Closed loop gain of an amplifier

In the following design pattern, Vin, input voltage is applied to the non-inverting input terminal that means we get “Positive” value of amplifier’s output gain as compared to the "Inverting Amplifier" where we got the “Negative” value of output gain as per our pervious discussion.

As an outcome of this, the input signal is “in-phase” with the output signal. By applying a minor portion of output voltage signal through voltage divider network Rf - R2 to the inverting input i.e. by creating negative feedback, a feedback control system is accomplished.As illustrated the in figure, this closed-loop configuration generates a non-inverting amplifier circuit with very high input impedance,  Rin approaching infinity, low output impedance, Rout ,very good stability as under ideally no current is flowing in the positive input terminal.

The same potential ( V1 ) is applied at the intersection of feedback and input signal. That means at summing point junction is a "virtual earth".  Resisters Rf and R2  are used to develop a simple potential divider network in the non-inverting amplifier. As illustrated below the ratio of resisters Rf  and R2, decides the output voltage gain of the circuit. Let’s calculate the Closed-loop voltage gain ( A V ) by applying the method to find out the output voltage of a potential divider network of Non-inverting Amplifier.

Fig. Equivalent Potential Divider Network
 For a Non-inverting Amplifier, closed loop voltage gain is given as:

For a Non-inverting Amplifier, closed loop voltage gain is a positive value which is always greater than unity (1) and is dependent on the ratio of Rf and R2. The gain of the amplifier will be right equal to unity (1) only if the Rf=0 i.e. value of the feedback resistor is zero. The gain of the amplifier will approaching infinity if the R2=0 . i.e. value of the feedback resistor is zero.   If resistor R2 is zero the gain will approach infinity, but usually it is restricted to the ( Ao ), open-loop differential gain of an Op-Amp. If we change input connections as illustrated below then we can translate the system configuration from operational amplifier into a non-inverting amplifier. 

Nonlinearities in a system

Homogeneity and superposition are the two properties of linear system. With superposition property we mean, output response of a system to the sum of inputs is the sum of the responses to the individual inputs. Therefore, if an input r1(t) gives an output for C1 (t) and output for C2(r) is provided by r2(t) then with r1(t)- r2(t) as input, we will get C1(t) + C2(t) as output. The superposition property can be expressed as the system response with respect to product of input and scalar. Saying it in specific terms, for a linear system the homogeneity property can be concluded as if for an input of r1(i) that yields an output of c(t), an input of Ar1(t) yields an output of Ac1(t); that is, multiplication of an input by a scalar yields a response that is multiplied by the same scalar.

Linearity can be visualized with the help of Fig. 1. Figure 1(a) represents a linear system in which input is twice of the output or you can say that f(x) = 0.5x; without considering value of X.  For instance, output will be ½ and 1 for input of 1 and 2 respectively. By using the superposition property, sum of the original inputs give inputs or 3 and it will give output as sum of original outputs means 1.5. For testing the property of homogeneity, we will take input as 2 and it will provide 1 as output value. Similarly multiplying the input by 2, also doubles the output.

Figure 1(a) the input of 4 is not producing output of 2. In this way, the readers can check that linearity property is not verified here and is not applicable to the relationship depicted in the Fig. 1(b). Some useful examples of physical nonlinearities are shown in the Fig. 2.  An Electronic Amplifier satisfies the property of linearity over specific range but at high value of input it exhibits non linearity known as saturation at high value of input voltages. A motor offers no response for low input value for voltage because of frictional forces which exhibits dead zone that is non linearity. Nonlinearity known as backlash is exhibited by gears which do not fit tightly. The input shifts only over a small range of values.

FIGURE 1 a. Linear system; b. nonlinear system

FIGURE 2 some physical nonlinearity
Without responding of output, the readers are required to verify that the curves represented in the Figure 2. does not justify the linearity definitions over complete range. Phase Detector is one more example of nonlinear subsystem. It is used in the FM radio receiver as part of phase locked loop. The output response for input is sine of the input. Linear approximations can be made by the designer for a nonlinear system. Design and analysis are simplified with linear approximations. It can be used till good approximation to reality is yielded.  For instance, if the range of input values about that point is small and translation of origin can be done to the point. Linear Amplification having little excursions about any point can be demonstrated by a physical device like electronic amplifiers.

Time response of a dynamic system

The information about how the system responds to certain inputs is provided by the time response of a dynamic system. To determine the stability of the system and the performance of the controller, we can analyze the time response.

Numerically integration of the system model in time is involved by obtaining the time response of a system. The time response of the control system can be found by finding its equation of motion and for that we need to model the overall system dynamics.

The system could be composed of the mechanical, electrical, or other sub-systems. In addition, each sub-system is expected to have sensors and actuators to sense the environment and to interact with it.
In order to find the transfer function of all the sub-components and use the block diagram, we can use the Laplace transforms. We can use the signal flow diagrams in order to find the interactions among the system components.

According to the objectives, the manipulation of the system final response by adding the feedback or poles and zeros to the system block diagram, can be done. By using the inverse Laplace transforms, the overall transfer function of the system can be found and we can obtain the time response of the system to a test input normally a step input.

The Lab VIEW Control Design and the Simulation Module provides Vis to help us to find these time-domain solutions. To analyze the response of a system to step and impulse inputs, we can use this Time Response Vis. Initial conditions can be applied to both of these responses.

To simulate the response of the system to an arbitrary input, we can use the Time Response Vis.

Rise time (tr) — It is the time required for the dynamic system response to rise from the lower threshold to an upper threshold. The default values are 10% for the lower threshold and90% for the upper threshold.

Maximum overshoot (Mp)—the dynamic system response value that most exceeds the unity, expressed as a percent.

Peak time (tp) — it is the time required to reach the peak value of the first overshoot by the dynamic system response.

Settling time (ts) — it is the time required for the dynamic system response to reach and stay within a threshold of the final value. The default threshold is 1%.

Steady state gain— it is the final value around which the dynamic system response settles to a step input.

Peak value (yp) — it is the value at which the maximum absolute value of the time response occurs.

Routh Table - Entire row ZERO!

Entire row is zero
  • If sometimes making a Routh table we find an entire row having zero as there is an even polynomial that is factor of original polynomial.
  • It is a different case from the case of zero in first column only. We will see an example how to construct and interpret the Routh table when an entire row is zero.
  • Further if the entire row is zero is appeared in Routh table when a purely even or purely odd polynomial is a factor of the original polynomial. Example s4 + 5s2 + 7 are an even polynomial as it has only powers of s.
  • Even polynomial only have roots that are in symmetry to the origin. Three conditions can occur .a) roots are symmetrical and real, b)roots are quadrantal, c)roots are symmetric and imaginary.
  • Figure 6.5 shows these cases. Each of the cases will generate even polynomial. It is an even polynomial that causes the row of zeros.
  • The row of zeros shows the existence of an even polynomial which has roots that are symmetric about the origin. Some of these roots can be on the j?-axis.
  • Now however the j? roots are symmetric about the origin, if there are no rows of zeros, we cannot have j? roots.
  • Another feature of Routh table is the row previous to the row that of the zero row has an even polynomial which is the factor of the original polynomial.

Transient Response, Steady-State Response and Stability

1. Transient Response

Transient response has a very important role in the system. For example if we apply it to the elevator we can see that with a slow transient response passengers will become impatient whereas when it is very rapid then it can be present very unpleasant and scary situation for them.  A confusing situation may occur in case of elevator's oscillations just before the arrival of the floor.  Some major structural changes can also occur with the transient response: a very fast transient response can result in permanent physical damage. Transient response also contributes for the time that is required for a computer to write to or to read from the computer's disk storage. As we know that it is not possible to read and write until head stops , hence head's read/write movement affects the computer's overall speed.  With the help of this book, we will try to establish the quantitative definitions of the transient response. So that we can analyze the existing transient response of a system. Our primary Analysis and design objective is to get a preferred transient response and for that we can make the adjustments in the design components or parameters.

2. Steady-State Response

The Steady state response is the another goal of the analysis and design. This response seems like input and is something that remains when the transients are decayed to zero. For instance, this is a response seen when the elevator has stopped very close to the fourth floor or when the disk drive's head has stopped moving after the right track. The accuracy of the steady state response is a major concern for us.

For a comfortable exit of passengers it is important that an elevator should stop at a proper level with the floor. When the write or read tack is not placed over the command track then there are chances of getting computer errors. An antenna tracking of a satellite is done in order to maintain the satellite in its beam width so that it don't lose the track. In the text we will define the steady state errors on quantative basis, try to analyze the steady state error of the system and then corrective action are designed to decrease steady state error. This is our second analysis and design objective.
And objective of design.

3. Stability

Without the stability, the discussion made in steady state error and on the transient response is doubtful. To give details about the stability we shall begin with the information that the sum of the forced response and the natural response makes up the system's total the time of studying the linear differential equation, many of you may have referred to such responses as the homogeneous and particular solutions, respectively. The method by which a system acquires and dissipates energy was described by the natural response. The system and not the input is the factor that determines the response's nature or form. Whereas the input is responsible for the nature or form of the forced response. So to describe a linear system we can use:
Total response = Natural response + Forced response (1.1)2

In order to make a control system useful, it is necessary for the natural response to reach zero, and so only the forced response should be left. However there are some cases where the natural response without any bound grows and rather than diminish to zero oscillate. Hence the natural response is so much higher to the forced response that it cannot be controlled and the situation is known as instability. Ultimately, this type of condition can end up with self destruction of the physical device until the device has no limit stops as its part. For instance, there are chances of an elevator to crash down through the ceiling or the floor, an aircraft can take uncontrollable roll, there are chances that the antennae that is supposed to point to the target can rotate, it may line up with the target and then can start oscillating for about the target with the increasing oscillations and velocity they can reach to an output limit till antenna are structurally damaged. One can also see this transient response in an unstable system which increases without any proof of steady state response.

It is very important to design a control system to achieve stability. This can be achieved only when the natural response will decay up to the zero level when the time reaches the infinity or oscillate. In most of the systems we will find that the transient response and the natural response are directly proportional to each other.  So, when the natural response degrades with the time approaching to infinity, transient response is going to die out, and only the forced response is left behind. When the system is stable then proper designing of  steady state characteristics and transient response can be done.  The third objective of analysis and design is stability.

Performance measurement of a control system

The time domain characteristics are used to measure the control system' performance in a more realistic way. And the explanation for this is that most control system's performance is assessed on the basis of the time responses because of certain test signals. This is in distinction to the Communication system's analysis and design in which the frequency response is more important, as the maximum of the signals need to be processed are either composed of sinusoidal components or sinusoidal. In case of design problems, no unified methods of arriving at a designed system are there that fulfills the time-domain performance specifications, like rise time, maximum overshoot, settling time, delay time, and many more. Other than this, in the frequency domain, the graphical method's wealth is available that are not restricted to the low-order systems. 

It is necessary to understand that the time-domain performances and the frequency-domain are interconnected with each other in a linear system, hence the predictions on the system's time-domain properties can be done depending on the on the frequency-domain characteristics. The domain of frequency is very convenient for measuring the sensitivity of a system to parameter variations and noise. Having such concepts, we think that convenience and the availability of the existing analytical tools are main motivations for conducting control systems analysis and design in the frequency domain. Another reason is that it has given a second point of view to the problems of control-system, which usually gives important information in the control system's complex analysis and design. So, to conduct the linear control system's frequency-domain analysis does not mean that the system is only sinusoidal input's subject.

With the help of the frequency-response studies,we can project the system's time-domain performance. the system's schematic and demonstrated this step for a position control system. In order to get a schematic, the engineers of control system have to make many simplifying assumptions so as to keep the ensuing model manageable and also can approximate the physical reality. The other step is to make the mathematical models from the schematics of physical systems. Here we will be discussing two methods: (1) transfer functions in the frequency domain and (2) state equations in the time domain.

As we move further, we will see that the first step in every case for developing a mathematical model is the use of the fundamental physical  laws of science and engineering. For instance, when electrical networks are modeled, Kirchhoff's laws and ohm's law, which are the important basic electronic network's laws should be applied first. We will add voltages in a loop or will sum the currents at a node. At the time of studying the mechanical systems, we have to use Newton's laws as the fundamental guiding principles. Here we will add up all the torques and forces. With all these equations we will get the relationship between the input and output of the system.

The structure of the differential equation with its coefficients are the description of the system. Even though with the use of the differential equation the relation between the system's input and output can be made but, it is an unsatisfying representation with a system's perception. We can see that the parameters of system, like the coefficient, the inputs,r(t) and the output,c(t), are seen throughout the equation. It is better to go for the mathematical representation like in Figure 2.1(a), where the system, output and the input are the separate and distinct parts. Also we can represent easily the connection between many subsystems.

Like, we can show cascaded interconnections, as shown in Figure 2.1(b), where there will be a mathematical function, known as the transfer function, is within each block and the block functions can be combined to get Figure 2.1 (a) for analysis and design. This convenience cannot be obtained with the differential equation.

Steady-State Error for Unity Feedback Systems

Consider the following unity feedback system                 

Recall the closed-loop transfer function of the system, T(s), is

The system output can then be expressed as

The error, E(s), between the system input, R(s), and the system output, C(s), is     

Using the final value theorem, the steady-state error is