As shown in Figure 2(d), with 10% negative feedback, the gain AV drops to half of AV. In other words, negative feedback reduces the overall gain of an amplifier. Therefore it is called degenerative feedback. Here, a portion of the amplifier output is fed back to the input in such way that it opposes the input. Consider the case when AV = AV/100. Since the gain in deci­bels is defined as 20 log10(VO/VI), the reduction of the gain by a factor of 100 means a loss of 20 log10 iio = —40 dB. Thus, expressed in decibels,

AV = AV — 40 dB

Now we can revisit Fig. 1 and study the frequency re­sponse with and without negative feedback. This is shown in Fig. 3. We can easily see that the bandwidth of the amplifier has been increased with negative feedback. This is the great­est advantage of negative-feedback amplifiers. In addition, negative feedback results in stabilized amplifier gain, ex­tended bandwidth, and reduced noise and distortion. In other words, it is possible to achieve predictable voltage gains with negative feedback amplifiers. Besides, the resulting amplifi­ers are very stable.

It is possible to prove that series voltage feedback in­creases the input impedance of the circuit. The input imped­ance can be reduced by incorporating a parallel current feedback.

We can rewrite the previously considered equation incorpo­rating the negative sign as

A’ =

V 1 + My

Without feedback

For very large values of AV the above equation becomes


Consider a case when AV2 = (1 + 0.707AV = 1.707/3. Re­calculate the new closed-loop gain with negative feedback:

V 2

Ay2 1 + M



1 + (1.707в/в)

= 0. 707/в = 0.707AV

Observe that the above equation is independent of AV, which is the gain without feedback. (also sometimes called the open – loop gain). In other words, even though the open-loop gain falls by a factor 1.707, the closed-loop gain falls only by 3 dB. Similarly, we can prove

Percentage distortion with negative feedback

Percentage distortion without negative feedback 1 + pAv

Negative feedback also helps in reducing the circuit noise. Thus, the signal-to-noise ratio is greatly improved:

= (1 + вAv )(S/N)


For a simple amplifier the voltage gain is defined as the ratio of output voltage to input voltage. This is written as AV = VO/VI as shown in Fig. 2(a). Addition of a feedback of magni­tude 3, as shown in Fig. 2(b), will result in a modified value for the voltage gain given by the equation: AV = AV/(1 — 3AV). The term /SAV, called the feedback factor, can be either positive or negative. A study of the variation of AV with posi­tive as well as negative values of 3 is shown in Fig. 2(c, d). It is observed that the value of AV becomes infinite with only 10% of positive feedback. However, this should not be viewed as advantageous, because positive feedback greatly increases distortion in the output. Mathematically it is true that the gain approaches infinity; however, in reality, the circuit be­gins to oscillate. Positive feedback does not find many applica­tions.

voltage Vj

voltage VO




Figure 2. (a) Block diagram of an amplifier with AV = VO/Vi. (b) Block diagram of an amplifier with feedback. The dashed line en­closes the entire amplifier including the feedback; its gain is AV = VO/Vi.


Midfrequency level




Figure 1. Frequency response curve of an audio amplifier.

Consider the case when PO = JPj. Then the gain in decibels is 101og10 = -3.0103

Therefore, for audio engineers the point of interest lies where the gain falls by 3 dB. The frequencies at which this occurs are called half-power frequencies. Let the flat portion of the


Av/V2 = AV-3 dB a’v

AV -3 dB = A’v/V2

With negative feedback


Frequency (log scale)

Bandwidth _________

(no feedback)

Bandwidth (with feedback)




(S/N )f,


However, in 1932, Harry Nyquist of Bell Telephone Labo­ratories extended this theory at length and published his fa­mous paper on regenerative theory. His principles laid the foundation for the development of feedback oscillators. There­fore, positive feedback is also called regenerative feedback. While designing frequency-selective feedback circuits for audio amplifiers, one may use positive feedback either as a bass or as a treble boost.


When analyzing amplifiers mathematically, it is convenient to assume that the gain calculations are not affected by the reactive elements that might be present in the circuit. How­ever, in reality, capacitances and inductances play a major role in determining how the amplifier performs over a range of frequencies. The effect of inductances can be minimized but it is impossible to ignore the presence of capacitances. This effect is more pronounced particularly while analyzing multistage amplifiers. Coupling capacitors and bypass capaci­tors can reduce the gain of an amplifier at lower or higher frequencies, because the capacitive reactance is inversely pro­portional to the frequency. In other words, as the frequency increases, the capacitive reactance decreases because


x =

j(2n fC)

Therefore, if there is a grounded bypass capacitor, signal cur­rents may be inadvertently diverted to ground instead of be­ing transmitted to the output. This is because bypass capaci­tors offer low reactances to signal currents at higher frequencies. However, the bypass capacitors offer high re­actances to signals at lower frequencies, and therefore diver­sion of such currents to ground does not pose a major problem.

Figure 1 is a representation of a frequency plot of an am­plifier. Here, the output voltage or power gain is plotted against a range of frequencies (for a given constant input volt­age). The frequency axis is normally plotted on a logarithmic scale. The unit for the y axis is decibels (dB); the number of decibels of gain is given by

201ogi° ^


Table 1. Bandwidth Values for Selected Electronic Signals

Signal Type

Frequency Range


0.05 to 100 Hz

Audio signals (human ear)

20 Hz to 15,000 Hz

AM radio waves

550 kHz to 1600 kHz

FM radio waves

88 MHz to 100 MHz

Microwave and satellite signals

1 GHz to 50 GHz

amplifier characteristic be assigned a voltage level of Vflat. Then the frequencies at which voltage levels have dropped to

0. 707Vflat are denoted by fL and fH. The range of frequencies that lies between f L and f H is known as the bandwidth. In other words, the bandwidth can be defined as the frequency range over which the amplifier gain remains within 29.3% of its maximum value (3 dB level, or 1 — 0.707 = 0.293).

The bandwidth of an amplifier depends upon the applica­tions and signal type involved. Bandwidth values for some selected electronic signals are given in Table 1.


High-Precision Instruments for Voltage Standards

• Datron Systems Division, 200 West Los Angeles Ave., Simi Valley, CA 93065-1650. Phone: 805-584-1717. Fax: 805-526-0885.

• The Eppley Laboratory, Inc., 12 Sheffield Avenue, New­port, RI 02840. Phone: 401-847-1020. Fax: 401-847-1031.

• Fluke Corporation, MS 250, P. O. Box 9090, Everett, WA 98206-9090. Phone: 800-44F-LUKE or 425-347-6100. Fax: 425-356-5116.

• Hewlett Packard Co., Electronic Measurement Systems, 815 14th Street SW, P. O. Box 301, Loveland, CO 80538. Phone: 970-679-5000. Fax: 970-679-5954.

• Julie Research Laboratories, Inc., 508 West 26th Street, New York, NY 10001. Phone: 212-633-6625. Fax: 212­691-3320.

• Keithley Instruments, 28775 Aurora Road, Cleveland, OH 44139. Phone: 800-552-1115 or 440-248-0400. Fax: 440-248-6168.

Voltage Reference Integrated Circuits

• Analog Devices Inc., 1 Technology Way, P. O. Box 9106, Norwood, MA 02062-9106. Phone: 781-329-4700. Fax: 781-326-8703.

• Burr Brown Corp., International Airport Industrial Park, 6730 South Tucson Boulevard, P. O. Box 11400, Tucson, AZ 85734. Phone: 800-548-6132 or 520-746-1111. Fax: 520-889-1510.

• LTC (Linear Technology Corp.), 1630 McCarthy Boule­vard, Milpitas, CA 95035-7417. Phone: 408-432-1900. Fax: 408-434-0507.

• Maxim Integrated Products, 120 San Gabriel Drive, Sun­nyvale, CA 94086-9892. Phone: 408-737-7600. Fax: 408­737-7194.

• NSC (National Semiconductor Corp.), MS D2565, 2900 Semiconductor Drive, Santa Clara, CA 95051. Phone: 408-721-8165 or 800-272-9959. Fax: 800-737-7018.

• Thaler Corp., 2015 North Forbes #109, Tucson, AZ 85745. Phone: 800-827-6006 or 520-882-4000. Fax: 520­770-9222.


In classical metrology, one uses a precision (six-digit, seven­digit, or eight-digit) voltage divider, known as a potentio­meter. This has very little in common with the variable resis­tor often called a “potentiometer” or ‘‘pot’’—but it does act as a voltage divider. When such a precision potentiometer is used with a null meter, any voltage can be compared with a standard or reference voltage. The unknown voltage is thus well determined, according to the ratio of the potentiometer and the standard voltage (allowing for its uncertainty.) How­ever, most precision potentiometers are not guaranteed to maintain their linearity to 1 LSD (Least Significant Digit) for long-term accuracy, after their resistive dividers are trimmed and calibrated. A good potentiometer may hold better than 1 X 10—6 linearity per year, but it is not guaranteed that switching from 0.499999 to 0.500000 will not cause a decrease of its output. Further, an inexperienced user may find it very time-consuming to use such a divider. When taking a large number of data, long-term system drift may cause errors that could be avoided by taking data more quickly.

The author’s recommendation is to use a good six-digit or seven-digit multislope integrating digital voltmeter (DVM), with 1 X 10—6 inherent differential linearity and excellent overall (end-to-end) linearity. The author has had excellent experience with HP 3456, 3457, 3468, and other similar inte­grating voltmeters. Differential nonlinearity has never been observed to exceed 1 X 10—6 of full scale, on 10 V scales. Noise, offsets, and gain errors are usually acceptably small. For best absolute accuracy, the DVM’s full-scale factor should be com­pared with other stable references. Note that not all six-digit or seven-digit DVMs have this inherent linearity.


Since most advances in references are designed by IC manu­facturers on a commercial basis, to be aware of good new products, one must inquire of the IC manufacturers, to see what is available. A list of IC makers is provided here, as well as a list of companies making precision references and measuring equipment.