When we study the analysis of the transient state and steady state response of control system it is very essential to know a few basic terms and these are described below. Standard Input Signals : These are also known as test input signals. The input signal is very complex in nature, it is complex because it may be a combination of various other signals. Thus it is very difficult to analyze characteristic performance of any system by applying these signals. So we use test signals or standard input signals which are very easy to deal with. We can easily analyze the characteristic performance of any system more easily as compared to non standard input signals. Now there are various types of standard input signals and they are written below: Unit Impulse Signal : In the time domain it is represented by ∂(t). The Laplace transformation of unit impulse function is 1 and the corresponding waveform associated with the unit impulse function is shown below. Unit Step Signal : In the time domain it is represented by u (t). The Laplace transformation of unit step function is 1/s and the corresponding waveform associated with the unit step function is shown below. Unit Ramp signal : In the time domain it is represented by r (t). The Laplace transformation of unit ramp function is 1/s 2 and the corresponding waveform associated with the unit ramp function is shown below. Unit Ramp Signal Parabolic Type Signal : In the time domain it is represented by t 2 / 2. The Laplace transformation of parabolic type of the function is 1 / s 3 and the corresponding waveform associated with the parabolic type of the function is shown below. Sinusoidal Type Signal : In the time domain it is represented by sin (ωt).The Laplace transformation of sinusoidal type of the function is ω / (s 2 + ω 2 ) and the corresponding waveform associated with the sinusoidal type of the function is shown below. Cosine Type of Signal : In the time domain it is represented by cos (ωt). The Laplace transformation of the cosine type of the function is ω / (s 2 + ω 2 ) and the corresponding waveform associated with the cosine type of the function is shown below, Now are in a position to describe the two types of responses which are a function of time. Transient Response of Control System As the name suggests transient response of control system means changing so, this occurs mainly after two conditions and these two conditions are written as follows- Condition one : Just after switching ‘on’ the system that means at the time of application of an input signal to the system. Condition second : Just after any abnormal conditions. Abnormal conditions may include sudden change in the load, short circuiting etc. Steady State Response of Control System Steady state occurs after the system becomes settled and at the steady system starts working normally. Steady state response of control system is a function of input signal and it is also called as forced response. Now the transient state response of control system gives a clear description of how the system functions during transient state and steady state response of control system gives a clear description of how the system functions during steady state. Therefore the time analysis of both states is very essential. We will separately analyze both the types of responses. Let us first analyze the transient response. In order to analyze the transient response, we have some time specifications and they are written as follows: Delay Time : This time is represented by t d . The time required by the response to reach fifty percent of the final value for the first time, this time is known as delay time. Delay time is clearly shown in the time response specification curve. Rise Time : This time is represented by t r . We define rise time in two cases: In case of under damped systems where the value of ζ is less than one, in this case rise time is defined as the time required by the response to reach from zero value to hundred percent value of final value. In case of over damped systems where the value of ζ is greater than one, in this case rise time is defined as the time required by the response to reach from ten percent value to ninety percent value of final value. Peak Time : This time is represented by t p . The time required by the response to reach the peak value for the first time, this time is known as peak time. Peak time is clearly shown in the time response specification curve. Settling Time : This time is represented by t s . The time required by the response to reach and within the specified range of about (two percent to five percent) of its final value for the first time, this time is known as settling time. Settling time is clearly shown in the time response specification curve. Maximum Overshoot : It is expressed (in general) in percentage of the steady state value and it is defined as the maximum positive deviation of the response from its desired value. Here desired value is steady state value. Steady State Error : It can be defined as the difference between the actual output and the desired output as time tends to infinity. Now we are in position we to do a time response analysis of a first order system. Transient State and Steady State Response of First Order Control System Let us consider the block diagram of the first order system. From this block diagram we can find overall transfer function which is linear in nature. The transfer function of the first order system is 1/((sT+1)). We are going to analyze the steady state and transient response of control system for the following standard signal. Unit impulse. Unit step. Unit ramp. Unit impulse response : We have Laplace transform of the unit impulse is 1. Now let us give this standard input to a first order system, we have Now taking the inverse Laplace transform of the above equation, we have It is clear that the steady state response of control system depends only on the time constant ‘T’ and it is decaying in nature. Unit step response : We have Laplace transform of the unit impulse is 1/s. Now let us give this standard input to first order system, we have With the help of partial fraction, taking the inverse Laplace transform of the above equation, we have It is clear that the time response depends only on the time constant ‘T’. In this case the steady state error is zero by putting the limit t is tending to zero. Unit ramp response : We have Laplace transform of the unit impulse is 1/s 2 . Now let us give this standard input to first order system, we have With the help of partial fraction, taking the inverse Laplace transform of the above equation we have On plotting the exponential function of time we have ‘T’ by putting the limit t is tending to zero. Transient State and Steady State Response of Second Order Control System Let us consider the block diagram of the second order system. From this block diagram we can find overall transfer function which is nonlinear in nature. The transfer function of the second order system is (ω 2 ) / ( s ( s + 2ζω )). We are going to analyze the transient state response of control system for the following standard signal. Unit impulse response : We have Laplace transform of the unit impulse is 1. Now let us give this standard input to second order system, we have Where ω is natural frequency in rad/sec and ζ is damping ratio. Unit step response : We have Laplace transform of the unit impulse is 1/s. Now let us give this standard input to first order system, we have With the help of partial fraction, taking the inverse Laplace transform of the above equation we have Now we will see the effect of different values of ζ on the response. We have three types of systems on the basis of different values of ζ. Under damped system : A system is said to be under damped system when the value of ζ is less than one. In this case roots are complex in nature and the real parts are always negative. System is asymptotically stable. Rise time is lesser than the other system with the presence of finite overshoot. Critically damped system : A system is said to be critically damped system when the value of ζ is one. In this case roots are real in nature and the real parts are always repetitive in nature. System is asymptotically stable. Rise time is less in this system and there is no presence of finite overshoot. Over damped system : A system is said to be over damped system when the value of ζ is greater than one. In this case roots are real and distinct in nature and the real parts are always negative. System is asymptotically stable. Rise time is greater than the other system and there is no presence of finite overshoot. Sustained Oscillations : A system is said to be sustain damped system when the value of zeta is zero. No damping occurs in this case. Now let us derive the expressions for rise time, peak time, maximum overshoot, settling time and steady state error with a unit step input for second order system. Rise time : In order to derive the expression for the rise time we have to equate the expression for c(t) = 1. From the above we have On solving above equation we have expression for rise time equal to Peak Time : On differentiating the expression of c(t) we can obtain the expression for peak time. dc(t)/ dt = 0 we have expression for peak time, Maximum overshoot : Now it is clear from the figure that the maximum overshoot will occur at peak time tp hence on putting the valye of peak time we will get maximum overshoot as Settling Time : Settling time is given by the expression Steady state error : The steady state error is diffrerence between the actual output and the desired output hence at time tending to infinity the steady state error is zero.

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Before I introduce you about various controllers in detail, it is very essential to know the uses of controllers in the theory of control systems. The important uses of the controllers are written below: Controllers improve steady state accuracy by decreasing the steady state errors. As the steady state accuracy improves, the stability also improves. They also help in reducing the offsets produced in the system. Maximum overshoot of the system can be controlled using these controllers. They also help in reducing the noise signals produced in the system. Slow response of the over damped system can be made faster with the help of these controllers. Now what are controllers? A controller is one which compares controlled values with the desired values and has a function to correct the deviation produced. Types of Controllers Let us classify the controllers. There are mainly two types of controllers and they are written below: Continuous Controllers: The main feature of continuous controllers is that the controlled variable (also known as the manipulated variable) can have any value within the range of controller’s output. Now in the continuous controller’s theory, there are three basic modes on which the whole control action takes place and these modes are written below. We will use the combination of these modes in order to have a desired and accurate output. Proportional controllers . Integral controllers . Derivative controllers . Combinations of these three controllers are written below: Proportional and integral controllers. Proportional and derivative controllers. Now we will discuss each of these modes in detail. Proportional Controllers We cannot use types of controllers at anywhere, with each type controller, there are certain conditions that must be fulfilled. With proportional controllers there are two conditions and these are written below: Deviation should not be large, it means there should be less deviation between the input and output. Deviation should not be sudden. Now we are in a condition to discuss proportional controllers, as the name suggests in a proportional controller the output (also called the actuating signal) is directly proportional to the error signal. Now let us analyze proportional controller mathematically. As we know in proportional controller output is directly proportional to error signal, writing this mathematically we have, Removing the sign of proportionality we have, Where K p is proportional constant also known as controller gain. It is recommended that K p should be kept greater than unity. If the value of K p is greater than unity, then it will amplify the error signal and thus the amplified error signal can be detected easily. Advantages of Proportional Controller Now let us discuss some advantages of proportional controller. Proportional controller helps in reducing the steady state error, thus makes the system more stable. Slow response of the over damped system can be made faster with the help of these controllers. Disadvantages of Proportional Controller Now there are some serious disadvantages of these controllers and these are written as follows: Due to presence of these controllers we some offsets in the system. Proportional controllers also increase the maximum overshoot of the system. Integral Controllers As the name suggests in integral controllers the output (also called the actuating signal) is directly proportional to the integral of the error signal. Now let us analyze integral controller mathematically. As we know in an integral controller output is directly proportional to the integration of the error signal, writing this mathematically we have, Removing the sign of proportionality we have, Where K i is integral constant also known as controller gain. Integral controller is also known as reset controller. Advantages of Integral Controller Due to their unique ability they can return the controlled variable back to the exact set point following a disturbance that’s why these are known as reset controllers. Disadvantages of Integral Controller It tends to make the system unstable because it responds slowly towards the produced error. Derivative Controllers We never use derivative controllers alone. It should be used in combinations with other modes of controllers because of its few disadvantages which are written below: It never improves the steady state error. It produces saturation effects and also amplifies the noise signals produced in the system. Now, as the name suggests in a derivative controller the output (also called the actuating signal) is directly proportional to the derivative of the error signal. Now let us analyze derivative controller mathematically. As we know in a derivative controller output is directly proportional to the derivative of the error signal, writing this mathematically we have, Removing the sign of proportionality we have, Where K d is proportional constant also known as controller gain. Derivative controller is also known as rate controller. Advantages of Derivative Controller The major advantage of derivative controller is that it improves the transient response of the system. Proportional and Integral Controller As the name suggests it is a combination of proportional and an integral controller the output (also called the actuating signal) is equal to the summation of proportional and integral of the error signal. Now let us analyze proportional and integral controller mathematically. As we know in a proportional and integral controller output is directly proportional to the summation of proportional of error and integration of the error signal, writing this mathematically we have, Removing the sign of proportionality we have, Where K i and k p proportional constant and integral constant respectively. Advantages and disadvantages are the combinations of the advantages and disadvantages of proportional and integral controllers. Proportional and Derivative Controller As the name suggests it is a combination of proportional and a derivative controller the output (also called the actuating signal) is equals to the summation of proportional and derivative of the error signal. Now let us analyze proportional and derivative controller mathematically. As we know in a proportional and derivative controller output is directly proportional to summation of proportional of error and differentiation of the error signal, writing this mathematically we have, Removing the sign of proportionality we have, Where K d and k p proportional constant and derivative constant respectively. Advantages and disadvantages are the combinations of advantages and disadvantages of proportional and derivative controllers

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Before I introduce you about the concept of state space analysis of control system , it is very important to discuss here the differences between the conventional theory of control system and modern theory of control system. The conventional control theory is completely based on the frequency domain approach while the modern control system approach is based on time domain approach. In the conventional theory of control system we have linear and time invariant single input single output (SISO) systems only but with the help of theory of modern control system we can easily do the analysis of even non linear and time variant multiple inputs multiple outputs (MIMO) systems also. In the modern theory of control system the stability analysis and time response analysis can be done by both graphical and analytically method very easily. Now state space analysis of control system is based on the modern theory which is applicable to all types of systems like single input single output systems, multiple inputs and multiple outputs systems, linear and non linear systems, time varying and time invariant systems. Let us consider few basic terms related to state space analysis of modern theory of control systems. State in State Space Analysis : It refers to smallest set of variables whose knowledge at t=t 0 together with the knowledge of input for t ≥ t 0 gives the complete knowledge of the behavior of the system at any time t ≥ t 0 . State Variables in State Space analysis : It refers to the smallest set of variables which help us to determine the state of the dynamic system. State variables are defined by x 1 (t), x 2 (t)……..X n (t). State Vector : Suppose there is a requirement of n state variables in order to describe the complete behavior of the given system, then these n state variables are considered to be n components of a vector x(t). Such a vector is known as state vector. State Space : It refers to the n dimensional space which has x 1 axis, x 2 axis ………x n axis. State Space Equations Let us derive state space equations for the system which is linear and time invariant. Let us consider multiple inputs and multiple outputs system which has r inputs and m outputs. Where r=u 1 , u 2 , u 3 ……….. u r . And m = y 1 , y 2 ……….. y m . Now we are taking n state variables to describe the given system hence n = x 1 , x 2 , ……….. x n . Also we define input and output vectors as, Transpose of input vectors, Where T is transpose of the matrix. Transpose of output vectors, Where T is transpose of the matrix. Transpose of state vectors, Where T is transpose of the matrix. These variables are related by a set of equations which are written below and are known as state space equations Representation of State Model using Transfer Function Decomposition : It is defined as the process of obtaining the state model from the given transfer function. Now we can decompose the transfer function using three different ways: Direct decomposition, Cascade or series decomposition, Parallel decomposition. In all the above decomposition methods we first convert the given transfer function into the differential equations also called the dynamic equations. After converting into differential equations we will take inverse Laplace transform of the above equation then corresponding to the type of decomposition we can create model. We can represent any type of transfer function in state model. We have various types of model like electrical model, mechanical model etc. Expression of Transfer Matrix in terms of A, B, C and D. We define transfer matrix as the Laplace transform of output to the Laplace transform of input. On writing the state equations again and taking the Laplace transform of both the state equation (assuming initial conditions equal to zero) we have We can write the equation as Where I is an identity matrix. Now substituting the value of X(s) in the equation Y(s) and putting D=0 (means is a null matrix) we have Inverse of matrix can substitute by adj of matrix divided by the determinant of the matrix, now on rewriting the expression we have of |sI-A| is also known as characteristic equation when equated to zero. Concept of Eigen Values and Eigen Vectors The roots of characteristic equation that we have described above are known as eigen values or eigen values of matrix A. Now there are some properties related to eigen values and these properties are written below- Any square matrix A and its transpose At have the same eigen values. Sum of eigen values of any matrix A is equal to the trace of the matrix A. Product of the eigen values of any matrix A is equal to the determinant of the matrix A. If we multiply a scalar quantity to matrix A then the eigen values are also get multiplied by the same value of scalar. If we inverse the given matrix A then its eigen values are also get inverses. If all the elements of the matrix are real then the eigen values corresponding to that matrix are either real or exists in complex conjugate pair. Now there exists one eigen vector corresponding to one Eigen value, if it satisfy the following condition ( ek × I – A )Pk = 0. Where k = 1, 2, 3, ……..n. State Transition Matrix and Zero State Response We are here interested in deriving the expressions for the state transition matrix and zero state response. Again taking the state equations that we have derived above and taking their Laplace transformation we have, Now on rewriting the above equation we have Let [sI-A] -1 = θ(s) and taking the inverse Laplace of the above equation we have The expression θ(t) is known as state transition matrix. L -1 .θ(t)BU(s) = zero state response. Now let us discuss some of the properties of the state transition matrix. If we substitute t = 0 in the above equation then we will get 1.Mathematically we can write θ(0) =1. If we substitute t = -t in the θ(t) then we will get inverse of θ(t). Mathematically we can write θ(-t) = [θ(t)] -1 . We also another important property [θ(t)] n = θ(nt).

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Let us recall the basic definition of linear system. Linear systems are those where principle of superposition ( if the two inputs are applied simultaneously, then output will be the sum of two outputs) is applicable but in case of highly non linear system we cannot apply principle of superposition. Analysis of different non linear system is very difficult because of non linear behavior. Also we cannot use pole zero method in order to analyse non linear systems because pole zero method is strictly restricted to linear systems. Instead of these disadvantages, there are some advantages of having non linear systems and they are written below: Non linear systems can perform better than the linear systems. Non linear systems are less costly than the linear systems. They are usually small and compact in size as compared to linear system. In practice, all physical systems have some kind of non linearity. Sometimes it may even be desirable to introduce a non linearity deliberately in order to improve the performance of a system and making its operation safer. At it result, the system is more economical than linear system. One of the simplest examples of a system with an intentionally introduced non linearity is a relay controlled or ON/OFF system. For instance, in a typical home heating system, a furnace is turned ON when the temperature falls below a certain specified value and OFF when the temperature exceeds another given value. Here we are going to discuss two different types of analysis or method for analyzing the non linear systems. The two methods are written below and briefly discussed with the help of example. Describing function method in control system Phase plane method in control system Common Non Linearities In most control systems we can not avoid the presence of certain types of non linearities in control system . These can be classified as static or dynamic. A system for which there is a nonlinear relationship between input and output, that does not involve a differential equation is called a static non linearity. On the other hand, the input and output may be related through a non linear differential equation. Such a system is called a dynamic non linearity. Now we are going to discuss various types of non linearities in control system : Saturation non linerities Friction non linerities Dead zone non linerities Relay (ON/OFF controller) non linerities Backlash non linerities Saturation Non Linearities We have seen this type of non linearity many times. For example we have seen saturation in the magnetizing curve of DC motor. In order to understand this type of non linearity let us discuss saturation curve or magnetizing curve which is given below:  From the above curve we can see that the output showing linear behavior in the beginning but after that there is a saturation in the curve which one kind of non linearity in the system. We have also shown approximated curve. Same type of saturation non linearity also we can see in an amplifier for which the output is proportional to the input only for a limited range of values of input. When the input exceeds this range, the output tends to become non linearity. Non Linear Friction Anything which opposes the relative motion of the body is called friction. It is a kind of non linearity present in the system. The common example in an electric motor in which we find coulomb friction drag due to the rubbing contact between the brushes and the commutator.  Friction may be of three types and they are written below: Static friction: In simple words, the static friction acts on the body when the body is at rest. Dynamic friction : Dynamic friction acts on the body when there is a relative motion between the surface and the body. Limiting friction: It is defined as the maximum value of limiting friction that acts on the body when it is at rest. Dynamic friction can also be classified as (a) Sliding friction (b) Rolling friction. Sliding friction acts when two bodies slides over each other while rolling acts when the bodies rolls over another body. In mechanical system we have two types of friction namely (a) Viscous friction (b) Static friction. Dead Zone Non Linearities This type of non linearity is shown by various electrical devices like motors, dc servomotors, actuators etc. Dead zone non linearities refer to a condition in which output becomes zero when the input crosses certain limiting value.   Relays ( ON/OFF Controller ) Non Linearities Electromechanical relays are frequently used in control systems where the control strategy requires control signal with only two or three states. This is also called as ON / OFF controller or two state controller.    Relay Non Linearity (a) ON/OFF (b) ON/OFF with Hysteresis (c) ON/OFF with Dead Zone. Fig (a) shows the ideal characteristics of a bidirectional relay. In practice relay will not respond instantaneously. For input currents between the two switching instants, the relay may be in one position or other depending upon the previous history of the input. This characteristic is called ON / OFF with hysteresis that shows in Fig (b). A relay also has a definite amount of dead zone in practice that show in Fig (c). The dead zone is caused by the fact that the relay field winding requires a finite amount of current to move the armature. Backlash Non Linearities Another important non linearity commonly occurring in physical system is hysteresis in mechanical transmission such as gear trains and linkages. This non linearity is somewhat different from magnetic hysteresis and is commonly reffered to as backlash non linearities . Backlash in fact is the play between the teeth of the drive gear and those of the driven gear. Consider a gear box as shown in below figure (a) having backlash as illustrated in fig (b).  Fig (b) shows the teeth A of the driven gear located midway between the teeth B 1 , B 2 of the driven gear. Fig (c) gives the relationship between input and output motions. As the teeth A is driven clockwise from this position, no output motion takes place until the tooth A makes contact with the tooth B 1 of the driven gear after travelling a distance x/2. This output motion corresponds to the segment mn of fig (c). After the contact is made the driven gear rotates counter clockwise through the same angle as the drive gear, if the gear ratio is assumed to be unity. This is illustrated by the line segment no. As the input motion is reversed, the contact between the teeth A and B 1 is lost and the driven gear immediately becomes stationary based on the assumption that the load is friction controlled with negligible inertia. The output motion therefore causes till tooth A has travelled a distance x in the reverse direction as shown in fig (c) by the segment op. After the tooth A establishes contact with the tooth B 2 , the driven gear now mores in clockwise direction as shown by segment pq. As the input motion is reversed the direction gear is again at standstill for the segment qr and then follows the drive gear along rn. Describing Function Method of Non Linear Control System The describing function method in control system was invented by Nikolay Mitrofanovich Kryloy and Nikolay Bogoliubov in year of 1930 and later it developed by Ralph Kochenburger. Describing function method is used for finding out the stability of a non linear system. Of all the analytical methods developed over the years for non linear control systems, this method is generally agreed upon as being the most practically useful. This method is basically an approximate extension of frequency response methods including Nyquist stability criterion to non linear system. The describing function method of a non linear system is defined to be the complex ratio of amplitudes and phase angle between fundamental harmonic components of output to input sinusoid. We can also called sinusoidal describing function. Mathematically, Where, N = describing function, X = amplitude of input sinusoid, Y = amplitude of fundamental harmonic component of output, φ 1 = phase shift of the fundamental harmonic component of output. Let us discuss the basic concept of describing function of non linear control system. Let us consider the below block diagram of a non linear system, where G 1 (s) and G 2 (s) represent the linear element and N represent the non linear element. Let us assume that input x to the non linear element is sinusoidal, i.e, For this input, the output y of the non linear element will be a non sinusoidal periodic function that may be expressed in terms of Fourier series as Most of non linearities are odd symmetrical or odd half wave symmetrical; the mean value Y 0 for all such case is zero and therefore output will be, As G 1 (s) G 2 (s) has low pass characteristics , it can be assumed to a good degree of approximation that all higher harmonics of y are filtered out in the process, and the input x to the nonlinear element N is mainly contributed by fundamental component of y i.e. first harmonics . So in the describing function analysis, we assume that only the fundamental harmonic component of the output. Since the higher harmonics in the output of a non linear system are often of smaller amplitude than the amplitude of fundamental harmonic component. Most control systems are low pass filters, with the result that the higher harmonics are very much attenuated compared with the fundamental harmonic component. Hence y 1 need only be considered. We can write y 1 (t) in the form , Where by using phasor, The coefficient A 1 and B 1 of the Fourier series are given by- From definition of describing function we have, Let us find out describing function for these non linearities. Describing Function for Saturation Non Linearity We have the characteristic curve for saturation as shown in the given figure. Let us take input function as Now from the curve we can define the output as : Let us first calculate Fourier series constant A 1 . On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A1 as zero. Similarly we can calculate the value of Fourier constant B 1 for the given output and the value of B 1 can be calculated as, The phase angle for the describing function can be calculated as Thus the describing function for saturation is Describing Function for Ideal Relay We have the characteristic curve for ideal relay as shown in the given figure. Let us take input function as Now from the curve we can define the output as The output periodic function has odd symmetry : Let us first calculate Fourier series constant A 1 . On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A 1 as zero. Similarly we can calculate the value of Fourier constant B 1 for the given output and the value of B 1 can be calculated as On substituting the value of the output in the above equation y(t) = Y we have the value of the constant B 1 And the phase angle for the describing function can be calculated as Thus the describing function for an ideal relay is Describing Function for Real Relay (Relay with Dead Zone) We have the characteristic curve for real realy as shown in the given figure. If X is less than dead zone Δ, then the relay produces no output; the first harmonic component of Fourier series is of course zero and describing function is also zero. If X > &Delta, the relay produces the output. Let us take input function as Now from the curve we can define the output as The output periodic function has odd symmetry : Let us first calculate Fourier series constant A 1 . On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A 1 as zero. Similarly we can calculate the value of Fourier constant B for the given output and the value of B can be calculated as Due to the symmetry of y, the coefficient B 1 can be calculated as follows, Therefore, the describing function is Describing Function for Backlash Non Linearity We have the characteristic curve for backlash as shown in the given figure. Let us take input function as  Now from the curve we can define the output as Let us first calculate Fourier series constant A 1 . On substituting the value of the output in the above equation and integrating the function from zero to 2π we have the value of the constant A 1 as Similarly we can calculate the value of Fourier constant B for the given output and the value of B 1 can be calculated as On substituting the value of the output in the above equation and integrating the function from zero to pi we have the value of the constant B 1 as We can easily calculate the describing function of backlash from below equation

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Sometimes, the control element has only two position either it is fully closed or fully open. This control element does not operate at any intermediate position, i.e. partly open or partly closed position. The control system made for controlling such elements, is known as on off control theory . In this control system, when process variable changes and crosses certain preset level, the output valve of the system is suddenly fully opened and gives 100% output. Generally in on off control system, the output causes change in process variable. Hence due to effect of output, the process variable again starts changing but in reverse direction. During this change, when process variable crosses certain predetermined level, the output valve of the system is immediately closed and output is suddenly reduced to 0%. As there is no output, the process variable again starts changing in its normal direction. When it crosses the preset level, the output valve of the system is again fully open to give 100% output. This cycle of closing and opening of output valve continues till the said on-off control system is in operation. A very common example of on-off control theory is fan controlling scheme of transformer cooling system. When transformer runs with such a load, the temperature of the electrical power transformer rises beyond the preset value at which the cooling fans start rotating with their full capacity. As the cooling fans run, the forced air (output of the cooling system) decreases the temperature of the transformer. When the temperature (process variable) comes down below a preset value, the control switch of fans trip and fans stop supplying forced air to the transformer. After that, as there is no cooling effect of fans, the temperature of the transformer again starts rising due to load. Again when during rising, the temperature crosses the preset value, the fans again start rotating to cool down the transformer. Theoretically, we assume that there is no lag in the control equipment. That means, there is no time day for on and off operation of control equipment. With this assumption if we draw series of operations of an ideal on off control system, we will get the graph given below. But in practical on off control, there is always a non zero time delay for closing and opening action of controller elements. This time delay is known as dead time. Because of this time delay the actual response curve differs from the above shown ideal response curve. Let us try to draw actual response curve of an on off control system. Say at time T O the temperature of the transformer starts rising. The measuring instrument of the temperature does not response instantly, as it requires some time delay for heating up and expansion of mercury in temperature sensor bulb say from instant T 1 the pointer of the temperature indicator starts rising. This rising is exponential in nature. Let us at point A, the controller system starts actuating for switching on cooling fans and finally after period of T 2 the fans starts delivering force air with its full capacity. Then the temperature of the transformer starts decreasing in exponential manner. At point B, the controller system starts actuating for switching off the cooling fans and finally after a period of T 3 the fans stop delivering force air. Then the temperature of the transformer again starts rising in same exponential manner. N.B.: Here during this operation we have assumed that, loading condition of the electrical power transformer, ambient temperature and all other conditions of surrounding are fixed and constant.

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When a number of elements are combined together to form a system to produce desired output then the system is referred as control system. As this system controls the output, it is so referred. Each element connected to the system has its own effect on the output. Definition of Control System A control system is a system of devices or set of devices, that manages, commands, directs or regulates the behavior of other device(s) or system(s) to achieve desire results. In other words the definition of control system can be rewritten as A control system is a system, which controls other system . As the human civilization is being modernized day by day the demand of automation is increasing accordingly. Automation highly requires control of devices. In recent years, control systems plays main role in the development and advancement of modern technology and civilization. Practically every aspects of our day-to-day life is affected less or more by some control system. A bathroom toilet tank, a refrigerator, an air conditioner, a geezer, an automatic iron, an automobile all are control system. These systems are also used in industrial process for more output. We find control system in quality control of products, weapons system, transportation systems, power system, space technology, robotics and many more. The principles of control theory is applicable to engineering and non engineering field both. Feature of Control System The main feature of control system is, there should be a clear mathematical relation between input and output of the system. When the relation between input and output of the system can be represented by a linear proportionality, the system is called linear control system. Again when the relation between input and output cannot be represented by single linear proportionality, rather the input and output are related by some non-linear relation, the system is referred as non-linear control system. Requirement of Good Control System Accuracy: Accuracy is the measurement tolerance of the instrument and defines the limits of the errors made when the instrument is used in normal operating conditions. Accuracy can be improved by using feedback elements. To increase accuracy of any control system error detector should be present in control system. Sensitivity: The parameters of control system are always changing with change in surrounding conditions, internal disturbance or any other parameters. This change can be expressed in terms of sensitivity. Any control system should be insensitive to such parameters but sensitive to input signals only. Noise: An undesired input signal is known as noise. A good control system should be able to reduce the noise effect for better performance. Stability: It is an important characteristic of control system. For the bounded input signal, the output must be bounded and if input is zero then output must be zero then such a control system is said to be stable system. Bandwidth: An operating frequency range decides the bandwidth of control system. Bandwidth should be large as possible for frequency response of good control system. Speed: It is the time taken by control system to achieve its stable output. A good control system possesses high speed. The transient period for such system is very small. Oscillation: A small numbers of oscillation or constant oscillation of output tend to system to be stable. Types of Control Systems There are various types of control system but all of them are created to control outputs. The system used for controlling the position, velocity, acceleration, temperature, pressure, voltage and current etc. are examples of control systems. Let us take an example of simple temperature controller of the room, to clear the concept. Suppose there is a simple heating element, which is heated up as long as the electric power supply is switched on. As long as the power supply switch of the heater is on the temperature of the room rises and after achieving the desired temperature of the room, the power supply is switched off. Again due to ambient temperature, the room temperature falls and then manually the heater element is switched on to achieve the desired room temperature again. In this way one can manually control the room temperature at desired level. This is an example of manual control system . This system can further be improved by using timer switching arrangement of the power supply where the supply to the heating element is switched on and off in a predetermined interval to achieve desired temperature level of the room. There is another improved way of controlling the temperature of the room. Here one sensor measures the difference between actual temperature and desired temperature. If there is any difference between them, the heating element functions to reduce the difference and when the difference becomes lower than a predetermined level, the heating elements stop functioning. Both forms of the system are automatic control system . In former one the input of the system is entirely independent of the output of the system. Temperature of the room (output) increases as long as the power supply switch is kept on. That means heating element produces heat as long as the power supply is kept on and final room temperature does not have any control to the input power supply of the system. This system is referred as open loop control system . But in the later case, the heating elements of the system function, depending upon the difference between, actual temperature and desired temperature. This difference is called error of the system. This error signal is fed back to the system to control the input. As the input to output path and the error feedback path create a closed loop, this type of control system is referred as closed loop control system . Hence, there are two main types of control system . They are as follow Open loop control system Closed loop control system Open Loop Control System A control system in which the control action is totally independent of output of the system then it is called open loop control system . Manual control system is also an open loop control system. Fig – 1 shows the block diagram of open loop control system in which process output is totally independent of controller action. Practical Examples of Open Loop Control System Electric Hand Drier – Hot air (output) comes out as long as you keep your hand under the machine, irrespective of how much your hand is dried. Automatic Washing Machine – This machine runs according to the pre-set time irrespective of washing is completed or not. Bread Toaster – This machine runs as per adjusted time irrespective of toasting is completed or not. Automatic Tea/Coffee Maker – These machines also function for pre adjusted time only. Timer Based Clothes Drier – This machine dries wet clothes for pre – adjusted time, it does not matter how much the clothes are dried. Light Switch – lamps glow whenever light switch is on irrespective of light is required or not. Volume on Stereo System – Volume is adjusted manually irrespective of output volume level. Advantages of Open Loop Control System Simple in construction and design. Economical. Easy to maintain. Generally stable. Convenient to use as output is difficult to measure. Disadvantages of Open Loop Control System They are inaccurate. They are unreliable. Any change in output cannot be corrected automatically. Closed Loop Control System Control system in which the output has an effect on the input quantity in such a manner that the input quantity will adjust itself based on the output generated is called closed loop control system . Open loop control system can be converted in to closed loop control system by providing a feedback. This feedback automatically makes the suitable changes in the output due to external disturbance. In this way closed loop control system is called automatic control system. Figure below shows the block diagram of closed loop control system in which feedback is taken from output and fed in to input. Practical Examples of Closed Loop Control System Automatic Electric Iron – Heating elements are controlled by output temperature of the iron. Servo Voltage Stabilizer – Voltage controller operates depending upon output voltage of the system. Water Level Controller – Input water is controlled by water level of the reservoir. Missile Launched & Auto Tracked by Radar – The direction of missile is controlled by comparing the target and position of the missile. An Air Conditioner – An air conditioner functions depending upon the temperature of the room. Cooling System in Car – It operates depending upon the temperature which it controls. Advantages of Closed Loop Control System Closed loop control systems are more accurate even in the presence of non-linearity. Highly accurate as any error arising is corrected due to presence of feedback signal. Bandwidth range is large. Facilitates automation. The sensitivity of system may be made small to make system more stable. This system is less affected by noise. Disadvantages of Closed Loop Control System They are costlier. They are complicated to design. Required more maintenance. Feedback leads to oscillatory response. Overall gain is reduced due to presence of feedback. Stability is the major problem and more care is needed to design a stable closed loop system. Comparison of Closed Loop And Open Loop Control System Sr. No. Open loop control system Closed loop control system 1 The feedback element is absent. The feedback element is always present. 2 An error detector is not present. An error detector is always present. 3 It is stable one. It may become unstable. 4 Easy to construct. Complicated construction. 5 It is an economical. It is costly. 6 Having small bandwidth. Having large bandwidth. 7 It is inaccurate. It is accurate. 8 Less maintenance. More maintenance. 9 It is unreliable. It is reliable. 10 Examples: Hand drier, tea maker Examples: Servo voltage stabilizer, perspiration Feedback Loop of Control System A feedback is a common and powerful tool when designing a control system. Feedback loop is the tool which take the system output into consideration and enables the system to adjust its performance to meet a desired result of system. In any control system, output is affected due to change in environmental condition or any kind of disturbance. So one signal is taken from output and is fed back to the input. This signal is compared with reference input and then error signal is generated. This error signal is applied to controller and output is corrected. Such a system is called feedback system. Figure below shows the block diagram of feedback system. When feedback signal is positive then system called positive feedback system. For positive feedback system, the error signal is the addition of reference input signal and feedback signal. When feedback signal is negative then system is called negative feedback system. For negative feedback system, the error signal is given by difference of reference input signal and feedback signal. Effect of Feedback Refer figure beside, which represents feedback system where R = Input signal E = Error signal G = forward path gain H = Feedback C = Output signal B = Feedback signal Error between system input and system output is reduced. System gain is reduced by a factor 1/(1±GH). Improvement in sensitivity. Stability may be affected. Improve the speed of response.

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Theory of Network Synthesis Network Functions As the name suggests, in theory of network synthesis we are going to study about the synthesis of various networks which consists of both the active (resistors) and passive elements (inductors and capacitors). Let us know what is a network function ? In the frequency domain, network functions are defined as the quotient obtained by dividing the phasor corresponding to the circuit output by the phasor corresponding to the circuit input. In simple words, network functions are the ratio of output phasor to the input phasor when phasors exists in frequency domain. The general form of network functions are given below: Now with the help of the above general network function we are in position to describe the necessary conditions of the stability of all the network functions. There are three mains necessary conditions for the stability of these network functions and they are written below: The degree of the numerator of F(s) should not exceed the degree of denominator by more than unity. In other words (m – n) should be less than or equal to one. F(s) should not have multiple poles on the jω-axis or the y-axis of the pole-zero plot. F(s) should not have poles on the right half of the s-plane. Hurwitz Polynomial If above all the stability criteria are fulfilled (i.e. we have stable network function) then the denominator of the F(s) is called the Hurwitz polynomial . Where Q(s) is a Hurwitz polynomial . Properties of Hurwitz Polynomials There are five important properties of Hurwitz polynomials and they are written below: For all real values of s value of the function P(s) should be real. The real part of every root should be either zero or negative. Let us consider the coefficients of denominator of F(s) is b n , b (n-1) , b (n-2) . . . . b 0 . Here it should be noted that b n , b (n-1) , b 0 must be positive and b n and b (n-1) should not be equal to zero simultaneously. The continued fraction expansion of even to the odd part of the Hurwitz polynomial should give all positive quotient terms, if even degree is higher or the continued fraction expansion of odd to the even part of the Hurwitz polynomial should give all positive quotient terms, if odd degree is higher. In case of purely even or purely odd polynomial, we must do continued fraction with the of derivative of the purely even or purely odd polynomial and rest of the procedure is same as mentioned in the point number (4). From the above discussion we conclude one very simple result, If all the coefficients of the quadratic polynomial are real and positive then that quadratic polynomial is always a Hurwitz polynomial. Positive Real Functions Any function which is in the form of F(s) will be called as a positive real function if fulfill these four important conditions: F(s) should give real values for all real values of s. P(s) should be a Hurwitz polynomial. If we substitute s = jω then on separating the real and imaginary parts, the real part of the function should be greater than or equal to zero, means it should be non negative. This most important condition and we will frequently use this condition in order to find out the whether the function is positive real or not. On substituting s = jω, F(s) should posses simple poles and the residues should be real and positive. Properties of Positive Real Function There are four very important properties of positive real functions and they are written below: Both the numerator and denominator of F(s) should be Hurwitz polynomials. The degree of the numerator of F(s) should not exceed the degree of denominator by more than unity. In other words (m-n) should be less than or equal to one. If F(s) is positive real function then reciprocal of F(s) should also be positive real function. Remember the summation of two or more positive real function is also a positive real function but in case of the difference it may or may not be positive real function. Following are the four necessary but not the sufficient conditions for the functions to be a positive real function and they are written below: The coefficient of the polynomial must be real and positive. The degree of the numerator of F(s) should not exceed the degree of denominator by more than unity. In other words (m – n) should be less than or equal to one. Poles and zeros on the imaginary axis should be simple. Let us consider the coefficients of denominator of F(s) is b n , b (n-1) , b (n-2) . . . . b 0 .Here it should be noted that b n , b (n-1) , b 0 must be positive and b n and b (n-1) should not be equal to zero simultaneously. Now there two necessary and sufficient conditions for the functions to be a positive real function and they are written below: F(s) should have simple poles on the jω axis and the residues of these poles must be real and positive. Summation of both numerator and denominator of F(s) must be a Hurwitz polynomial. 

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Before I introduce you the theory of control system it is very essential to know the various types of control systems . Now there are various types of systems, we are going to discuss only those types of systems that will help us to understand the theory of control system and detail description of these types of system are given below: Linear Control Systems In order to understand the linear control system , we should know the principle of superposition. The principle of superposition theorem includes two the important properties and they are explained below: Homogeneity: A system is said to be homogeneous, if we multiply input with some constant ‘A’ then output will also be multiplied by the same value of constant (i.e. A). Additivity: Suppose we have a system ‘S’ and we are giving the input to this system as ‘a 1 ’ for the first time and we are getting output as ‘b 1 ’ corresponding to input ‘a 1 ’. On second time we are giving input ‘a 2 ’ and correspond to this we are getting output as ‘b 2 ’. Now suppose this time we giving input as summation of the previous inputs ( i.e. a 1 + a 2 ) and corresponding to this input suppose we are getting output as (b 1 + b 2 ) then we can say that system ‘S’ is following the property of additivity. Now we are able to define the linear control systems as those types of control systems which follow the principle of homogeneity and additivity. Examples of Linear Control System Consider a purely resistive network with a constant dc source. This circuit follows the principle of homogeneity and additivity. All the undesired effects are neglected and assuming ideal behavior of each element in the network, we say that we will get linear voltage and current characteristic. This is the example of linear control system . Non-linear Systems We can simply define non linear control system as all those system which do not follow the principle of homogeneity. In practical life all the systems are non-linear system. Examples of Non-linear System A well known example of non-linear system is magnetization curve or no load curve of a dc machine. We will discuss briefly no load curve of dc machines here: No load curve gives us the relationship between the air gap flux and the field winding mmf. It is very clear from the curve given below that in the beginning there is a linear relationship between winding mmf and the air gap flux but after this, saturation has come which shows the non linear behavior of the curve or characteristics of the non linear control system . Analog or Continuous System In these types of control system we have continuous signal as the input to the system. These signals are the continuous function of time. We may have various sources of continuous input signal like sinusoidal type signal input source, square type of signal input source, signal may be in the form of continuous triangle etc. Digital or Discrete System In these types of control system we have discrete signal (or signal may be in the form of pulse) as the input to the system. These signals have the discrete interval of time. We can convert various sources of continuous input signal like sinusoidal type signal input source, square type of signal input source etc into discrete form using the switch. Now there are various advantages of discrete or digital system over the analog system and these advantages are written below: Digital systems can handle non linear control systems more effectively than the analog type of systems. Power requirement in case of discrete or digital system is less as compared to analog systems. Digital system has higher rate of accuracy and can perform various complex computations easily as compared to analog systems. Reliability of digital system is more as compared to analog system. They also have small and compact size. Digital system works on the logical operations which increases their accuracy many times. Losses in case of discrete systems are less as compared to analog systems in general. Single Input Single Output Systems These are also known as SISO type of system. In this the system has single input for single output. Various example of this kind of system may include temperature control, position control system etc. Multiple Input Multiple Output Systems These are also known as MIMO type of system. In this the system has multiple outputs for multiple inputs. Various example of this kind of system may include PLC type system etc. Lumped Parameter System In these types of control systems the various active (resistor) and passive parameters (like inductor and capacitor) are assumed to be concentrated at a point and that’s why these are called lumped parameter type of system. Analysis of such type of system is very easy which includes differential equations. Distributed Parameter System In these types of control systems the various active (resistor) and passive parameters (like inductor and capacitor) are assumed to be distributed uniformly along the length and that’s why these are called distributed parameter type of system. Analysis of such type of system is slightly difficult which includes partial differential equations.

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Mathematical Modelling of Control System There are various types of physical systems namely we have Mechanical system. Electrical system. Electronic system. Thermal system. Hydraulic system. Chemical system etc. Before I describe these systems in detail let us know, what is the meaning of modeling of the system? Mathematical modelling of control system is the process of drawing the block diagram for these types of systems in order to determine the performance and the transfer functions. Now let us describe mechanical and electrical type of systems in detail. We will derive analogies between mechanical and electrical system only which are most important in understanding the theory of control system. Mathematical Modelling of Mechanical Systems We have two types of mechanical systems. Mechanical system may be a linear mechanical system or it may be a rotational mechanical type of system. In linear mechanical type of systems we have three variables – Force which is represented by ‘F’. Velocity which is represented by ‘V’. Linear displacement represented by ‘X’ And also we have three parameters- Mass which is represented by ‘M’. Coefficient of viscous friction which is represented by ‘B’. Spring constant which is represented by ‘K’. In rotational mechanical type of systems we have three variables- Torque which is represented by ‘T’. Angular velocity which is represented by ‘ω’ Angular displacement represented by ‘θ’ And also we have two parameters – Moment of inertia which is represented by ‘J’. Coefficient of viscous friction which is represented by ‘B’. Now let us consider the linear displacement mechanical system which is shown below- We have already marked various variables in the diagram itself. We have x is the displacement as shown in the diagram. From the Newton’s second law of motion, we can write force as- From the diagram we can see that the, On substituting the values of F 1 , F 2 and F 3 in the above equation and taking the Laplace transform we have the transfer function as, This equation is mathematical modelling of mechanical control system . Mathematical Modelling of Electrical System In electrical type of systems we have three variables – Voltage which is represented by ‘V’. Current which is represented by ‘I’. Charge which is represented by ‘Q’. And also we have three parameters which are active and passive elements – Resistance which is represented by ‘R’. Capacitance which is represented by ‘C’. Inductance which is represented by ‘L’. Now we are in condition to derive analogy between electrical and mechanical types of systems. There are two types of analogies and they are written below: Force Voltage Analogy : In order to understand this type of analogy, let us consider a circuit which consists of series combination of resistor, inductor and capacitor. A voltage V is connected in series with these elements as shown in the circuit diagram. Now from the circuit diagram and with the help of KVL equation we write the expression for voltage in terms of charge, resistance, capacitor and inductor as, Now comparing the above with that we have derived for the mechanical system we find that- Mass (M) is analogous to inductance (L). Force is analogous to voltage V. Displacement (x) is analogous to charge (Q). Coefficient of friction (B) is analogous to resistance R and Spring constant is analogous to inverse of the capacitor (C). This analogy is known as force voltage analogy. Force Current Analogy : In order to understand this type of analogy, let us consider a circuit which consists of parallel combination of resistor, inductor and capacitor. A voltage E is connected in parallel with these elements as shown in the circuit diagram. Now from the circuit diagram and with the help of KCL equation we write the expression for current in terms of flux, resistance, capacitor and inductor as, Now comparing the above with that we have derived for the mechanical system we find that, Mass (M) is analogous to Capacitor (C). Force is analogous to current I. Displacement (x) is analogous to flux (ψ). Coefficient of friction (B) is analogous to resistance 1/ R and Spring constant K is analogous to inverse of the inductor (L). This analogy is known as force current analogy. Now let us consider the rotational mechanical type of system which is shown below we have already marked various variables in the diagram itself. We have θ is the angular displacement as shown in the diagram. From the mechanical system, we can write equation for torque (which is analogous to force) as torque as, From the diagram we can see that the, On substituting the values of T 1 , T 2 and T 3 in the above equation and taking the Laplace transform we have the transfer function as, This equation is mathematical modelling of electrical control system.

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Magnesium is used as anode materials in primary battery because of its high standard potential. It is a light metal. It is also easily available low cost metal. Magnesium/manganese dioxide (Mg/MnO 2 ) battery has twice the service life i.e. capacity of the zinc/manganese dioxide (Zn/MnO 2 ) battery of same size. It has also the ability to retain its capacity, during storage, even at high temperatures. Magnesium battery is very durable and storable since it has always a protective cover which is naturally formed on the surface of the magnesium anode. The magnesium battery generally loses its storability once it has been partially discharged and that is why it is not very suitable for using in long-term intermittent applications. This is the main reason, why magnesium battery is losing its popularity and lithium battery are occupying its market. Chemistry of Magnesium Battery In magnesium primary battery , magnesium alloy is used as anode; manganese dioxide is used as cathode material. But manganese dioxide cannot provide required conductivity to the cathode and that is why acetylene black is mixed with manganese dioxide to achieve required conductivity. Magnesium per-chlorate is used as electrolyte. Barium and lithium chromate are added to electrolyte for preventing corrosion. Magnesium hydroxide is also added to this mixture as buffering agent to improve storability. The oxidation reaction occurs in the anode is, The reduction reaction occurs in cathode is, Overall reaction, The open circuit voltage, this cell gives around 2 volt but the theoretical value of the cell potential is 2.8 volt. The chance of corrosion of magnesium is very less even under extreme environmental conditions. Actually raw magnesium reacts with moisture and form a coating of thin film of Mg(OH) 2 on its surface. This thin film of magnesium peroxide serves as a corrosion protective layer over the magnesium. In addition to that chromate treatment on magnesium improves this protection to very large extent. But when this protective film of magnesium peroxide is puncher or removed due to discharge of battery , corrosion takes place with formation of hydrogen gas. This is the basic chemistry of magnesium battery . Construction of Magnesium Battery Construction wise a cylindrical magnesium battery cell is similar to a cylindrical zinc carbon battery cell. Here an alloy of magnesium is used as main container of the battery . This alloy is formed by magnesium and small quantity of aluminum and zinc. Here, manganese dioxide is used as cathode material. As the manganese dioxide has poor conductivity, acetylene black is mixed with this to improve its conductivity. This also helps to retain water inside cathode. In this cathode mixture barium chromate is added as an inhibitor, and also magnesium hydroxide is added as a pH buffer. Magnesium per-chlorate with lithium chromate mixed in water is used as electrolyte. A carbon is inserted in the cathode mix as current collector. Kraft papers, absorbed with electrolyte solution are placed in between cathode and anode materials as separators. Special attention is to be given during designing of sealing arrangement in magnesium battery . The sealing of the battery should not be so porous that the moisture inside the battery will be evaporated during storing of the battery and it should not be so nonporous that the hydrogen gas formed during discharge cannot escape from the battery . So the seal of the battery should retain the moisture inside it and at the same time it gives sufficient vent to the hydrogen gas formed during discharge. This can be done by providing a small hole on the top of the plastic seal washed under the Retainer ring. When excess gas comes out from the hole this retainer ring is deformed due to pressure and resulting escaping of the gas. Generally magnesium anode forms the outer cover of the battery but another construction of magnesium battery is also available where carbon forms the outer container of the battery . Here a typical shaped container is formed from highly conductive carbon. This container is formed in a cylindrical cup shape and one rod like shape is projected from its center as shown in the picture. Anode of the battery is formed by a cylinder or drum of magnesium. The diameter of the anode cylindrical is about half of the carbon cup. The cathode mix is placed inside this anode cylinder and separated from the inner wall of the cylinder by a paper separator. The space between inner surface of the carbon cup and outer surface of the anode cylinder also filled with cathode mix and here also the outer surface of the anode cylinder is separated from cathode mix by a paper separator. The cathode mix is produced by mixing manganese dioxide, carbon black, and small quantity of aqueous magnesium bromide or per-chlorate as the electrolyte. Positive terminal is connected to the end of the carbon cup. The negative terminal is connected to the end of anode drum. The entire system is encapsulated in a crimped tin-plated steel jacket. Advantage of Magnesium Battery It has very good self life; it can be stored for long even under high-temperature. These battery can be stored up to 5 years at the temperature 20°C. It has twice capacity compared to equivalent size Leclanche battery . Higher battery voltage than zinc-carbon battery . Cost is also moderate. Disadvantages of Magnesium Battery Delayed action.(voltage delay) Evolution of hydrogen during discharge. Heat generated during use. Poor storage after partial discharge. The battery are no longer manufactured commercially. Sizes And Types Of Mg/Mno 2 Batteries Cylindrical Magnesium Primary Batteries Battery type Diameter in mm Height in mm Weight in gm Capacity in Ah N 11 31 5 0.5 B 19.2 53 26.5 2 C 25.4 49.7 45 3 1LM 22.8 84.2 59 4.5 D 33.6 60.5 105 7 FD 41.7 49.1 125 8 No. 6 63.5 159 1000 65

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