Transmission Lines


The transmission lines are like the arteries of the power system. Transmission lines are important part of power network and are also loosely called as power lines. It should be remembered that the term power line also includes distribution line. As in this article our main concern is transmission line so we will not use the less specific term power line. Transmission lines act as medium for carrying bulk energy from one substation to other. Different voltage levels are standardized by energy authorities of the respective countries for transmission lines to operate. The electric energy transmission is carried out at High and Extra High Voltages (EHV).  Some common AC transmission voltage standards are 110 kV, 115 kV, 132 kV , 138 kV, 220 kV,  230 kV, 345kV, 380 kV, 400 kV, 500 kV, 765 kV, 1100 kV etc. . Voltage above 220 kV is usually referred as Extra High Voltage. For very long transmission line, transmission at High Voltage  DC  (HVDC) has proved to be economical. HVDC transmission also has several other advantages. Here we will mainly concentrate on basics of AC transmission. Here we will remain silent about HVDC transmission.
The electric substations are interconnected by transmission lines and form a network. It is also called Power Grid or simply Grid. The modern Grid is very large. It spans regions and countries.

The transmission lines can be constructed over head or under ground. The overhead lines are bare conductors with proper  clearances from earthed structures and  between the phase conductors. The underground transmission  is carried out by means of cables. Gas insulated lines are also increasingly found as transmission medium. Here we will discuss about over head transmission system which is the primary means of electric energy transmission.

The over head transmission is cheaper in comparison to underground transmission. The selection of conductor  for transmission line depends on several factors. The main factors being electrical and Mechanical it should also be economical. The corrosion of conductor due to environment is also kept in mind. The transmission line should be able to carry the electrical load and should be strong enough to sustain the mechanical stress under different conditions.

Obviously the transmission line is required to have good conductivity and high strength to prevent snapping under normal and abnormal conditions. Although copper has high conductivity it is not used for transmission purposes primarily due to its higher cost and being heavier than aluminium. Aluminium is used because it is cheap, light and of course has good conductivity  . But because of low strength, aluminum alone is not used for transmission purposes. Instead of a single rounded conductor stranded conductors made from conductors of smaller radii are used. The stranded conductor has few advantages over single rounded conductor as listed below.
  • The stranded conductor has larger surface area so the electric field strength  near the conductor surface is reduced. As a result the stranded conductor is less likely to experience corona. Corona is the phenomenon that happens due to the ionization of air adjacent to the conductor surface due to high electric field strength. 
  • The stranded conductor is more flexible than a single rounded conductor of equal capacity. For higher capacity conductor the conductor diameter may become large enough that use of one single conductor of large diameter becomes non-flexible.
  • The strength of stranded conductor  is increased considerably by using few strands of steel or alloy core at the center as required.
 Three types of conductors are mainly used for transmission purposes
  • All Aluminum Alloy Conductor (AAAC)
  • Aluminum Conductor Steel Reinforced (ACSR)
  • Aluminum Conductor Alloy Reinforced (ACAR)

In case of AAAC all the strands of conductor are alloys of aluminum. So the conductor has good tensile strength in comparison to pure aluminum. The ACAR conductor has central core comprised of strands of aluminum alloy and the surrounding strands are of aluminum.
The ACSR conductor has a steel central core of few strands and surrounded by aluminum strands in one or more layers. The alternate layer of strands are laid in helical form would in opposite direction (otherwise how can the strands remain intact). The steel core provides the strength and surrounded aluminum conductors carry the current. So the ACSR  satisfies the requirement of a good transmission line, i.e. it has both good conductivity and sufficient strength. More tensile strength in ACSR results in less sagging so greater span length between towers. You will mostly find ACSR conductors for high voltage transmission purposes. In Fig-A is shown the cross section of a ACSR conductor. It has seven strands of central steel core surrounded by two layers of aluminum conductors.

Bundled Conductor

In high capacity transmission lines instead of one conductor (ACSR/AAAC conductor) per phase, two, three or four conductors are used. This is called bundled conductor. Here the conductors are separated by spacers. The two conductor bundle is called twin conductor bundle. When three conductors are used they take the position of vertices of equilateral triangle, if four nos of conductors (quad conductor) are used they take the position of the corners of a square. In Fig-B below is shown twin and triple conductor bundles.

Main benefits of conductor bundling.
  •  the transmission losses are reduced due to reduced resistance. 
  • The reactance of the line is also reduced so the transmission capacity of the line is increased. 
  • The main benefit is that the line corona is reduced due to increase in surface area, so reducing the loss due to corona. Reduction in chances of corona also means reduction of chances of interference with communication networks. See figure-B for bundled conductors.

Need For Transmission Lines

The  need for construction of transmission line mainly falls into one of the four categories mentioned below .
  • A new generating station/unit is established and the power from the generating station is required to be evacuated. The transmission lines are built for injecting power from the generating station to the already existing high voltage network (also called Grid). See Fig-C. These lines are most important types and get priority. The generating stations cannot generate without the commissioning of these lines

  • Two already existing HV or EHV substations of the grid are interconnected by transmission lines for strengthening the network.
  • It is usually seen that two adjacent power transmission companies agree to interconnect their systems by tie lines. By interconnecting the two adjacent systems, power needed by one company as agreed by both sides( or emergency requirement) is supplied by other side. It is done to improve the reliability and stability of the systems. See Fig-D .

  • A transmission line may be required to be built to serve an upcoming large load.  As an example if a new industry and/or its township is built (usually in new place) then transmission line is built to feed the distribution network of new industry/township from the existing grid. The power is received at the receiving station of load center and stepped down by transformer, which feeds the primary high voltage distribution network. See Fig-E. To enhance the availability of supply the receiving station at load center may be connected to two separate substations of the Grid forming.

For planning transmission line addition or alteration, computer simulated load flow study is carried out . The  study is carried out for several alternative configurations under peak and off-peak loading and generation conditions. From several alternative configurations the most suitable one is finalised for construction.

Electric Substation Types

The transmission and distribution lines carrying electric energy are terminated at the substations. One substation may be connected to one or more substations by transmission lines or to a generating station. These lines operate at  voltage levels as adopted by a particular region or nation. An actual transmission substation is shown below.

Substations Types

The main purposes for establishment of a substation are:

  • Need for connecting a generator to the transmission system:  We have already discussed that generators generate electricity at a low voltage mostly below 22 kV. But bulk quantity of power can be transmitted to very long distance only at high or extra high voltage. So generator transformers (GT) are used at the power station to step up the voltage to transmission level. Here the substation is not a separate entity rather it is switchyard of the Power Station. These substations are quite complex. One can find here more numbers of transformers in comparison to other types of substations.
  • To inter-connect two or more transmission lines at two or more different voltage levels: In a transmission system it is often planned or it is required to facilitate power flow between  transmission lines at different voltage levels. These are  transmission substations. Inter connecting Transformers (ICT) are used for this purpose. The transmission substations operate at Extra High Voltage (EHV). See Figure below.
  • To step down the voltage to a lower level for further transmission (sub-transmission) at an economic rate: Sometimes for shorter distance transmission of comparatively less power, the transmission company may feel that it is economical to transmit at a voltage level in-between the standard transmission and distribution level. Hence the requirement of establishing the substation.
  • To step down the voltage to a lower level for feeding a high voltage distribution network: These are the substations where the voltage is stepped down to a voltage level suitable for forming the major distribution network for a township.
  • Step down to low voltage for supply to consumers: These are area substations, where the voltage is again stepped down to a level suitable for house holds consumption. 
  • HVDC substation: There are two main purposes of a HVDC substation. 1) Two  dissimilar systems with different frequency can be connected by HVDC link to facilitate power transfer from one side to the other side. 2) Direct current long distance bulk power transmission. HVDC substations have many  advantages.
  • Switching substation:  These are the substations meant for switching purposes only. The Switching substation does not have a transformer. In a power system these are few in numbers. The switching substations are meant for disconnecting and connecting a part of network and facilitating  maintenance works.

Here I reproduce the SLD of a basic imaginary power system where we can see few types of substations and their relative configuration.

A substation may not be exclusively for one of the above purposes. A substation may be established primarily for the purpose of bulk power transmission but at the same time may have another transformer installed for stepping down the voltage to a lower level and distribute locally. Sometimes any combination from the above is adopted depending on the requirements.

The substations can be Air Insulated Substation (AIS)  or Gas Insulated substation (GIS). It can be over ground or Under ground.

The other important major elements usually found in the substation are Capacitor bank and Reactor.

A substation has switching arrangements to connect or disconnect these high voltage equipments or elements to the system.

Some important functions that are carried out in the substation are controlling the flow of energy, VAR compensation, voltage control, power factor improvement and measurement and recording of system parameters and initiating proper action.

The switching arrangement is chosen as per the requirement.
A substation has many equipments, some of these are circuit breaker, isolator, current transformer, potential transformer, surge absorber, measuring and recording instruments, relay and protective devices, batteries, battery chargers, fire fighting equipments etc.

The substation transformers can be connected several ways.  We have already discussed about transformer connection and Vector Group. We will discuss about all types of substations in future.

Symmetrical Components in Electrical Engineering


Symmetrical components method was discovered by C. L. Fortescue. The knowledge of symmetrical components is very useful for the study of unsymmetrical faults in three phase power networks. The concept  is also useful for studying the three phase machine behavior under unbalanced condition.

This article requires a little bit  knowledge about phasor representation that we already discussed.

Symmetrical Components

Why symmetrical components? Symmetrical component technique is used for analyzing unbalanced three phase systems. When the system is balanced, analysis is very simple. We do not analyze for all the three phases instead we analyze it as single phase system. So the three phase system is reduced to simpler single phase system. Symmetrical component method helps to apply the single phase analysis tools also to the unbalanced three phase system. How?

In Symmetrical component method, any unbalanced three phase system can be resolved into three sets of symmetrical components. These three sets are positive sequence, negative sequence and zero sequence.

Considering counterclockwise abc sequence as positive sequence, then acb will be negative sequence (See Fig-A). Both the positive and negative sequence components are balanced. It means that the three phasors have the same magnitude and the phase angle between any two phasors is 120 degrees. The three phasors of zero sequence are of same magnitude and aligned in the same direction. So, in case of zero sequence the angle between any two phasors is zero. All these phasors rotate counterclockwise with frequency of the system. So the relative position between the phasors remains the same. .

For identification purposes we have used the symbols +, - and 0 for positive, negative and zero sequence components respectively.

Every three phase unbalanced system can be decomposed to three balanced systems as in Fig-A.

A three phase unbalanced system is shown in Fig-B.
The unbalnced system in Fig-B can be resolved to symmetrical components like Fig-A. In Figure-C just see  how each unbalanced component is made up of +ve, -ve and 0 sequence components.

From the diagram above it is easy to verify the below equations.

Now is the time to apply the phasor operator  a  that we learned previously.
(Phasor operator when applied to a phasor rotates the phasor anticlockwise by 120 degrees).

So the above equation can be written as below

In the above equation we have eliminated both b and c phase positive, negative and zero sequence components.

When the unbalanced system is known. we know Va, Vb and Vc.
Of course we also know the value of phasor operator a which is constant.

So the above three equations has three unknown Va+, Va- and Va0.
We can solve the equations and find the three unknowns by using school maths.

Now from Va+, Va- and Va0 that we calculated we can construct the full symmetrical components as in Fig-A. It will simplify for per phase analysis.

Operators j and a in Electrical Engineering

We have already discussed about phasors and its simple properties. Perhaps now it is the time that we want to explore a little more. Every effort is made to keep it as simple as our previous article. 

Before proceeding further I want to clarify that here we are mainly concerned about phasor multiplication and 'j' and 'a' operators.This article will also help us better appreciate the use of symmetrical components ( for analysis of unbalanced 3-phase systems)  and subsequently other phenomena  in transformer and AC circuits.

We know that phasor in the form x+j y is drawn as an arrow from origin to (x, y) point.

Till now I represented the phasor in x+j y form also called rectangular form. A phasor can also be represented in polar form. In the polar form we also need two parameters, these are length of phasor (r) and angle(phi) it makes with the +ve horizontal axis . See the Figure-A.

Phasor Multiplication

I have already discussed the use of j in phasor representation.
We know that j is equal to square root of -1.

or    j = sqrt(-1) so j.j = -1

Now consider two phasors A = 2 + j 3 and B = -1 + j 2

Let us multiply A and B

 A.B = (2+j 3) . (-1 + j2) = -2 + j 4 -j 3 + j.j (3.2) = -2 + j 1 - 6 = -8 + j 1 

Directly multiply each of real and imaginary parts from A with each of B. It is simple!

It is even easier to multiply in polar form. See the example below. As illustrated in figure-A,  we represent below the phasors A and B in the polar form. For phasor A, 4 is its length and it makes 20 degrees with x-axis. Similarly B is of length 3 units and it makes 40 degrees with +ve horizontal axis

( 20, 40 and 60 are angles in degree)

Representing in polar form, the multiplication has become extremely easy.
Just multiply the lengths and add the angles to get the new phasor. 
You can convert it back to the rectangular form.

                                      A.B = 12(cos 60 + sin 60)   
j and a Operators

What we will get, if a phasor is multiplied with j?

for example
      if    A = 3 + j 4
Then    j A =j(3 + j 4) = j 3 + j.j 4 = -4 + j 3   ( As j.j =-1)

Now draw the phasor -4 + j 3. It will be observed that the angle between 3 + j 4 and -4 + j 3 is 90 degrees.

Any phasor when multiplied by j  will rotate the original phasor by 90 degrees in anticlockwise direction. Now if the resultant phasor is again multiplied by j then the phasor is again rotated by 90 degrees in anticlockwise direction, so on.
In our example j(-4 + j 3) = -j4 -3 = -(3 + j 4), which is in opposite direction to (p + j q). So clearly the phasor has again undergone 90 degrees anticlockwise rotation. See the figure. Every time we apply j, we rotate the phasor counterclockwise by 90 degrees.

Now let us consider about another operator ' a ' (standard symbol). It has the capacity to rotate a phasor counterclockwise by 120 degrees. applying ' a ' twice the phasor is rotated by 240 degrees, by applying thrice the original phasor is rotated 360 degrees or one complete rotation, so the original phasor.

It is clear that as the phasor is rotated 120 degrees (magnitude remains the same) then in polar form

                              a = 1/120deg  

in rectangular form   a = 1.cos 120 + j 1.sin 120
                          or  a = -0.5 + j 0.866

see the Fig-C how a phasor A is rotated by 120 degrees when applied with operator a.

I colored them red green and blue to recall our balanced three phase system.

Clearly we are able to get the phasors B and C  by applying the operator a repeatedly on phasor A.
Otherwise we can say that, the balanced system of A-B-C sequence can be equally represented in terms of 'a' and A only. The operator a will be used more in our article symmetrical components.

Fourier Series in Electrical Engineering

The Fourier Series deals with periodic waves and named after J. Fourier who discovered it.

The knowledge of Fourier Series is essential to understand some very useful concepts in Electrical Engineering.Fourier Series is very useful for circuit analysis, electronics, signal processing etc. . The study of Fourier Series is the backbone of Harmonic analysis. We know that harmonic analysis is used for filter design, noise and signal analysis. etc.. Harmonic analysis is also very important in power system studies. In power network, harmonics are mainly generated by non-linear elements and switching equipment. Although it is a applied mathematics topic but like our previous article here also we will try to minimise the maths and depict in a simpler way.

The Fourier Series deals with periodic waves and named after J. Fourier who discovered it.

Fourier Series 

The Fourier series is concerned with periodic waves. The periodic wave may be rectangular, triangular, saw tooth or any other periodic form(single valued). Here I will also call this periodic wave as signal wave.Any periodic signal wave can be represented as a sum of a series of sinusoidal waves of different  frequencies and amplitudes.

Otherwise we can also say that a series of sinusoidal waves of different frequencies and amplitudes add up to give a periodic wave of non-sinusoidal form.

Let us consider a periodic signal wave v . According to the definition of Fourier series, this periodic signal wave can be written as sum of sinusoidal waves as below.

Now the question is how can we find the coefficients

I am not going to show you the details of how I obtained the formulas, but you can remember the three general  formulas as shown below for obtaining the coefficients. Many of you may not require to find the harmonics by using the formulas below, but those who are interested can use these formulas. Of course, students are required to remember these formulas.

                                                         n = 1, 2, 3, 4 .....

Putting the values of n we obtain different coefficients. For example if n=1 we get a1 and b1.

Some knowledge about the properties of the Fourier series will immensely help you. In most cases signal waves maintain symmetry. Depending on the symmetry of the wave we may not be always required to find all the sine and cosine terms coefficients. So now I will guide you through some important properties that you should remember so that just looking at the signal wave you can immediately say which coefficients should be present in the series.

Let us first of all talk a little about harmonics.

What is harmonic ?

Every periodic wave has a Time period(T) which is one complete cycle. The whole signal is the repetition of this period. Frequency (N) of the wave is the number of complete cycles in one second.

clearly  N = 1 / T

Observe the general Fourier series, it has a component (a1 sin wt), this sinusoid has the same frequency as the actual signal wave and it is called the fundamental component. The next component is (a2 sin 2wt), its frequency is twice that of fundamental (or original signal wave), this sinusoid is called second harmonic. So also the third, fourth etc.

As an example if the fundamental wave has a frequency of 60 Hz, then the frequency of second harmonic is 120 (2* 60) Hz, third harmonic 180 Hz, 9th harmonic will be 540 Hz etc. The frequency of nth harmonic is (n.60).

Properties of Fourier Series

We will now consider some important and very useful properties of Fourier series
  • In the fourier Series the constant term a0 will not appear if the signal wave average value in one period is zero. (one period is T which is equal to 2PI)

               Looking at the figure it is clear that area bounded by the Square wave above and below t-axis are
               A1 and A2 respectively. Here A1=A2, so the average is zero. The wave has a0 equal to zero. You
               can also confirm by using the formula above. But shifting the same wave in vertical direction as
               redrawn in Fig-B  will automatically introduce a0 in Fourier series.

              Yes by integrating using the formula above you will find a0. By shifting I have actually disturbed  the
              symmetry around horizontal axis and introduced the average value a0.  Here in this figure a0 is exactly how
              much I shifted the  whole wave in vertical direction. This ais also called  DC component of the signal
    • The Fourier series having sine and cosine terms can be combined. 
             We know that if,    p =  a1 sin x +b1 cos x        then it can be written in the form
                                          p = r sin (x + phi)               phi is the angle displacement from sin x.
             Using school trigonometry r & phi are found from a1 and b1
             (Remember that same harmonics are combined in this form. sin x with cos 2x or cos x with sin 2x etc.
             are not combined in above form).

    • Considering the square wave in the above Fig-A we will get only terms with sine function. There will be  no cosine terms because the coefficients a1, a2 etc will be zero. This you can verify from above formula also. The Fourier series is:

             As the above square wave maintains symmetry about origin so it will be composed of sine waves only.
            (sine waves are symmetric about origin). In Fig-C, I have reproduced the square wave. Here I have  
            calculated only three terms of the Fourier series ' v ', the fundamental, third harmonic and fifth harmonic. 
            The blue curve is the sum of these three terms of the Fourier series of the square wave shown. You can 
            see how just by considering three terms of the Fourier series, the blue curve approximate the square 
            wave. In Fig-D the sum of Fourier series(in blue) is drawn by calculating more coefficients( 9 ). Observe
            how the blue curve approximates the suare wave so that I had to remove the square wave from the figure for clarity. Taking more terms the curve will be even smoother.

    • Again consider the square wave. I have now chosen the vertical axis by shifting pi/2 to the right  and redrawn in Fig-E. Now applying the above equations for getting the coefficients, we will find that the a0 and the sine term coefficients b1, b2...etc has vanished and only cosine terms are there in the series. 
              The Fourier series is:

             It is very important to compare Figure-C and Figure-E. In both cases the square waves are identical.
             Only the vertical axes are chosen differently in each cases. Comparing the individual harmonics and their
             sum (blue)  in both the figures it is clear that actually choosing the position of vertical axis does not
             change the component harmonics but what previously you were getting as sine waves have now
             become cosine waves automatically. This is only due to the choice of vertical axis. 
             Both are sinusoidal waves and both are correct. If some intermediate position of axis is chosen, we will
             get both sine and cosine terms. The sine and cosine terms can be combined as above to get only either
             phase shifted sine(or cosine) terms for each harmonic.

             So the choice of axis does not change the shape or number of harmonics, only mathematically the series
             is different.

             So it is important to realize that for a particular shape of signal wave the harmonics are fixed, whether
             you want to show them as series of sine terms or cosine terms or both, it does not matter. Of course in
            exam you may not be allowed to choose axis. 
    • If the signal wave has quarter wave symmetry, then the Fourier expansion will contain only odd harmonics like 3rd, 5th etc.. To test quarter wave symmetry look at the signal wave at Fig-A. Draw an imaginary vertical line passing through T/4. For the +ve half wave if one side of this vertical line is mirror image of other side then the wave shows quarter wave symmetry.
             Note: A sine wave is symmetric with respect to origin (odd function) and cosine is symmetric with
                       respect to y-axis (even function). Combination of only sine waves will be symmetric with respect
                      to origin (odd function) and combination of cosine waves will remain symmetric with respect to
                      y-axis (even function)

    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.

    Transformer Zig-Zag Connection

    We have already discussed all the main configurations of transformer vector groups. In the table of groups we also included transformer zig-zag  connections. Now I will show you how that is achieved. The transformer primary or secondary can be connected in zig-zag . Zig-zag connection is sometimes desired as the disadvantages in star and delta connection can be overcome by the use of zig-zag connection of transformer. Here we discuss only the techniques of achieving some connection.

    Transformer connection is a good place for confusion. Applying these few points that we adopted will help understand the connections better.

    ·   Coloring is done to boost the visualization.
    ·   Windings of same color are on the same limb of core. For example all the three red color windings     are on one limb of core.
    ·   The voltage developed across the windings of same color are in phase (zero phase displacement) so they are drawn parallel to each other.
    ·   The analysis is done here considering anti-clockwise ABC phase sequence.
    ·   In any limb of core, windings terminals marked with even subscript are of one polarity and odd subscript are of other polarity. So for windings A1A2, a1a2 and a3a4, the terminals A1, a1 and a3 are of one polarity and A2, a2 and a4 are of other polarity.

    Let us first consider  delta-zigzag (Dz0) connection

    Delta-zigzag (Dz0) connection

    In zig zag transformer connection, there are three windings on each of the three limbs of the core, one for primary and two for secondary. Both the windings of secondary are of equal turns.

    The windings A1A2, a1a2 and a3a4 are wound on the same limb of the core hence they are all colored ‘red’. Similarly the other windings.

    Although looking at the diagram and applying IEC coding, you can easily verify the Dz0 connection, still you might find it difficult to draw. How can we obtain Dz0 connection?

    The easy way is first draw the phasor diagram and then derive the windings connection required for getting the desired phasor diagram or vector group.

    First question is what we need? Here for Dz0, the phase difference between primary and secondary is 0 degree. We have to connect the windings in such a way so that it will give zero degree phase displacement between the primary and secondary (or say the primary and secondary are in phase).

    See the diagram how connections are done to achieve a zigzag connection in the secondary.

    In the primary side A2, B2 and C2 are the terminals brought out at the transformer bushings. In the secondary side a4, b4 and c4 are the terminals brought out at the transformer bushings. Other terminals are internally connected.

    Actually for realizing the connection in secondary side the following sequence will help you.

    ·    Connect the primary side in Delta as usual or as we did in last article.

    ·    Then draw Delta(primary) side phasors.

    ·    NA2 phasor corresponds to a phase voltage in primary side. N is the virtual neutral, which does not exist physically in delta side, but found geometrically from the diagram. For obtaining the Dz0 configuration, the secondary ‘z’ side phasor diagram should be such so that the corresponding phase voltage (na4 here) phasor should be parallel to NA2.  This zero phase displacement can be obtained by connecting the windings a3a4 with b2b1 (in other limb) in series. Clearly b1 should be connected to a3 and not the other ways.

    ·    The resultant phasor is the sum of the two phasors.
                  na4 = b2b1 + a3a4
    Voltage phasors b2b1 and B1B2 are out of phase (180 degree phase difference), so the arrow head direction. Above addition is the phasor addition and not the arithmetic one.

    ·    Similarly obtain the resultant phasors for other two phases by recognizing the symmetry.

    This way we get the secondary neutral point 'n' by connecting a2, b2 and c2 together. Of course unlike the primary side neutral 'N',  here the secondary side neutral 'n' is real and brought out at the bushing.

    So in this way we can realize the connection of windings from the phasor diagram. Accordingly the windings are connected. It is important that we realised the connection from the phasor diagram and rearranged the windings in Delta and Zigzag shape for better view.

    Delta-Zigzag (Dz6) connection 

    Here the primary side connection is same as previous case. In the secondary side we have just reversed the direction of phasors in previous case and automatically get Dz6 vector group. Of course now the phasors are rearranged to obey the rules of phasor addition.

    For phase 'a', na3=b1b2+a4a3. Compare it with previous Dz0 case.

    • Now the phasors are reversed and b2 is required to be connected to a4 (in Dz0 case b1 is connected to a3
    • Terminal a3 not (a4) is brought out at the transformer bushing
    • Secondary side neutral 'n' is obtained by connecting a1, b1 and c1 together
    Now the secondary side windings are connected as per the phasor diagram. See fig-B

    Star-Zigzag (Yz1) connection

    Below (fig-C) is the connection and phasor diagram of Yz1 notation. It is left for the reader to verify the vector group. The reader should also practice the connections for other vector groups and corresponding phasor diagrams.

    Transformer Vector Groups

    The three phase transformer windings can be connected several ways. Based on the windings' connection, the vector group of the transformer is determined.

    The transformer vector group is indicated on the Name Plate of transformer by the manufacturer.
    The vector group indicates the phase difference between the primary and secondary sides, introduced due to that particular configuration of transformer windings connection.

    The Determination of vector group of transformers is very important before connecting two or more transformers in parallel. If two transformers of different vector groups are connected in parallel then phase difference exist between the secondaries of the transformers and large circulating current flows between the two transformers which is very detrimental.

    The three phase transformer primary and secondary windings are mainly connected in the following ways
    • Wye - Wye (also called Star-Star)
    • Wye - Delta (also called Star-Delta)
    • Delta -Wye ( also called Delta-Star)
    • Delta - Delta
    The Star connection is also called Wye as it resembles the English letter 'Y'. As both the names Star and Wye are equally used we have the freedom to use them interchangeably. Of course some people also use the term 'Mesh' in place of 'Delta'. Let us first consider the Wye - Delta type where three primary windings are  connected in Wye and the three secondary windings in Delta.

    For this whole article you have to remember few points below to enhance learning. It is applicable for both single unit type and single-phase bank of transformer type.
    • The windings A1A2 and a1a2 are wound on the same limb of core. So also the other two sets of windings. (In case of 3-phase bank of transformers the two windings correspond to same single phase transformer). 
    • The primary and secondary  windings on the same limb of the core are shown with same color. 
    • The windings on Delta and Star sides are diagrammatically rearranged in Delta and Star like shapes(according to connection) respectively just to enhance learning.
    • The voltage developed in the windings shown with same color(placed on same limb of core) are in phase(or zero phase displacement). Hence the corresponding phasors are drawn parallel to each other.

    Wye - Delta (Star-Delta) transformer

    The windings in the primary are connected in Wye(Star) and the secondary windings are connected in Delta.

    In the primary side the three windings are A1-A2, B1-B2 and C1-C2.
    Similarly the three secondary windings are a1-a2, b1-b2 and c1-c2.

    It should be noted that both the windings A1-A2 of primary and a1-a2  of secondary are wound on the same limb of core. The naming of the terminals has been done according to their polarity. Other wise you can imagine that when A2 is positive with respect to A1, then also a2 is positive with respect to a1. Think similarly for the other windings.

    See carefully the diagram below. A2,B2,C2 and a2,b2,c2 are respectively the primary and Secondary side terminals taken out side of transformer.

    In the primary side the three windings are connected in star. So we have shorted A1, B1 and C1. This is the primary side (star side) neutral 'N'. In the secondary side the three windings are connected in delta. Here windings a1-a2 and A1-A2 are wound on the same limb of the core, so the corresponding voltage waves are in phase. Hence we have drawn a1-a2 parallel to A1-A2. similarly windings b1-b2 is drawn parallel to B1-B2 and c1-c2 drawn parallel to C1-C2. To see the actual physical placing of the windings on the core limbs of transformer see my (archived) article Three PhaseTransformer Basics. There also you can find one example for a bank of three single phase transformers used as three phase transformer.

    In the phasor diagrams we have drawn primary side voltage phasors A1A2, B1B2 and C1C2. As usual for three phase system, these are the phasors displaced 120 degree from each other.Similarly in the secondary side voltage phasors a1a2, b1b2 and c1c2 are drawn. Just observe that a1a2 is parallel to A1A2, b1b2 is parallel to B1B2 and c1c2 is parallel to C1C2. I repeat here, that, this is because a1a2 and A1 A2 are in phase (as they are wound on the same limb of core). Similarly b1b2 and B1B2 are in phase and also c1c2 and C1C2 are in phase.

    In the delta side we have so arranged that the phasors form the Delta. In the winding connection diagram a2 is connected to b1 so in the phasor diagram a2 and b1 are joined. Similarly by joining other two phasors according to their winding connection, we will automatically get the above phasor diagram.

    The neutral (star point) physically exist in the star side . In the delta side physically the neutral point does not exist so it cannot be brought out. The delta side neutral is the imaginary point 'n' (geometrically found) which is equidistant from a2, b2 and c2.

    c2a2, a2b2 and b2c2 are the line voltages in secondary delta side. So na2, nb2 and nc2 are the phase voltages in secondary side.

    Now compare the primary side vector diagram and secondary side vector diagram. From the diagram it is clear that as if the secondary side  phasor triad  has been rotated counterclockwise with respect to primary side. From the geometry it can be confirmed that this angle is 30 degree. As the phasors are rotating counterclockwise, so the secondary side phasor a2n (phase voltage) lead the primary side phasor A2N (phase voltage) by 30 degree.

    The transformers are classified into different Vector Groups depending on this phase difference between the primary and secondary sides, obtained due to different connection philosophy.

    IEC has devised the standard code for determination of transformer vector group.
    According to IEC the code for vector group consist of 2 or more letters followed by one or two digits.

    • The first letter is Capital letter which may be Y, D or Z, which stands for High voltage side Star, Delta or interconnected Star windings respectively.
    • The second letter  is a small letter which may be y, d or z which stands for low voltage side Star, Delta or interconnected Star windings respectively.
    • The third is the digits which stands for the phase difference between the high voltage and low voltage sides.

    From the above three points, the first two are quite straightforward. The third one follows the clock convention as described below.

             In this convention the  transformer high voltage side phase voltage (line to Neutral) represented by Minute hand is fixed at 12 O'clock position and the low voltage side phase voltage (line to neutral) is represented by the Hour hand which is free to move. Clearly when the minute hand is fixed at 12 position the hour hand can take only twelve numbers of discrete positions 1, 2, 3 ... upto 12 (think it  twice). The angle between any two consecutive numbers in a clock is 30 degrees (360/12). Hence the angle between hour and minute hands can only be multiples of 30 degrees. See the figure.

    Note: Remember that in star and zig-zag connection the neutral point exist physically and in delta connection the neutral does not exist physically and called virtual. But the line to neutral voltage can always be calculated algebraically/geometrically.

    Now back to our discussion of Star-Delta transformer. We have already shown that the low voltage secondary side phasor a2n leads the high voltage primary side phasor A2N by 30 degree. (remember that the comparison is between the phase voltages). According to the clock convention this specific case represent 11 O' clock. So the above transformer connection can be represented by the symbol Yd11(or YNd11). N or n may be used for a brought out neutral. Here we will keep the material simple and will not mention the neutral symbol.

    Let us change the connection slightly to get the Yd1 vector group. See Fig-B, here the primary side is as before, but in the secondary side a1 is connected to b2 etc. (compare with previous diagram).

    In the above diagram the individual phasors are still the same as in Yd11 case. Here we have only rearranged the phasors of delta side, only to satisfy the connection changes in the secondary side. Here the clock face indicate One O' clock. As a result we obtain the Yd1 vector symbol.

    Let us consider another important connection, Primary in Delta and Secondary Star connected.

    Delta-Wye (Delta-Star) connection

    Here the primary windings are connected in Delta and the secondary windings are connected in Star or Wye. The naming convention is similar to the Wye-Delta transformer.
      In the figure-C see how the windings of primary and secondary sides are connected in Delta and Star respectively. Also see the corresponding phasors.  In the Delta side each winding is subjected to line voltage, but in Wye side each winding is subjected to phase voltage (V/1.73).

      As already told and shown, although the neutral is not physically available in Delta side, but neutral point 'n' can be found geometrically . The arrow NA2 is the phasor representing phase voltage of high voltage side (primary). In the Star side(low voltage side) arrow na2 is clearly the phasor representing the phase voltage of low voltage side.

      From the diagram applying school geometry it is clear that na2 phasor lags NA2 phasor by 30 degrees.

      Applying IEC coding:
      NA2 is minute hand fixed at 12 O' clock and na2 is hour hand at 1 O' clock (as the angle between the two is 30 degrees)

      So the transformer is identified with Dy1 symbol.

      Similarly just slightly modifying the connection above we can get Dy11 notation. Here we have rearranged the windings in the primary side for connection modification and convenience. See Fig-D.

      If you understand the above examples then identifying Star-Star and Delta-Delta vector group are very easy. One can reasonably say that the phase difference between the primary and secondary sides of both these cases is zero. The vector group symbols will be Yy0 and Dd0.

      Remember the connections can be two different ways. Consider the Wye-Wye connection. In Yy0 (zero phase displacement between primary and secondary) secondary side neutral is obtained by shorting the terminals a1, b1 & c1 and a2,b2 & c2 are brought out terminals. In  Yy6 (180 degree phase  displaced) the neutral is obtained by shorting a2,b2 & c2 and a1,b1&c1 are brought out terminals. See Fig-E and Fig-F.

      Transformer connections are categorised into four main groups as tabulated below

      Undoubtedly transformers belonging to the same group can be operated in parallel without any difficulty.
       It is impossible to run in parallel, transformers in Group1 and 2 with transformers in Group3 and 4.You consider any one from group 1 or 2 and any one from group 3 or 4 and see the phase difference, which inhibit their paralleling.

      Also transformers in group1 and group2 cannot be operated in parallel as there is 180 degree phase difference between the two secondary windings. This can only be rectified by changing internal connection.

      Similarly if group3 and group4 transformers will be connected in parallel then there will be 60 degrees phase difference between their secondary windings. But with transformer external connection modification the phase difference of secondaries can be made zero. So group3 and group4 transformers can be operated in parallel with some external modification.