Equations of Reactance and its Implication

What is Reactance?

• Reactance is the opposition exhibited to the flow of alternating current in an inductor or capacitor due to inductance or capacitance. In a few ways, reactance is like an alternating current equivalent of direct current resistance.

• While resistance disperses energy as heat, when there is reactance to the flow of alternating current in an inductor, capacitor or conductor, the energy is stored and released from a magnetic or an electric field.

• There are two types of structures of reactance. The reactance is inductive if the field is magnetic. The reactance is capacitive if the field is electric.

• Similar to resistance, ohms is used to measure reactance with inductive structure of reactance being indicated by positive values and capacitive structure of reactance being indicated by negative values. It is represented  by the symbol X.

• An inductor or capacitor is said to be ideal if it has zero resistance whereas a resistor is said to be ideal if it has zero reactance.

Formulas of Reactance

Formula of Reactance – Inductive

• Consider an AC source with voltage (V) linked to an inductor with inductance (L). It is assumed that the resistance of the inductor is negligible and that the AC circuit is completely inductive.

• If V = V_msinwt  is the applied voltage.  (1)

• Due to the lack of a resistor in the circuit, there will exist a self-induced EMF in the inductor which will resist the change in the magnetic flux that causes it.

• V – L(di/dt) = 0   (2)

Where e =L(di/dt)

E is the self-induced EMF.

V is the source voltage.

L is the self-inductance of the inductor.

• Combining equations (1) and (2):

di/dt= V/L= V_msinwt/L   (3)

• When the current is a function of time, the slope di/dt must be sinusoidally altering with the equal phase of the voltage at the source with vm/L being the amplitude. 

• di/dt  is integrated with respect to time to obtain the current

∫di /dt dt= V_m /L∫ sin(ωt) dt

and get

i =-V_m/ωL cos(ωt) + constant

• The constant of integration has the same dimension as current and is independent of time. The current sustained by the source oscillates symmetrically about zero because the source has an EMF which oscillates symmetrically about zero. Therefore, the current does not have any time independent component, nor does any constant exist. Therefore, the integration constant is zero.

• By using -cos(ωt)= sin (ωt-π/2), we get 

i =i_m sin (ωt-π/2)+ constant (4)

Where i_m = (V_m/ ω L) is the amplitude of the current. 

• The quantity ω L is similar to resistance and is called inductive reactance, denoted by XL.

The formula of inductive reactance is therefore, XL = ω L  (5).

• The amplitude of the current is then

i_m=V_m/  XL (6)

• A comparison of Equations 1 and 6 for the source voltage and the current in an inductor shows that the current lags the voltage by /2 or one-quarter (1/4) cycle.

• The dimensional structure of inductive reactance is the same as that of resistance and its SI unit is ohm (W). 

• Similar to the way the current is limited by resistance in a purely resistive circuit, the current in a purely inductive circuit is limited by the inductive reactance.

• The inductive reactance is directly proportional to the frequency of the current and the inductance.

Formula of Reactance – Capacitive

• Consider an AC source with voltage (V) linked to an inductor with capacitance (C). It is assumed that the resistance of the inductor is negligible and that it is a purely capacitive AC circuit. 

• The source voltage or applied voltage V = Vmsinωt (1)

• When connected to an AC source, the capacitor does not completely prevent the flow of charge but limits and regulates the current. As the current is reversed every half cycle, the capacitor gets charged and discharged alternatively.

• At any instant t, the charge on the capacitor is kept as q. The voltage V across the capacitor at any instant is   

V=q/ (2)

• When the Kirchhoff’s loop rule is applied, it is seen that the voltage across the capacitor and the source are equal.

Vm sinωt=q/C (3)

• To find the current, we use the relation t=dq/dt

t=d/dt(V_m Csinωt)=ωCV_mcos(ωt)  

• By using the relation, cos(ωt)= sin(ωt+π/2), we have

t=t_m sin(ωt+π/2) (4)

where the amplitude of the oscillating current is t_m=ωCV_m.

It can be rewritten as t_m=V_m /1/ωC

• Equating it to t_m=V_m/R for a completely resistive circuit, it is seen that 1/ωC takes the place of resistance. It is given the name of capacitive reactance and is denoted by Xc

X_C=1/ωC (5)

• The current amplitude becomes t_m=V_m /V_m (6)

The dimensional structure of capacitive reactance is the same as that of resistance and its SI unit is ohm (W). 

Similar to the way the current is limited by resistance in a purely resistive circuit, the current in a purely capacitive circuit is limited by the capacitive reactance. 

It is inversely proportional to the frequency and capacitance. 

Comparing the equations (1) and (6), it can be seen that the current is /2 ahead of voltage.

Conclusion

Reactance is the opposition exhibited to the flow of alternating current in an inductor or capacitor due to inductance or capacitance. There are two types of reactance depending on the type of field.  The reactance is inductive if the field is magnetic. The reactance is capacitive if the field is electric. The current in a purely inductive circuit is limited by the inductive reactance. It is given as ω L is directly proportional to the frequency of the current and the inductance.

The current in a purely capacitive circuit is limited by the capacitive reactance. It is given by 1/ωC and inversely proportional to the capacitance and frequency.