Fig. 1.1: Simple Block Diagram of a Feedback Amplifier
A typical feedback connection is shown in Figure 1.1. The input signal, , is applied to a mixer network, where it is combined with a feedback signal, . The difference of these signals, , is connected to the feedback network which provides a reduced portion of the output as feedback signal to the input mixer network.
Fig. 1.2: Amplifier Block Diagram
For an ordinary amplifier i.e. one without feedback, the voltage gain is given by the ratio of the output voltage and input voltage As shown in the block diagram of Fig. 1.2, the input voltage is amplified by a factor of A to the value Vo of the output voltage.
This gain A is often called open-loop gain.
Fig. 1.3: Feedback Loop
Suppose a feedback loop is added to the amplifier (Fig. 1.3). If is the output voltage with feedback, then a fraction (it is not the same as the of a transistor) of this voltage is applied to the input voltage which, therefore, becomes ( ) depending on whether the feedback voltage is in phase or anti-phase with it. Assuming positive feedback, the input voltage will become When amplified A times, it becomes
The amplifier gain with feedback is given by
Positive feedback
Negative feedback
The term ‘βA’ is called feedback factor whereas β is known as feedback ratio. The expression is called loop gain. The amplifier gain A´ with feedback is also referred to as closed-loop gain because it is the gain obtained after the feedback loop is closed. The sacrifice factor is defined as
2.0 OBJECTIVES
At the end of the unit, you should be able to:
• understand the concept of feedback
• establish the fact that positive feedback is a criterion for oscillation
• understand the Nyquist criterion.
3.0 MAIN CONTENT 3.1 Positive Feedback
So far we have considered the operation of a feedback amplifier in which the feedback signal was opposite to the input signal - negative feedback (refer to unit 1 of this module). In any practical circuit, this condition occurs only for some mid-frequency range of operation. We know that an amplifier gain will change with frequency, dropping off at high frequencies from the mid-frequency value. In addition, the phase shift of an amplifier will also change with frequency.
If, as the frequency increases, the phase shift changes then some of the feedback signal will add to the input signal. It is then possible for the amplifier to break into oscillation due to positive feedback. If the amplifier oscillates at some low or high frequency, it is no longer useful as an amplifier. Proper feedback-amplifier design requires that the circuit be stable at all frequencies, not merely those in the range of interest. Otherwise a transient disturbance could cause a seemingly stable amplifier to suddenly start oscillating.
3.2 Nyquist Criterion
In judging the stability of a feedback amplifier, as a function of frequency, the βA product and the phase shift between input and output are the determining factors. One of the most popular techniques used to investigate stability is the Nyquist method. A Nyquist diagram is used to plot gain and phase shift as a function of frequency on a complex plane.
The Nyquist plot, in effect, combine the two Bode plots of gain versus frequency and phase shift versus frequency and phase shift versus frequency on a single plot. A Nyquist plot is used to quickly show whether an amplifier is stable for all frequencies and how stable the amplifier is relative to some gain or phase-shift criteria.
Fig. 3.1: Complex Plane Showing Typical Gain-phase Points As a start, consider the complex plane shown in fig 3.1. A few points of various gain (βA) values are shown at a few different phase-shift angles.
By using the positive real axis as reference (0o), a magnitude of βA = 2 is shown at a phase shift of 0o at point 1. Additionally, a magnitude of βA = 3 at a phase shift of -135o is shown at point 2 and a magnitude/phase of βA =1 at 180o is shown at point 3. Thus points on this plot can represent both gain magnitude of βA and phase shift. If the point representing gain and phase shift for an amplifier circuit can be plotted at increasing frequency, then a Nyquist plot is obtained as shown by the plot in fig 3.2.
At the origin, the gain is 0 at a frequency of 0 (for RC-type of coupling).
At increasing frequency, point f1, f2, and f3 and the phase shift increased, as did the magnitude of βA. At a representing frequency f4, the value of A is the vector length from the origin to point f4 and the phase shift is the angle . At a frequency f5, the phase shift is 180o. At higher frequencies, the gain is shown to decrease back to 0.
Fig 3.2: Nyquist plot
The Nyquist criterion for stability can be stated as follows:
• The amplifier is unstable if the Nyquist curve plotted encloses (encircles) the -1 point, and it is stable otherwise.
The use of positive feedback that results in a feedback amplifier having closed-loop gain greater than 1 and satisfies the phase condition will result in operation as an oscillator circuit. An oscillator circuit then provides a varying output signal. If the output signal varies sinusoidally, the circuit is referred to as sinusoidal oscillator. If the output voltage rises quickly to one voltage level and later drops quickly to another voltage level, the circuit is generally referred to as a pulse or square- wave oscillator.
Fig 3.3: Feedback circuit used as an oscillator
To understand how a feedback circuit performs as an oscillator, consider the feedback circuit of Fig 3.3. When the switch at the amplifier input is open, no oscillation occurs. Consider that we have a fictitious voltage at
the amplifier input . This results in an output voltage after
the amplifier stage and in a voltage after the feedback stage.
Thus, we have a feedback voltage where is referred to as the loop gain. If the circuits of the base amplifier and feedback network provide of a correct magnitude and phase, can be made equal to . Then, when the switch is closed and the fictitious voltage is removed, the circuit will continue operating since the feedback voltage is sufficient to drive the amplifier and feedback circuits resulting in a proper input voltage to sustain the loop operation. The output waveform will still exist after the switch is closed if the condition
is met. This is known as Barkhausen criterion for oscillation.
In reality, no input signal is needed to start the oscillator going. Only the condition must be satisfied for self-sustained oscillations to result. In practice, is made greater than and the system is started
oscillating by amplifying noise voltage, which is always present.
Saturation factors in the practical circuit provide an average value of of . The resulting waveforms are never exactly sinusoidal. However, the closer the value of is to exactly , the more nearly sinusoidal is the waveform.
3.3 Gain and Phase Margins
From the Nyquist criterion, we know that a feedback amplifier is stable if the loop gain is less than unity when its phase angle is . We can additionally determine some margin of stability to indicate how close to instability the amplifier is. That is, if the gain is less than unity but, say, in value, this would not be as relatively stable as another amplifier having, say, (both measured at ). Of course, amplifier with loop gain and are both stable, but one is closer to instability, if the loop gain increases, than the other.
We can define the following terms:
decibels at the frequency at which the phase angle is . Thus,
0 dB, equal to a value of , is on the border of stability and any negative decibel value is stable.
• Phase margin (PM) is defined as the angle of minus the magnitude of the angle at which the value of is unity
4.0 CONCLUSION
In this unit, you have been introduced to the basic concept of positive feedback, Nyquist criterion and the Barkhausen criterion as the basic criteria for oscillation.
5.0 SUMMARY
Positive feedback drives a circuit into oscillation as in various types of oscillator circuits and Nyquist criterion which states that The amplifier is unstable if the Nyquist curve plotted encloses (encircles) the point, and it is stable otherwise provides the condition for stability. The output waveform will still exist after the switch is closed if the condition
is met. This is known as Barkhausen criterion for oscillation.
6.0 TUTOR-MARKED ASSIGNMENT
i. With the aid of diagram(s), derive the expression for an amplifier with positive feedback and negative feedback. Indicate the feedback factor and feedback ratio in the expression.
ii. What is Nyquist Criterion? Explain.
iii. State the Barkhausen Criterion for oscillation.
iv. Define the following:
a. Gain Margin.
b. Phase margin.
7.0 REFERENCES/FURTHER READING
Robert, L. B. (1999). Electronic Devices and Circuit Theory. 7th Edition Prentice-Hall Inc. New Jersey.
Theraja, B. L. and Theraja, A. K. (2010). Electrical Technology. S. C.
Chand, New Delhi, India.
www.wikipedia.com www.howstuffworks.com www.worksheet.com
www.electronicstutorials.com