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Stability of system In linear time invariant system we have: 1.     If input is zero output also zero. 2.     It should be independent of any non linear operator like square , cube ,root, sine, log etc. on either input side or output side. 3.     Differential operator is linear operator .   There should not be any time scaling i.e. coefficient should be   independent of time. Stability of LTI system may be defined as :   When system is subjected to bounded input output must be bounded. That is it must follow the BIBO (bounded input and bounded output) criterion. The stability of system is depends upon roots of characteristics equation i.e.                     1+G(s) *H(s) = 0 A.     Addition of pole always results in stability. B.     Addition of zero results in stability. ...

Control systems types Linear & Non-Linear System. A system is said to be linear if it follows the  superposition principle and homogeneity whereas a  Non-Linear System   does not follow at least one of these rules. Mathematically, it can be written as: Additivity   If the input signals are time varying   x1(t) and x2(t) then   corresponds to output signals y1(t) and y2(t), then  x1(t) + x2(t) = y1(t) + y2(t)  ( additivity ) Homogeneity   If the input signals x1(t) corresponds to output signal y1(t) , and “a” be any scalar then their product will be a.x1(t)=a.y1(t)  ( Homogeneity ) Digital or Discrete System In these types of control systems, we have a discrete signal (or signal may be in the form of pulse) as the input to the system. We can convert various input signal like sinusoidal type signal into square type signal etc into a discrete form using the electronic switch. Sin...

Transfer Function The transfer function of a linear , time-invariant , differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. Above fig. shows the Block Diagram of Closed Loop Control System, where all variables are in laplace form i.e. E(s)  = Error Signal G(s) =  Forward Path Transfer Function. Y(s) =  Output Signal X(s) =  Reference Input Signal               H(s) =  Feedback Transfer Function. B(s) =  Feedback Signal. Transfer function of system is   From the block diagram,         Y(s) = G(s).E(s) ........1 B(s) = H(s).Y(s) ........2 E(s) = X(s) + B(s) .......3     (For positive feedback) ...

Linear Systems. A system is called linear if the principle of superposition applies. The principle of superposition states that the response produced by the simultaneous application of two different forcing functions is the sum of the two individual responses.  Hence, for the linear system, the response to several inputs can be calculated by treating one input at a time and adding the results. It is this principle that allows one to build up complicated solutions to the linear differential equation from simple solutions. Transfer Function. The transfer function of a linear, time-invariant, differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. click here to go superposition theorem click here to go power and energy  

Control systems Open-Loop Control Systems. Those systems in which the output has no effect on the control action are called open-loop control systems. In other words, in an open loop control system the output is neither measured nor fed back for comparison with the input. One practical example is a semi automatic washing machine. The machine does not measure the output signal, that is, the cleanliness of the clothes. Open loop system are highly sensitive and stable if no disturbances are given Closed-Loop Control Systems Feedback control systems are often referred to as closed-loop control systems.. In a closed-loop control system the actuating error signal, which is the difference between the input signal and the feedback signal, is fed to the controller so as to reduce the error and bring the output of the system to a desired value. The term closed-loop control always implies the use of feedback control action in order to reduce system error. This system ...

Equivalent circuit of no load transformer equivalent circuit  In the given figure we take primary referred equivalent. I 2 ’ = I 2 \a                       where,          a = turn ratio = N 1 \N 2 R 2 ’ = a 2 R 2                                    Gi = R 0        B m = X 0 X 2 ’= a 2 X 2 V 2 ’ = aV 2                     approximation circuit Full load equivalent phasor diagram for inductive load. The KVL equations for the primary and secondary circuits are V 1 = E 1 + I 1 R 1 + j I 1 X 1 V 2 = E 2 - I 2 R 2 - j I 2 X 2 ...

Types of losses in the transformers There are two types of losses in transformers. Iron loss and copper loss. Iron loss also called as a constant loss, while copper loss is also called as a variable loss these are depends upon load current. A. Copper loss   Copper losses. Copper losses (I2R) are the resistive heating losses in the primary and secondary windings of the transformer. They are proportional to the square of the current in the windings. B. Iron loss These are also in two types  1. Eddy current losses. Eddy current losses are resistive heating losses in the core of the transformer. They are also called swirling losses. They are proportional to the square of the voltage applied to the transformer. P e = K e f 2 (B max ) 2 t 2 \ ρ                     Where; K e = proportionality constant f     =   frequency B max = m...

No load equivalent circuit The primary winding draws a small amount of alternating current of instantaneous value I o , called the exciting current , from the voltage source. The exciting current establishes flux f in the core. When the primary side is connected to sinusoidal voltage source V 1 a current known magnetizing current I m flows in it. This current set up a alternating flux in the core. Hence it is called a magnetizing current. The flux Φ m is proportional to I m . I m lags behind the applied voltage v 1 by 90.    The flux Φ m links with both the primary and the secondary winding. When it links with primary winding it produce self induce EMF E 1 which is opposite to applied voltage. Similarly alternating flux links with secondary side its produce mutually induced EMF E 2 . Both E 1 and E 2 lag behind the Φ by 90 degree. This can be shown by the phasor diagram. Here the losses are neglected. No load equivale...

Main types of transformer 1.     Step Up Transformer - Used for step up the voltage level of power in  transmission and distribution power system. 2.       Step Down Transformer - Used for down the voltage level of power in transmission and distribution power system. 3.     Three Phase Transformer - It is generally used in three phase power system as it is cost effective single phase. 4.       Single Phase Transformer – It is used for single phase system and also in 3 phase system by employing 3 single phase transformers. 5.     Electrical Power Transformer – These are generally used in transmission network for stepping up or down the voltage level. It operates mainly during high or peak loads and has maximum efficiency at or near full load. 6.       Distribution Transformer - Distribution transformer steps down the voltage for distribution purpo...