Linear and Nonlinear Systems in Electrical Engineering


 Like any physical system, the Electrical Systems also work based on some well defined principles. For the study of the characteristics of any system, device or element, the block diagram notation (as shown below ) is used universally.
        

In the figure the element or system is represented by a rectangular block. Naturally the device will receive some input and give the output. The arrow is shown to represent the direction of input and output signal flow.

Basically any system can be  reduced to this simple form for studying its behavior.  When the block diagram represents a single element or simple device then instead of 'input' and 'output' we better use the terms 'cause' and 'effect'  respectively (even the terms 'excitation' and 'response' are equally used).  For example the cause may be voltage applied across the element and effect is the current that flows in the element. Here we will mainly use the terms 'input', 'output' and 'cause', 'effect'. I feel that the terms excitation and response are more specialized. Here we also use different terms like element, device and system interchangeably to satisfy people of different mindset. It should be remembered that these terms are not exactly of the same meaning. For example different elements or devices may be connected in a particular way to form a system etc.

The simplest way of studying the behaviour of a element or system is by applying a series of inputs and observing the respective outputs. That is what is our approach here.


Linear System


When the input(cause) and corresponding output(effect) are tied by the rules of linearity, then the device or system behavior is said to be linear, otherwise nonlinear.

Then what is exactly this linearity ?

Suppose we apply a series of inputs say x1, x2, x3 and x4. When input is x1 the output obtained is y1. Similarly when inputs are x2, x3 and x4 the respective outputs are y2, y3 and y4.

Now a graph is drawn taking points P1(x1, y1), P2(x2, y2), P3(x3, y3), P4(x4, y4). See the points marked with small circles in X-Y plane. In the X axis we take input values and in Y- axis output values.



Now join the points by smooth free hand curve, the result is figure-B(ii). The more points we take, the more accurate will be the curve.
Depending on the device behavior the curve of input versus output can be of any shape. some more cases are shown in fig-C(i) to (iv).
        

                                
Each of the curves may fit to some mathematical linear or nonlinear equation. For many devices the  relationship between input and output can be found directly analytically.

Remember that the graph (or curve) is drawn with input( X axis) versus output (Y axis) ( not input vs. time or output vs. time).


If a system or device or element satisfy the following two properties then it is called linear.

  • Superposition
          Suppose when  x1 input is applied the  y1 output is obtained,

                  and when x2 input is applied the y2 output is obtained.

          Now, if input applied is (x1+x2),  then the output obtained will be y1+y2 .
                (equivalently we say that if x1 and x2 are applied simultaneously then out put will be the sum of the
                           outputs obtained individually)
  • Homogeneity
          when  x1 is the input  applied the  y1 output is obtained,

          Then if  (kx1) input is applied, then output obtained will be ky1. Here k is any real number.
    
By examining all the input(X) vs output(Y) curves one can be sure that above two properties will be satisfied by only curve c(ii).
The equation of this curve is in the form y = mx. Here the constant 'm' may be any real number.

It is easy to remember that the input vs. output characteristic curve of a linear device should be straight line passing through the origin (center). Only in this case the above two laws of linearity are satisfied.

All the other curves in fig-C are non-linear and they do not satisfy the above two properties of linearity (check it). It should be realized that even the straight line in fig-C(i) does not represent a linear system. Students are often confused and consider any straight line relationship between the input and output as linear. But it is not. One can also verify that the above two properties are not satisfied by every straight line relationship.

In maths the linear equation is represented by the form y = mx + c, which is a straight line whose orientation depends on the values of m and c. The constants 'm' and 'c' can be any real numbers. In case of physical systems, all represented by these linear equations are not strictly linear systems. For the system to be linear, c must be zero.

Linear and Nonlinear elements/systems in Electrical engineering

Now is the time to consider few examples. The best example of a linear element is an ordinary resistance. If the voltage applied across the resistance is 'cause' and current flowing through the resistance as 'effect'. Then the graph of voltage versus current is a straight line (Ohms Law) passing through the origin. see fig-D. You can verify that it satisfies the linearity laws.



The next example is the simple Diode circuit (Fig-E). Look at the voltage versus current curve here. Although it passes through the origin it is not a straight line, hence one can immediately say that it's a nonlinear element.

Now we will consider the example of a magnetic circuit.

Consider the magnetic core in a transformer. Due to the current 'i' applied in the coil surrounding the core, flux     'phi' is established in the magnetic core. A curve is drawn between field intensity and flux density. The arrow direction in the curve indicates that two different values of flux density obtained for same value of field intensity, one is for when the excitation current is increasing and other when the excitation current is decreasing. The o-a-b path is only at  beginning of magnetisation. This type of behavior in magnetic circuit of transformer is called hysteresis. The closed loop formed by the curve is called hysteresis loop. Now just looking at the diagram one can be sure that the field intensity vs. flux density curve is non-linear.



This non-linearity in transformer characteristic gives rise to harmonics, which requires some techniques to handle.

Nonlinearities are often encountered in electrical devices and systems. Most of the elements show non-linearities to some degree beyond certain input range. There are some methods to deal with the nonlinearities. We can also say that every element or system behaves like a linear element or system within a small range of input variation. For small variations of input around the operating point, the curve can be approximated by a small straight line. The middle of this small straight line is to be treated as origin. See Fig-G. In the figure the portion a-b can be treated as linear and Q as operating point.

The truth is that in real world sometimes even the systems with much pronounced nonlinear characteristics is analysed using linear techniques. this is because the variation of system input and output is small enough around the operating point.



 'x' and 'y' are small changes around the operating point that is our new origin 'Q'. Again remember that the curve with respect to 'x' & 'y' coordinates is linear (within a-b range) while with respect to 'X' and 'Y' coordinates it is still nonlinear.

(While solving problems considering small changes around the new origin 'Q', we only concern about 'x', 'y' and 'Q', forgetting original 'X', 'Y' and 'O')

Let us consider one more example.
(This portion requires little more knowledge in curve tracing and electronics)

Consider the characteristics of a transistor (see Fig-H). The output 'Ic' versus 'Vce' (collector to emitter voltage)   is drawn for different values of base current 'Ib'.

   Ic = collector current
Vce = collector to emitter voltage
   Ib = base current



Examining the transfer curve (between Ib(cause) and Ic(effect)) it is clear that the curve of Ib versus Ic is non-linear. Close examination reveals that the curve seems to behave linearly in part of the curve (point a to point b). Then our aim is to fix 'Q' in such a position so that the variations ib and ic around Q lies within this linear region.

( Ib and Ic are total values but ib and ic are variations around Q)

Just look at the point Q that we fixed in the curve. If the circuit is allowed to operate around this point Q, so that its input value say Ib varies in small values, then this variation with respect to this operating point Q is linear  variation.
It should be noted that the overall variation with respect to origin 'O' is not linear. The linearity is with respect to point Q.

Then why do we need this linear variation? Yes, in the linear portion the output wave shape is not distorted. That means the shape of the wave is preserved. Operating in somewhat linear range gives birth to very little noise.

This is one important reason why we need the biasing in the transistor and set the quiescent point Q.

In the linear range the superposition law works.
If sinusoidal input signal is applied to the transistor, then actually the transistor operates at base value plus sinusoidal input.

So actually whether you apply input (ib) or not, IB,IC and VCE are always present as long as Vcc battery is connected as shown. These are the quiescent point values due to biasing, which helps the transistor operate in linear range. 
From the total collector current Ic,  the output ic is filtered out using a capacitor or signal transformer.

So, we applied ib and got ic. However ic is several times ib and so we got ic as amplified version of ib.

The linearisation method is used in many situations. This technique is also helpful in power system small signal stability studies and other small signal oscillation studies.




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