Equation on Negative Feed Back Loop Equation

Circuit Analysis

Morgan Jones , in Valve Amplifiers (Fourth Edition), 2012

Practical Limitations of the Feedback Equation

The feedback equation implies improved performance provided that βA ₀>>1. If, for any reason, the open loop gain of the amplifier is less than infinite, then βA ₀ will not be much greater than 1, and the approximation will no longer be true.

Practical amplifiers always have finite gain; moreover, this gain falls with frequency. A practical amplifier will always distort the input signal, and because the distortion-reducing ability of negative feedback falls with frequency, the closed loop distortion must rise with frequency.

Crossover distortion in Class B amplifiers can be considered to be a severe reduction of gain as the amplifier traverses the switching point of the transistors or valves. Because of the drastically reduced open loop gain in this region, negative feedback is not very effective at reducing crossover distortion.

Although not explicitly stated by the feedback equation, the phase of the feedback signal is crucially important. If the phase should change by 180°, then the feedback will no longer be negative, but positive, and our amplifier may turn into an oscillator.

We will explore the practical limitations of feedback in Chapter 6, but we should realise that, as with any weapon wielded carelessly, it is possible to shoot oneself in the foot.

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Linear Systems in the Frequency Domain

John Semmlow , in Signals and Systems for Bioengineers (Second Edition), 2012

Analysis: The solution given in Equation 5.10 is known as the feedback equation and is used in later analyses of more-complex systems. In this example, the two elements, G and H, were constants, but could have been anything as long as they can be treated algebraically. When the individual elements contain differential or integral operations in their input/output relationships, the techniques described below are required to encode these calculus operations into algebraic manipulations, but this equation, and the algebraic manipulations used in this example, still apply.

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Noise in the Luenberger Observer

George Ellis , in Observers in Control Systems, 2002

7.3.1.2 Transfer-Function Analysis of CN with Disturbance Decoupling

The sensitivity of the actual state to noise is the final measure on the control system. The next step will be to investigate this by building the transfer function of C(S)/N(S). To simplify transfer-function analysis, the disturbance-decoupled system is drawn in Figure 7-12 by substituting Equation 7.17 in place of the common implementation of the Luenberger observer. Just to be clear, the two implementations differ, but the analysis is valid because transfer functions are the same.

Figure 7-12. Noise in an observer-based disturbance-decoupled system redrawn according to Equation 7.17.

The transfer function from N(S)to C(S)can be written using Mason's signal flow graphs. There are two paths from N(S)to C(S), one through the observer (P ₁) and the other through the control law (P ₂). There are three loops: one through the control law (L ₁) and two passing through the observer (L ₂and L ₃).

(7.19) $P_1=G_{SEst}^{-1}\left(s\right)\times G_{PEst}^{-1}\left(s\right)\times G_{OLPF}\left(s\right)\times G_{PC}\left(s\right)\times G_p\left(s\right)$

(7.20) $P_2=G_C\left(s\right)\times G_{PC}\left(s\right)\times G_p\left(s\right)$

(7.21) $L_1=-G_C\left(s\right)\times G_{PC}\left(s\right)\times G_P\left(s\right)\times G_S\left(s\right)$

(7.22) $L_2=-G_{PC}\left(s\right)\times G_P\left(s\right)\times G_S\left(s\right)\times G_{SEst}^{-1}\left(s\right)\times G_{PEst}^{-1}\left(s\right)\times G_{OLPF}\left(s\right)$

(7.23) $L_3=G_{PC}\left(s\right)\times G_{OLPF}\left(s\right)$

Since all loops touch (via G _PC(S)), the denominator of the transfer function contains no combinations of loops. Since all forward paths touch all loops (again, via G _PC(S)), the cofactors, which appear in the numerator, are all 1. Thus, the transfer function is simply:

(7.24) $\frac{C\left(s\right)}{N\left(s\right)}=\frac{P_1+p_2}{1-L_1-L_2-L_3}.$

Two assumptions simplify the transfer-function analysis. Assuming that the estimated plant and estimated sensor are accurate, (G _P(S)≈G _PEst(S)and G _S (S) ≈G _{SES t}(S)),L ₂and L ₃cancel in the denominator. Also, Equation 7.19 simplifies to:

(7.25) $P_1\approx G_{SEst}^{-1}\left(s\right)\times G_{OLPF}\left(s\right)\times G_{PC}\left(s\right).$

The transfer function of noise sensitivity is then:

(7.26) $\frac{C\left(s\right)}{N\left(s\right)}\approx \frac{G_{SEst}^{-1}\times G_{OLPF}\left(s\right)\times G_{PC}\left(s\right)+G_C\left(s\right)\times G_{PC}\left(s\right)\times G_P\left(s\right)}{1+G_C\left(s\right)\times G_{PC}\left(s\right)\times G_P\left(s\right)\times G_S\left(s\right)}.$

Equation 7.26 indicates that the disturbance-decoupled system has much greater noise sensitivity than does the system with observer feedback (Equation 7.8). The denominators of Equations 7.8 and 7.26 are similar, but the first term in the numerator of Equation 7.26, which represents the decoupling path from noise to the actual state, bypasses the control law. ² The amplifying factor (G⁻¹ _SEst (S))lacks the attenuating term G _PEst(S). In practice, the noise passing through the disturbance-decoupling path,P ₁, will normally be much larger than the noise passing through the control-law path,P ₂.(In fact, in many cases,P ₂can be ignored.) This accounts for the disturbance-decoupled system's greatly increased noise sensitivity.

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The Power Amplifier

Morgan Jones , in Valve Amplifiers (Fourth Edition), 2012

Output Resistance with Both Terminals Equally Loaded (Class A1 Loading)

The concertina is a special case (R _k=R _a ) of an unbypassed common cathode amplifier with outputs taken from both anode and cathode. The general feedback equation is:

$A=\frac{A_0}{(1+\beta A_0)}$

The denominator of the feedback equation is the factor by which resistances are changed and is known as the feedback factor. Since we know the gain of the concertina and the gain of a simple triode amplifier, we can substitute them into the feedback equation to solve for the feedback factor:

$\frac{\mu R_{\mathrm{L}}}{R_{\mathrm{L}}(\mu +2)+r_{\mathrm{a}}}=\frac{(\mu R_{\mathrm{L}}/R_{\mathrm{L}}+r_{\mathrm{a}})}{\text{feedback}\text{factor}}$

Cross-multiplying to find the feedback factor:

$\text{Feedback}\text{factor}=\frac{R_{\mathrm{L}}(\mu +2)+r_{\mathrm{a}}}{R_{\mathrm{L}}+r_{\mathrm{a}}}$

The anode output resistance of a common cathode triode amplifier with no feedback is:

$r_{\text{out}}=\frac{R_{\mathrm{L}}{r}_{\mathrm{a}}}{R_{\mathrm{L}}+r_{\mathrm{a}}}$

The feedback works to reduce anode output resistance, so this value must be divided by the feedback factor (practically, we multiply by the inverse of the feedback factor):

${r΄}_{\text{out}}=\frac{R_{\mathrm{L}}{r}_{\mathrm{a}}}{R_{\mathrm{L}}+r_{\mathrm{a}}}·\frac{R_{\mathrm{L}}+r_{\mathrm{a}}}{R_{\mathrm{L}}(\mu +2)+r_{\mathrm{a}}}$

The (R _L+r _a) terms cancel, leaving:

${r΄}_{\text{out}}=\frac{R_{\mathrm{L}}{r}_{\mathrm{a}}}{R_{\mathrm{L}}(\mu +2)+r_{\mathrm{a}}}$

Initially, it seems most surprising that series feedback (R _k=R _a, after all) should reduce output resistance from the anode so that r _out ≈1/g _m, but this can be understood by considering an external capacitive load on each output. In the same way that R _k=R _a defines a gain of 1 at low frequencies, so X _C(k)=X _C(a) defines a gain of 1 at high frequencies, and changing this ratio of capacitances certainly would change the gain, or frequency response at high frequencies, since it would change the feedback ratio β.

Because Z _k=Z _a, the frequency response at each output is forced to be the same, so the output resistances must also be equal, and r _out(k)=r _out(a).

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Basic Building Blocks

Morgan Jones , in Valve Amplifiers (Fourth Edition), 2012

The Effect on AC Conditions of an Unbypassed Cathode Bias Resistor

Although the cathode bias resistor stabilised and set the DC conditions of the stage, it did so by means of negative feedback, so we should expect it to affect the AC conditions such as gain and output resistance. We can use the universal feedback equation to determine the effect it will have.

$A_{\text{fbk}}=\frac{A_0}{1+\beta ·A_0}$

The feedback fraction β in this case is the ratio R _k/R _L, so:

$A_{\text{fbk}}=\frac{72}{1+(1.56/175)72}=44$

The gain has been considerably reduced. The feedback is series-derived, series-applied, so it raises the input and output resistances. Since the input resistance of a valve is virtually infinite anyway, this won't be affected, but the anode resistance r _a will be raised.

Although the feedback equation is very handy for quickly determining the new gain, it is not quite so easily used for finding the new r _a.

Looking down through the anode, the only path to ground is the cathode, via the anode resistance r _a. Since, in this direction, resistances are multiplied by (μ+1), we see an effective anode resistance of:

${r΄}_{\mathrm{a}}=r_{\mathrm{a}}+(\mu +1)·R_{\mathrm{k}}=65+(100+1)1.56=223\mathrm{k}\mathrm{\Omega }$

The value of r _a rises from 65   k to 223   k. In parallel with R _L, this gives a new output resistance of 98   k, as opposed to 47   k. Incidentally, there is no reason why we should not calculate the new r _a first and use that new value in the standard gain formula to determine the new gain:

$A_v=m\left(\frac{{r΄}_{\mathrm{a}}}{R_{\mathrm{L}}+{r΄}_{\mathrm{a}}}\right)=100\left(\frac{175}{175+223}\right)=44$

It is most important to appreciate that the feedback affected only the valve's internal r _a. The anode load resistor R _L was external to the feedback, and therefore not affected.

Having evaluated the new values of gain and output resistance, we may decide that they are no longer satisfactory. We could either choose a new value of R _L, and try a new operating point, or we might even choose a new valve. However, there is another avenue open to us.

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Linear Systems in the Complex Frequency Domain

John Semmlow , in Circuits, Signals and Systems for Bioengineers (Third Edition), 2018

7.3.4 Integrator Element

As shown in Equation 7.7, integration in the Laplace domain is accomplished by dividing by s. So in the absence of initial conditions, the system representation of integration in the Laplace domain has a transfer function of

(7.24) $TF\left(s\right)=1/s$

This parallels the representation in the frequency domain where 1/s becomes 1/jω. The representation of this element is shown in Figure 7.4B.

Example 7.3

Find the Laplace and frequency domain transfer function of the system in Figure 7.5. The gain term k is a constant.

Figure 7.5. Laplace representation of a system used in Example 7.3.

Solution: The approach to finding the transfer function of a system in the Laplace domain is exactly the same as in the frequency domain used in Chapter 6. Here we are asked to determine both the Laplace and frequency domain transfer function. First we find the Laplace transfer function, then substitute jω for s, and rearrange the format for the frequency domain transfer function.

We could solve this several different ways, but this is clearly a feedback system so the easiest solution is to use the feedback equation, Equation 6.7. All we need to do is find the equivalent feedforward gain function, G(s), and the equivalent feedback function, H(s). The feedback function is H(s)   =   1 since all of the output feeds back to the input in this system. ³ The feedforward gain is the product of the two elements:

$G\left(s\right)=k\frac{1}{s}$

Substituting G(s) and H(s) into the feedback equation:

$TF\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)H\left(s\right)}=\frac{\frac{k}{s}}{1+\frac{k}{s}}=\frac{k}{s+k}$

Result: Again, the Laplace domain transfer function has the highest coefficient of complex frequency, in this case s, normalized to 1. Substituting s  = jω and rearranging the normalization:

$TF\left(\omega \right)=\frac{k}{j\omega +k}=\frac{1}{1+j\frac{\omega }{k}}$

This has the same form as a first-order transfer function given in Equation 6.31 and shown in Figure 6.12B, where the constant term ω ₁ is k.

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Feedback and Stability Theory

Ron Mancini , in Op Amps for Everyone (Third Edition), 2009

6.3 Feedback Equation and Stability

Figure 6.7 shows the canonical form of a feedback loop with control system and electronic system terms. The terms make no difference except that they have meaning to the system engineers, but the math does have meaning, and it is identical for both types of terms. The electronic terms and negative feedback sign are used in this analysis, because subsequent chapters deal with electronic applications. The output equation is written in Equation (6.1):

Figure 6.7. Comparison of control and electronic canonical feedback systems.

(6.1) $V_{\text{OUT}}=EA$

The error equation is written in Equation (6.2):

(6.2) $E=V_{\text{IN}}-\beta V_{\text{OUT}}$

Combining Equations (6.1) and (6.2) yields Equation (6.3):

(6.3) $\frac{V_{\text{OUT}}}{A}=V_{\text{IN}}-\beta V_{\text{OUT}}$

Collecting terms yields Equation (6.4):

(6.4) $\begin{array}{cc}{V}_{\text{OUT}}\left(\frac{1}{A}+\mathrm{\beta }\right)=V_{\text{IN}}& \end{array}$

Rearranging terms yields the classic form of the feedback, Equation (6.5):

(6.5) $\begin{array}{cc}\frac{V_{\text{OUT}}}{V_{\text{IN}}}=\frac{A}{1+A\mathrm{\beta }}& \end{array}$

When the quantity Aβ in Equation (6.5) becomes very large with respect to 1, the 1 can be neglected and Equation (6.5) reduces to Equation (6.6) , which is the ideal feedback equation. Under the conditions that Aβ ≫ 1, the system gain is determined by the feedback factor, β. Stable passive circuit components are used to implement the feedback factor; thus in the ideal situation, the closed loop gain is predictable and stable because β is predictable and stable.

(6.6) $\begin{array}{cc}\frac{V_{\text{OUT}}}{V_{\text{IN}}}=\frac{1}{\mathrm{\beta }}& \end{array}$

The quantity Aβ is so important that it has been given a special name, loop gain. In Figure 6.7, when the voltage inputs are grounded (current inputs are opened) and the loop is broken, the calculated gain is the loop gain, Aβ. Now, keep in mind that we are using complex numbers, which have magnitude and direction. When the loop gain approaches −1, or to express it mathematically, 1∠−180 °, Equation (6.5) approaches 1/0 ⇒ ∝. The circuit output heads for infinity as fast as it can using the equation of a straight line. If the output were not energy limited, the circuit would explode the world, but happily it is energy limited, so somewhere it comes up against the limit.

Active devices in electronic circuits exhibit nonlinear phenomena when their output approaches a power supply rail, and the nonlinearity reduces the gain to the point where the loop gain no longer equals 1∠−180 °. Now, the circuit can do two things: First, it can become stable at the power supply limit; second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail.

The first state, where the circuit becomes stable at a power supply limit, is named lockup: The circuit will remain in a locked up state until power is removed and reapplied. The second state, where the circuit bounces between power supply limits, is named oscillatory. Remember, the loop gain, Aβ, is the sole factor determining stability of the circuit or system. Inputs are grounded or disconnected, so they have no bearing on stability.

Equations (6.1) and (6.2) are combined and rearranged to yield Equation (6.7), which is the system or circuit error equation:

(6.7) $\begin{array}{cc}E=\frac{V_{\text{IN}}}{1+A\mathrm{\beta }}& \end{array}$

First, note that the error is proportional to the input signal. This is the expected result, because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. As the loop gain increases, the error decreases, thus large loop gains are attractive for minimizing errors.

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Basic Analog Electronics

John Semmlow , in Circuits, Signals and Systems for Bioengineers (Third Edition), 2018

15.3 The Operational Amplifier

The "operational amplifier," or "op amp," is a basic building block for a wide variety of analog circuits. One of its first uses was to perform mathematical operations, such as addition and integration, in analog computers, hence the name operational amplifier. Although the functions provided by analog computers are now performed by digital computers, the op amp remains a valuable, perhaps the most valuable, tool in analog circuit design.

In its idealized form, the op amp has the same properties as the ideal amplifier described previously except for one curious departure: it has infinite gain. Thus an ideal op amp has infinite input impedance (ideal load), zero output impedance (an ideal source), and a gain, A _v   →   ∞. (The symbols A _v and A _VOL are commonly used to represent the gain of an op amp.) Obviously an amplifier with a gain of infinity is of limited value, so an op amp is rarely used alone, but usually in conjunction with other elements that reduce its gain to a finite level.

Negative feedback can be used to limit the gain. Assume that the ideal amplifier is represented as A _V in the feedback system shown in Figure 15.5.

Figure 15.5. A basic feedback control system used to illustrate the use of feedback to set a finite gain in a system that has infinite feedforward gain, A _V .

The gain of the system can be found from the basic feedback equation introduced and derived in Example 6.1. When we insert A _V and β into the feedback equation, Equation 6.7, the overall system gain, G, becomes:

(15.4) $G=\frac{A_V}{1+A_V\beta }$

Now if we let the amplifier's gain, A _V , go to infinity:

(15.5) $G=\underset{A_V\to \infty }{\lim }|\frac{A_V}{1+A_V\beta }=\underset{A_V\to \infty }{\lim }|\frac{A_V}{A_V\beta }=\frac{1}{\beta }$

The overall gain expressed in decibel becomes:

(15.6) $G_{db}=20\log G=20\log \left(\frac{1}{\beta }\right)=-20\log \beta$

If β  <   1, then the gain of the feedback system G  = V _out /V _in is >1 and the system increases the signal amplitude. If β  =   1, then G  =   1 and the amplitude of the output signal is the same as the input signal. This may seem useless, but there are times when such a system is used because we still get the benefits of the ideal input and output properties of the amplifier. We rarely make β  >   1 because then G  <   1, and the feedback system actually reduces the signal amplitude. If a reduction in gain is desired, it is easier to use a passive voltage divider, a series resistor pair with one resistor to ground. So in real op amp circuits, the feedback gain, β, is ≤1 and G  ≥   1. This is fortunate, as all we need to produce a feedback gain <1 is a voltage divider network: one end of the two connected resistors goes to the output and the other end to ground, and the reduced feedback signal is taken from the intersection of the two resistors, Figure 15.6. A feedback gain of β  =   1 is even easier: just feed the entire output back to the input.

Figure 15.6. A voltage divider network that can be used to feed back a portion of the output signal as a negative feedback signal. To make the feedback signal, V _fbk , negative, it is fed to the inverting or negative input of an operational amplifier.

The approach of beginning with an amplifier that has infinite gain then reducing that gain to a finite level with the addition of feedback seems needlessly convoluted. Why not design the amplifier to have a finite fixed gain to begin with? The answer is summarized in two words: flexibility and stability. If feedback is used to set the gain of an op amp circuit, then only one basic amplifier needs to be produced: one with an infinite (or just very high) gain. Any desired gain can be achieved by modifying a simple two-resistor network. More importantly, the feedback network is almost always implemented using passive components: resistors and sometimes capacitors. Passive components are always more stable than transistor-based devices, that is, they are more immune to fluctuations owing to changes in temperature, humidity, age, and other environmental factors. Passive elements can also be more easily manufactured to tighter tolerances than active elements. For example, it is easy to buy resistors that have a 1% error in their values, whereas most common transistors vary in gain by a factor of two or more. Finally, back in the flexibility category, a wide variety of different feedback configurations can be used enabling one type of op amp to perform many different signal processing operations. Some of these different functions are explored in the section on op amp circuits at the end of this chapter.

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Development of the Nonideal Op Amp Equations

Bruce Carter , Ron Mancini , in Op Amps for Everyone (Fifth Edition), 2018

7.2 Review of the Canonical Equations

A block diagram for a generalized feedback system is repeated in Fig. 7.1. This simple block diagram is sufficient to determine the stability of any system.

Figure 7.1. Feedback system block diagram.

The output and error equation development is repeated below:

(7.1) ${\text{V}}_{\text{OUT}}=\text{EA}$

(7.2) $\text{E}={\text{V}}_{\text{IN}}={\text{V}}_{\text{OUT}}$

Combining Eqs. (7.1) and (7.2) yields Eq. (7.3):

(7.3) $\frac{{\text{V}}_{\text{OUT}}}{\text{A}}={\text{V}}_{\text{IN}}-\mathrm{\beta }{\text{V}}_{\text{OUT}}$

Collecting terms yields Eq. (7.4):

(7.4) ${\text{V}}_{\text{OUT}}\left(\frac{1}{\text{A}}+\mathrm{\beta }\right)={\text{V}}_{\text{IN}}$

Rearranging terms yields the classic form of the feedback equation.

(7.5) $\frac{{\text{V}}_{\text{OUT}}}{{\text{V}}_{\text{IN}}}=\frac{\text{A}}{1+\text{A}\mathrm{\beta }}$

Notice that Eq. (7.5) reduces to Eq. (7.6) when the quantity Aβ in Eq. (7.5) becomes very large with respect to one. Eq. (7.6) is called the ideal feedback equation because it depends on the assumption that Aβ   ≫   1, and it finds extensive use when amplifiers are assumed to have ideal qualities. Under the conditions that Aβ   ≫   1, the system gain is determined by the feedback factor β. Stable passive circuit components are used to implement the feedback factor, thus the ideal closed-loop gain is predictable and stable because β is predictable and stable.

(7.6) $\frac{{\text{V}}_{\text{OUT}}}{{\text{V}}_{\text{IN}}}=\frac{1}{\mathrm{\beta }}$

The quantity Aβ is so important that it has been given a special name, loop gain. Consider Fig. 7.2; when the voltage inputs are grounded (current inputs are opened) and the loop is broken, the calculated gain is the loop gain, Aβ. Now, keep in mind that this is a mathematics of complex numbers, which have magnitude and direction. When the loop gain approaches −1, or to express it mathematically 1 ∠–180   degrees, Eq. (7.5) approaches infinity because 1/0   ⇒   ∞. The circuit output heads for infinity as fast as it can use the equation of a straight line. If the output were not energy limited the circuit would explode the world, but it is energy limited by the power supplies so the world stays intact.

Figure 7.2. Feedback loop broken to calculate loop gain.

Active devices in electronic circuits exhibit nonlinear behavior when their output approaches a power supply rail, and the nonlinearity reduces the amplifier gain until the loop gain no longer equals 1 ∠–180   degrees. Now the circuit can do two things: first, it could become stable at the power supply limit or second, it can reverse direction (because stored charge keeps the output voltage changing) and head for the negative power supply rail.

The first state where the circuit becomes stable at a power supply limit is named lockup; the circuit will remain in the locked up state until power is removed. The second state where the circuit bounces between power supply limits is named oscillatory. Remember, the loop gain, Aβ, is the sole factor that determines stability for a circuit or system. Inputs are grounded or disconnected when the loop gain is calculated, so they have no effect on stability. The loop gain criteria are analyzed in depth later.

Eqs. (7.1) and (7.2) are combined and rearranged to yield Eq. (7.7), which gives an indication of system or circuit error.

(7.7) $\text{E}=\frac{{\text{V}}_{\text{IN}}}{1+\text{A}\mathrm{\beta }}$

First, notice that the error is proportional to the input signal. This is the expected result because a bigger input signal results in a bigger output signal, and bigger output signals require more drive voltage. Second, the loop gain is inversely proportional to the error. As the loop gain increases the error decreases, thus large loop gains are attractive for minimizing errors. Large loop gains also decrease stability, thus there is always a trade-off between error and stability.

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Counters and registers

B. HOLDSWORTH BSc (Eng), MSc, FIEE , R.C. WOODS MA, DPhil , in Digital Logic Design (Fourth Edition), 2002

7.25 Shift registers with XOR feedback

The 4-stage shift register shown in Figure 7.38(a) has XOR feedback from stages C and D such that the input to the first stage J_A = C ⊕ D. To determine the sequence of states for the register, it is assumed initially that the shift register is in the state D = 0, C = 0, B = 0 and A = 1, in which case J_A = 0 ⊕ 0, and on receipt of the next clock pulse the register enters the state D = 0, C = 0, B = 1 and A = 0. The complete sequence of states for the register is tabulated in Figure 7.38(b), the value of the feedback function for each state appearing in the right-hand column of the tabulation.

Figure 7.38. (a) Four-stage MLS shift register generator (b) MLS for four-stage shift register

In all, there are 15 states, and this is the maximum number a 4-stage register having XOR feedback can have. This sequence is termed the maximum length sequence (MLS). S₀ = 0000 is not included in the sequence since this is a lock-in state. If the register enters this state J_A = 0 ⊕ 0 = 0; it is unable to leave it when the next and subsequent clock pulses arrive. In general, the maximum length sequence for such a circuit is given by l = 2 ^N − 1 where N is the number of stages in the shift register.

Not all XOR connections result in a maximum length sequence. The table in Figure 7.39 gives the feedback functions which will give the maximum length sequence for values of N up to and including 18.

Figure 7.39. Feedback functions for maximum-length sequences

Other maximum length sequences are available with the same register length. For example, if the inverse of the XOR function C ⊕ D is used as feedback, then an alternative maximum length sequence is obtained and is tabulated in Figure 7.40 . Furthermore, an examination of the feedback equations in Figure 7.39 shows that one of the digits in the equation is always the Nth digit in the register, and the other digit (or digits) is obtained by looking back down the register. For example, for N = 4 the Nth digit is D, and the other digit in the equation, C, is the (N − 1)th digit. Two alternative maximum length sequences for a 4-stage register can be obtained by looking forward to the (N + 1)th digit which, in this case, is A. Hence the other two maximum length sequences are obtained by using the feedback A ⊕ D and A ⊙ D, and these sequences are shown tabulated in Figure 7.40.

Figure 7.40. (a) The MLS for a four-stage shift register with feedback C ⊙ D (b) A ⊕ D and (c) A ⊙ D

Clearly, the circuit shown in Figure 7.38(a) can be used as a binary sequence generator, the output sequence being taken directly from the output of one of the flip-flops in the register. In this case, the binary output sequence appearing at the output of FFD is 000100110101111. This kind of generator is sometimes referred to as a pseudo-random binary sequence generator because the digits in the sequence are in apparently random order. However, the randomness repeats itself every 2 ^N − 1 clock pulses. For a given clock frequency, the periodicity of the randomness increases very rapidly with the number of stages in the register.

$\text{If}N=10,(2^N-1)=1023$

and if the clock frequency is 1 MHz the sequence repeats itself every 1.023 ms.

$\text{If}N=20,2^N-1=1048575$

and the period of the sequence is 1.05 s.

$\text{If}N=30,(2^N-1=1073741823$

and the period of the sequence is 1073.74 s.

The design of pseudo-random sequence generators is based on the theory of finite fields developed by the French mathematician Evariste Galois. The algebra associated with finite field theory is frequently referred to as Galois field algebra. This type of binary sequence generator has a number of applications. Typical of these is the generation of repetitive noise for test circuits and also in the process of encrypting serial transmissions to ensure message security.

Non-maximum length sequences can be generated with a 4-stage register if an alternative XOR feedback is used. For example, if the feedback function is B ⊕ D, one of the sequences tabulated in Figure 7.41 will be generated. The form which the sequence takes will depend on the initial state of the register.

Figure 7.41. Non-maximum length sequences generated by a four-stage shift register with feedback B ⊕ D

The basic MLS generator shown in Figure 7.38 is not necessarily self-starting, since on switching on the initial state of the generator may be 0000. As the circuit stands, there is no way in which it can leave this state. With a slight modification to the feedback circuit it is possible to make the generator self-starting. The required modification is the logical addition of the term $\overline{A}\overline{B}\overline{C}\overline{D}$ to the feedback equation so that it becomes:

$f=C\oplus D+\overline{A}\overline{B}\overline{C}\overline{D}$

This function is plotted on the K-map shown in Figure 7.42(a) and, after simplification, it reduces to:

Figure 7.42. (a) K-map plot for a self-starting MLS generator (b) implementation of self-starting generator

$f=C\oplus D+\overline{A}\overline{B}\overline{D}$

The implementation of the self-starting generator is shown in Figure 7.42(b).

It is also possible to generate non-maximum length sequences by using a jump technique. The method of approach is to start with an MLS generator using XOR feedback and then reduce the length of the sequence by introducing additional feedback. The method will be described for the 4-stage shift register generator shown in Figure 7.43.

Figure 7.43. (a) State diagram of the four-stage MLS generator with modified feedback showing the jump (b) Modified MLS sequence (c) K-map plot of the feedback function f (d) Implementation of an MLS generator employing the 'jump' technique

It will be assumed that initially the generator is in the state DCBA = 0011 (S₃). If, when in this state, the feedback is a 0, then the next state of the generator will be DCBA = 0110 (S₆). Examination of the state table for the 4-stage MLS generator in Figure 7.38 shows that C ⊕ D = 0 when the generator is in state S₃, and the next state is S₆. If the feedback is modified to a 1 then the next state of the generator is S₇.

The state diagram for the MLS generator having four stages is shown in Figure 7.43(a), and it can be seen that by modifying the feedback, the states S₆, S₁₃, S₁₀, S₅ and S₁₁ will be omitted from the sequence, thus reducing its length from 15 to 10 states.

The modified sequence for the generator is shown in the state table in Figure 7.43(b) and the new value of the feedback function in state S₃ is shown encircled. The feedback function in conjunction with the unused states and the 'lock-in' state S₀ are plotted on a K-map and then simplified (see Figure 7.43(c)). This gives a modified feedback function of

$f=C\oplus D+AB\overline{D}+\overline{A}\overline{B}\overline{D}$

and the implementation of this self-starting non-maximum length sequence generator is shown in Figure 7.43(d).

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