Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of Inductive Reactance Vs Frequency

Variation of Inductive Reactance Vs Frequency

We know that inductive reactance XL= 2πfL means inductive reactance is directly proportional to frequency ( XL&prop ƒ ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit acts as a short circuit; but when frequency increases; inductive reactance also increases. At infinite frequency, inductive reactance becomes infinity and circuit behaves as open circuit. It means that, when frequency increases inductive reactance also increases and when frequency decreases, inductive reactance also decreases. So if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in the figure above.

Variation of Capacitive Reactance Vs Frequency

Variation of Capacitive Reactance Vs Frequency

It is clear from the formula of capacitive reactance XC = 1 / 2πfC that, frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behaves as open circuit and when frequency increases and becomes infinite, capacitive reactance decreases and becomes zero at infinite frequency, at that point the circuit acts as short circuit, so the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency, it is an hyperbolic curve as shown in figure above.

Inductive Reactance and Capacitive Reactance Vs Frequency

Inductive Reactance and Capacitive Reactance Vs Frequency

From the above discussion, it can be concluded that the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency, i.e at low frequency XL is low and XC is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency, then there must occur a point where these two graphs cut each other. At that point of intersection, the inductive and capacitive reactance becomes equal and the frequency at which these two reactances become equal, is called resonant frequency, fr.
At resonant frequency, XL = XL

At resonance f = fr and on solving above equation we get,

Variation of Impedance Vs Frequency

Variation of Impedance Vs Frequency

At resonance in series RLC circuit, two reactances become equal and cancel each other. So in resonant series RLC circuit, the opposition to the flow ofcurrent is due toresistance only. At resonance, the total impedance of series RLC circuit is equal toresistance i.e Z = R, impedance has only real part but no imaginary part and this impedance at resonant frequency is called dynamic impedance and this dynamic impedance is always less than impedance of series RLC circuit. Before series resonance i.e before frequency, fr capacitive reactance dominates and after resonance, inductive reactance dominates and at resonance the circuit acts purely as resistive circuit causing a large amount of current to circulate through the circuit.

Resonant Current

Resonant Current

In series RLC circuit, the total voltage is the phasor sum of voltage across resistor, inductorand capacitor. At resonance in series RLC circuit, both inductive and capacitive reactance cancel each other and we know that in series circuit, the current flowing through all the elements is same, So the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit,voltage across resistor is equal to supply voltage i.e V = Vr.

In series RLC circuit current, I = V / Z but at resonance current I = V / R , therefore the currentat resonant frequency is maximum as at resonance in impedance of circuit is resistance only and is minimum.

The above graph shows the plot between circuit current and frequency. At starting, when the frequency increases, the impedance Zc decreases and hence the circuit currentincreases. After some time frequency becomes equal to resonant frequency, at that point inductive reactance becomes equal to capacitive reactance and the impedance of circuit reduces and is equal to circuit resistance only. So at this point, the circuit current becomes maximum I = V / R. Now when the frequency is further increased, ZL increases and with increase in ZL, the circuit current reduces and then the current drops finally to zero as frequency becomes infinite.

Power Factor at Resonance

Power Factor at Resonance

At resonance, the inductive reactance is equal to capacitive reactance and hence the voltageacross inductor and capacitor cancel each other. The total impedance of circuit is resistanceonly. So, the circuit behaves like a pure resistive circuit and we know that in pure resistive circuit, voltage and the circuit current are in same phase i.e Vr , V and I are in same phase direction. Therefore, the phase angle between voltage and current is zero and the power factor is unity.

Application of Series RLC Resonant Circuit

Since resonance in series RLC circuit occurs at particular frequency, so it is used for filtering and tuning purpose as it does not allow unwanted oscillations that would otherwise cause signal distortion, noise and damage to circuit to pass through it.

Summary
For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance:

1. Inductive reactance XL is equal to capacitive reactance XC.
2. Total impedance of circuit becomes minimum which is equal to R i.e Z = R.
3. Circuit current becomes maximum as impedance reduces, I = V / R.
4. Voltage across inductor and capacitor cancels each other, so voltage across resistor Vr = V, supply voltage.
5. Since net reactance is zero, circuit becomes purely resistive circuit and hence thevoltage and the current are in same phase, so the phase angle between them is zero.
6. Power factor is unity.
7. Frequency at which resonance in series RLC circuit occurs is given by

Delta - Star Transformation
Star - Delta Transformation

Three branches in an electrical network can be connected in numbers of forms but most common among them is either star or delta form. In delta connection, three branches are so connected, that they form a closed loop. As these three branches are connected nose to tail, they form a triangular closed loop, this configuration is referred as delta connection. On the other hand, when either terminal of three branches is connected to a common point to form a Y like pattern is known as star connection. But these star and delta connections can be transformed from one form to anoother. For simplifying complex network, delta to star or star to delta transformation is often required.

Delta - Star Transformation

The replacement of delta or mesh by equivalent star connection is known as delta - star transformation. The two connections are equivalent or identical to each other if the impedance is measured between any pair of lines. That means, the value of impedance will be the same if it is measured between any pair of lines irrespective of whether the delta is connected between the lines or its equivalent star is connected between that lines.

Consider a delta system that's three corner points are A, B and C as shown in the figure.Electrical resistance of the branch between points A & B, B & C and C & A are R1, R2 and R3respectively. The resistance between the points A & B will be,

Now, one star system is connected to these points A, B, and C as shown in the figure. Three arms RA, RB and RC of the star system are connected with A, B and C respectively. Now if we measure the resistance value between points A and B, we will get,

Since the two systems are identical, resistance measured between terminals A and B in both systems must be equal.

Similarly, resistance between points B and C being equal in the two systems,

And resistance between points C and A being equal in the two systems,

Adding equations (I), (II) and (III) we get,

Subtracting equations (I), (II) and (III) from equation (IV) we get,

The relation of delta - star transformation can be expressed as follows.

The equivalent star resistance connected to a given terminal, is equal to the product of the two delta resistances connected to the same terminal divided by the sum of the delta connected resistances.

If the delta connected system has same resistance R at its three sides then equivalent starresistance r will be,

Star - Delta Transformation

For star - delta transformation we just multiply equations (v), (VI) & (VI), (VII) & (VII),(V) that is by doing (v)X(VI) + (VI)X(VII) + (VII)X(V) we get,

Now dividing equation (VIII) by equations (V), (VI) and equations (VII) separately we get,

Three Phase Vs Single Phase
Star Connection
Delta Connection

There are two types of system available in electric circuit, single phase and three phase system. In single phase circuit, there will be only one phase, i.e thecurrent will flow through only one wire and there will be one return path called neutral line to complete the circuit. So in single phase minimum amount of power can be transported. Here the generating station and load station will also be single phase. This is an old system using from previous time.

In 1882, new invention has been done on polyphase system, that more than one phase can be used for generating, transmitting and for load system. Three phase circuit is the polyphase system where three phases are send together from the generator to the load. Each phase are having a phase difference of 120°, i.e 120° angle electrically. So from the total of 360°, three phases are equally divided into 120° each. The power in three phase system is continuous as all the three phases are involved in generating the total power. The sinusoidal waves for 3 phase system is shown below

The three phases can be used as single phase each. So if the load is single phase, then one phase can be taken from the three phase circuit and the neutral can be used as ground to complete the circuit.

Why Three Phase is preferred Over Single Phase?

There are various reasons for this question because there are numbers of advantages over single phase circuit. The three phase system can be used as three single phase line so it can act as three single phase system. The three phase generation and single phase generation is same in the generator except the arrangement of coil in the generator to get 120° phase difference. The conductor needed in three phase circuit is 75% that of conductor needed in single phase circuit. And also the instantaneous power in single phase system falls down to zero as in single phase we can see from the sinusoidal curve but in three phase system the net power from all the phases gives a continuous power to the load.

Till now we can say that there are three voltage source connected together to form a three phase circuit. And actually it is inside the generator. The generator is having three voltage source s which are acting together in 120° phase difference. If we can arrange three single phase circuit with 120° phase difference, then it will become a three phase circuit. So 120° phase difference is must otherwise the circuit will not work, the three phase load will not be able to get active and it may also cause damage to the system.

The size or metal quantity of three phase devices is not having much difference. Now if we consider the transformer, it will be almost same size for both single phase and three phase because transformer will make only the linkage of flux. So the three phase system will have higher efficiency compared to single phase because for the same or little difference in mass of transformer, three phase line will be out whereas in single phase it will be only one. And losses will be minimum in three phase circuit. So overall in conclusion the three phase system will have better and higher efficiency compared to the single phase system.

In three phase circuit, connections can be given in two types:

1. Star connection
2. Delta connection

Star Connection

In star connection, there is four wire, three wires are phase wire and fourth is neutral which is taken from the star point. Star connection is preferred for long distance power transmission because it is having the neutral point. In this we need to come to the concept of balanced and unbalanced current in power system.

When equal current will flow through all the three phases, then it is called as balanced current. And when the current will not be equal in any of the phase, then it is unbalanced current. In this case, during balanced condition there will be no current flowing through the neutral line and hence there is no use of the neutral terminal. But when there will be unbalanced current flowing in the three phase circuit, neutral is having a vital role. It will take the unbalanced current through to the ground and protect the transformer. Unbalanced current affects transformer and it may also cause damage to the transformer and for this star connection is preferred for long distance transmission.
The star connection is shown below-

In star connection, the line voltage is √3 times of phase voltage. Line voltage is the voltagebetween two phases in three phase circuit and phase voltage is the voltage between one phase to the neutral line. And the current is same for both line and phase. It is shown as expression below

Delta Connection

In delta connection, there is three wires alone and no neutral terminal is taken. Normally delta connection is preferred for short distance due to the problem of unbalanced current in the circuit. The figure is shown below for delta connection. In the load station, ground can be used as neutral path if required.

In delta connection, the line voltage is same with that of phase voltaage. And the line currentis √3 times of phase current. It is shown as expression below,

In three phase circuit, star and delta connection can be arranged in four different ways-

1. Star-Star connection
2. Star-Delta connection
3. Delta-Star connection
4. Delta-Delta connection

But the power is independent of the circuit arrangement of the three phase system. The net power in the circuit will be same in both star and delta connection. The power in three phase circuit can be calculated from the equation below,

Since there is three phases, so the multiple of 3 is made in the normal power equation and the PF is power factor. Power factor is a very important factor in three phase system and some times due to certain error, it is corrected by using capacitors.