. Negative feedback is used in this way in many amplifiers and control systems.

Figure 1: Ideal negative feedback model

Classical feedback

Voltage amplifiers

Below, the gain of the amplifier with feedback, the closed-loop gain A_fb, is derived in terms of the gain of the amplifier without feedback, the open-loop gain A_OL and the feedback factor β, which governs how much of the output signal is applied to the input. See Figure 1, top right. The feedback parameter β is determined by the feedback network that is connected around the amplifier. For an operational amplifier two resistors may be used for the feedback network to set β between 0 and 1. This network may be modified using reactive elements like capacitors or inductors to (a) give frequency-dependent closed-loop gain as in equalization/tone-control circuits or (b) construct oscillators.
Consider a voltage amplifier with voltage feedback. Without feedback, the output voltage V_out = A_OL V_in, where the open-loop gain A_OL in general may be a function of both frequency and voltage. The open-loop gain A_OL is defined by:

$A_\mathrm{OL} = \frac{V_\mathrm{out}}{V_\mathrm{in}} \ ,$

where V_in is the input to the amplifier, assuming no feedback, and V_out is the amplifier output, again with no feedback.
Suppose we have a feedback loop so that a fraction β V_out of the output is subtracted from the input. The input to the amplifier is now V'_in, where

$V'_\mathrm{in} = V_\mathrm{in} - \beta \cdot V_\mathrm{out}$

The gain of the amplifier with feedback, called the closed-loop gain, A_fb is given by,

$A_\mathrm{fb} = \frac{V_\mathrm{out}}{V_\mathrm{in}}$

Substituting for V_in,

$A_\mathrm{fb} = \frac{V_\mathrm{out}}{V'_\mathrm{in} + \beta \cdot V_\mathrm{out}}$

Dividing numerator and denominator by V'_in,

$A_\mathrm{fb} = \frac{ \frac{V_\mathrm{out}}{V'_\mathrm{in}} }{ 1 + \beta \cdot \frac{V_\mathrm{out}}{V'_\mathrm{in}} }$

But since

$A_\mathrm{OL} = \frac{V_\mathrm{out}}{V'_\mathrm{in}}$ ,

then

$A_\mathrm{fb} = \frac{A_\mathrm{OL}}{1 + \beta \cdot A_\mathrm{OL}}$

If A_OL >> 1, then A_fb ≈ 1 / β and the effective amplification (or closed-loop gain) A_fb is set by the feedback constant β, and hence set by the feedback network, usually a simple reproducible network, thus making linearizing and stabilizing the amplification characteristics straightforward. Note also that if there are conditions where β A_OL = −1, the amplifier has infinite amplification – it has become an oscillator, and the system is unstable. The stability characteristics of the gain feedback product β A_OL are often displayed and investigated on a Nyquist plot (a polar plot of the gain/phase shift as a parametric function of frequency). A simpler, but less general technique, uses Bode plots.
The combination L = β A_OL appears commonly in feedback analysis and is called the loop gain. The combination ( 1 + β A_OL ) also appears commonly and is variously named as the desensitivity factor or the improvement factor.
Bandwidth extension

Figure 2: Gain vs. frequency for a single-pole amplifier with and without feedback; corner frequencies are labeled.

Feedback can be used to extend the bandwidth of an amplifier (speed it up) at the cost of lowering the amplifier gain. Figure 2 shows such a comparison. The figure is understood as follows. Without feedback the so-called open-loop gain in this example has a single time constant frequency response given by

$A_{OL}(f) = \frac {A_0} { 1+ j f / f_C } \ ,$

where f_C is the cutoff or corner frequency of the amplifier: in this example f_C = 10⁴ Hz and the gain at zero frequency A₀ = 10⁵ V/V. The figure shows the gain is flat out to the corner frequency and then drops. When feedback is present the so-called closed-loop gain, as shown in the formula of the previous section, becomes,
$A_{fb} (f) = \frac { A_{OL} } { 1 + \beta A_{OL} }$
$= \frac { A_0/(1+jf/f_C) } { 1 + \beta A_0/(1+jf/f_C) }$

$= \frac {A_0} {1+ jf/f_C + \beta A_0}$

$= \frac {A_0} {(1 + \beta A_0) \left(1+j \frac {f} {(1+ \beta A_0) f_C } \right)} \ .$

The last expression shows the feedback amplifier still has a single time constant behavior, but the corner frequency is now increased by the improvement factor ( 1 + β A₀ ), and the gain at zero frequency has dropped by exactly the same factor. This behavior is called the gain-bandwidth tradeoff. In Figure 2, ( 1 + β A₀ ) = 10³, so A_fb(0)= 10⁵ / 10³ = 100 V/V, and f_C increases to 10⁴ × 10³ = 10⁷ Hz.

Multiple poles

When the open-loop gain has several poles, rather than the single pole of the above example, feedback can result in complex poles (real and imaginary parts). In a two-pole case, the result is peaking in the frequency response of the feedback amplifier near its corner frequency, and ringing and overshoot in its step response. In the case of more than two poles, the feedback amplifier can become unstable, and oscillate. See the discussion of gain margin and phase margin. For a complete discussion, see Sansen.
Feedback and amplifier type
Amplifiers use current or voltage as input and output, so four types of amplifier are possible. See classification of amplifiers. Any of these four choices may be the open-loop amplifier used to construct the feedback amplifier. The objective for the feedback amplifier also may be any one of the four types of amplifier, not necessarily the same type as the open-loop amplifier. For example, an op amp (voltage amplifier) can be arranged to make a current amplifier instead. The conversion from one type to another is implemented using different feedback connections, usually referred to as series or shunt (parallel) connections. See the table below.

The feedback can be implemented using a two-port network. There are four types of two-port network, and the selection depends upon the type of feedback. For example, for a current feedback amplifier, current at the output is sampled and combined with current at the input. Therefore, the feedback ideally is performed using an (output) current-controlled current source (CCCS), and its imperfect realization using a two-port network also must incorporate a CCCS, that is, the appropriate choice for feedback network is a g-parameter two-port.

Two-port analysis of feedback

One approach to feedback is the use of return ratio. Here an alternative method used in most textbooks is presented by means of an example treated in the article on asymptotic gain model.

Figure 3: A shunt-series feedback amplifier

Figure 3 shows a two-transistor amplifier with a feedback resistor R_f. The aim is to analyze this circuit to find three items: the gain, the output impedance looking into the amplifier from the load, and the input impedance looking into the amplifier from the source.

Replacement of the feedback network with a two-port

The first step is replacement of the feedback network by a two-port. Just what components go into the two-port?
On the input side of the two-port we have R_f. If the voltage at the right side of R_f changes, it changes the current in R_f that is subtracted from the current entering the base of the input transistor. That is, the input side of the two-port is a dependent current source controlled by the voltage at the top of resistor R₂.
One might say the second stage of the amplifier is just a voltage follower, transmitting the voltage at the collector of the input transistor to the top of R₂. That is, the monitored output signal is really the voltage at the collector of the input transistor. That view is legitimate, but then the voltage follower stage becomes part of the feedback network. That makes analysis of feedback more complicated.

Figure 4: The g-parameter feedback network

An alternative view is that the voltage at the top of R₂ is set by the emitter current of the output transistor. That view leads to an entirely passive feedback network made up of R₂ and R_f. The variable controlling the feedback is the emitter current, so the feedback is a current-controlled current source (CCCS). We search through the four available two-port networks and find the only one with a CCCS is the g-parameter two-port, shown in Figure 4. The next task is to select the g-parameters so that the two-port of Figure 4 is electrically equivalent to the L-section made up of R₂ and R_f. That selection is an algebraic procedure made most simply by looking at two individual cases: the case with V₁ = 0, which makes the VCVS on the right side of the two-port a short-circuit; and the case with I₂ = 0. which makes the CCCS on the left side an open circuit. The algebra in these two cases is simple, much easier than solving for all variables at once. The choice of g-parameters that make the two-port and the L-section behave the same way are shown in the table below.