Alternating Current Concept Page - 2

Diagram

Phasor Diagram for a resistive circuit

Formula

Current through a resistor when AC voltage is applied

i=vmRsin(ωt)$i=\frac{v_m}{R}sin(\omega t)$ where vm$v_m$ is the amplitude of the oscillating potential and ω$\omega$ is the angular frequency

Definition

Inductive reactance

The opposition to current flow through an inductor is proportional to both the amount of inductance and to the frequency of the current in the inductor. This opposition to current flow is called INDUCTIVE REACTANCE . It is written as
XL=ωL$X_L=\omega L$
where XL$X_L$ is inductive reactance, ω$\omega$ is the angular frequency and L is the inductance.

Diagram

Phasor Diagram for a inductive circuit

Formula

Instantaneous and average power for an inductive circuit

The instantaneous power supplied to the inductor is
PL=iv=imsin(ωt−π2)×vmsin(ωt)=−imvmcos(ωt)sin(ωt)=−imvm2sin(2ωt)$P_L=iv=i_{m}sin(\omega t-{\displaystyle \frac{\pi }{2}})\times v_{m}sin(\omega t)=-i_m{v}_{m}cos(\omega t)sin(\omega t)=-{\displaystyle \frac{i_m{v}_m}{2}}sin(2\omega t)$
The average power supplied to an inductor over one complete cycle is zero since average of sin(2ωt)$\sin (2\omega t)$ over a complete cycle is zero.

Definition

Explain the magnetisation and demagnetisation of an inductor on application of AC

1. 0-1: ENERGY IS ABSORBED FROM THE SOURCE.
2. 1-2: ENERGY IS BEING RETURNED TO SOURCE.
3. 2-3: ENERGY IS ABSORBED.
4. 3-4: ENERGY ABSORBED DURING THE 14${\displaystyle \frac{1}{4}}$ CYCLE 2-3 IS RETURNED TO THE SOURCE.

Formula

Current through a inductor when AC voltage is applied

i=imsin(ωt−π2)$i=i_{m}sin(\omega t-{\displaystyle \frac{\pi }{2}})$
where i$i$ is the current through the inductor,
im=vmωL$i_m={\displaystyle \frac{v_m}{\omega L}}$ is the amplitude of the current, where ω$\omega$ is the angular frequency and L$L$ is the inductance.

Definition

Capacitive reactance

Capacitive reactance is an opposition to the change of voltage across a capacitance. It is written as 
XC=1ωC$X_C=\frac{1}{\omega C}$,
where XC$X_C$ is capacitive reactance, ω$\omega$ is angular frequency and C is the capacitance.

Diagram

Phasor Diagram for a capacitive circuit

Formula

Instantaneous and average power for an capacitive circuit

The instantaneous power supplied to the capacitor is Pc=iv=imcos(ωt)vmsin(ωt)=imvmcos(ωt)sin(ωt)=imvm2sin(2ωt)$P_c=iv=i_{m}cos(\omega t)v_{m}sin(\omega t)=i_m{v}_{m}cos(\omega t)sin(\omega t)=\frac{i_m{v}_m}{2}sin(2\omega t)$
The average power is 0$0$ since sin(2ωt)$sin(2\omega t)$ is zero over a complete cycle.

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