AC Voltage

AC (alternating current) voltage is a form of electrical voltage that reverses direction periodically, in contrast to DC (direct current) voltage, which flows continuously in one direction. The most prevalent waveform for AC voltage is the sine wave, which changes smoothly over time. AC voltage is commonly utilized in electrical power distribution due to its capacity to be transformed to different voltage levels with transformers, enhancing the efficiency of long-distance transmission. This ability to adjust AC voltages enables its versatile application across a wide range of uses.

1.0AC Voltage Across Resistor

• Let at any instant t, the current in the circuit is I
• Potential difference across the resistance =IR
• With the help of Kirchhoff's circuit law,

$R-IR=0\Rightarrow E_{0}sin\omega t=IR$

$I=\frac{E_0}{R}\mathrm{Sin}\omega t=I_0\mathrm{Sin}\omega t\left(I_0=\frac{E_0}{R}=\text{Peak or Maximum Value of Current }\right)$

• Alternating current developed in a pure resistance is also of the sinusoidal nature.  
• In a.c. circuits containing pure resistance, the voltage and current are in the same phase. 
• The vector or phasor diagram which represents the phase relationship between alternating current and alternating e.m.f.. 

• In the a.c. circuit having R only, as current and voltage are in the same phase, hence in fig. Both phasors E₀ and I₀ are in the same direction, making an angle with OX. Their projections on the Y-axis represent the instantaneous values of alternating current and voltage. 

$I=I_{0}sin\omega t$

$E=E_{0}sin\omega t$

$I_0=\frac{E_0}{R}$,

hence

$\frac{I_0}{\sqrt{2}}=\frac{E_0}{R\sqrt{2}}\Rightarrow I_{RMS}=\frac{E_{RMS}}{R}$

2.0AC Voltage Across Capacitor

• A circuit containing an ideal capacitor of capacitance C connected with a source of alternating emf as shown in fig. The alternating e.m.f. in the circuit $E=E_{0}sin\omega t$
• When alternating e.m.f. is applied across the capacitor a similarly varying alternating current flows in the circuit. 
• The two plates of the capacitor become alternately positively and negatively charged and the magnitude of the charge on the plates of the capacitor varies sinusoidally with time. Also the electric field between the plates of the capacitor varies sinusoidally with time. Let at any instant t charge on the capacitor = q
• Instantaneous potential difference across the capacitor, $E=\frac{q}{c}$

$q=CE\Rightarrow q=C{E}_0\mathrm{Sin}\omega t$

The instantaneous value of current

$I=\frac{dq}{dt}=\frac{d}{dt}\left(C{E}_0\mathrm{Sin}\omega t\right)=C{E}_0\omega \mathrm{Cos}\omega t$

$\Rightarrow I_0\mathrm{Sin}\left(\omega t+\frac{\pi }{2}\right)\text{ where }{I}_0=\omega C{E}_0$

• In a pure capacitive circuit, the current always leads the e.m.f by a phase angle of $\frac{\pi }{2}$. The alternating e.m.f lags behind the alternating current by a phase angle of $\frac{\pi }{2}$

• $\frac{E}{I}$ is the resistance R when booth E and I are in phase, in present case they differ in phase by $\frac{\pi }{2}$, hence $\frac{1}{\omega C}$ is not the resistance of the capacitor, the capacitor offers opposition to the flow of A.C.
• This non-resistive opposition to the flow of A.C in a pure capacitive circuit is known as capacitive reactance XC. $X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}$
• Unit of $X_c$: Ohm
• It is inversely proportional to frequency of A.C 
• XC decreases as the frequency increases.    

• For d.c circuit f=0

$X_C=\frac{1}{2\pi fC}=\infty$ but has a very small value for a.c

This shows that the capacitor blocks the flow of d.c but provides an easy path for individual components(R or L or C).

3.0AC Voltage Across Inductor

• A circuit containing a pure inductance L (having zero ohmic resistance)connected with a source of alternating emf. Let the alternating e.m.f. $E=E_0\sin \omega t$
• When a.c. flows through the circuit, emf induced across inductance = $-L\frac{dI}{dt}$

Note: Negative sign indicates that induced emf acts in opposite direction to that of applied emf.Because there is no other circuit element present in the circuit other than inductance so with the help of Kirchhoff's Circuit Law,

$E+\left(-L\frac{dI}{dt}\right)=0\Rightarrow E=L\frac{dI}{dt}$

$I=\frac{E_0}{\omega L}\mathrm{Sin}\left(\omega t-\frac{\pi }{2}\right)$

Maximum Current

$I_0=\frac{E_0}{\omega L}\times 1=\frac{E_0}{\omega L}$

Hence

$I=I_0\mathrm{Sin}\left(\omega t-\frac{\pi }{2}\right)$

• In a pure inductive circuit current always lags behind the emf by $\frac{\pi }{2}$ or alternating emf leads the a. c. by a phase angle of $\frac{\pi }{2}$

• Expression $I_0=\frac{E_0}{\omega L}$ resembles the expression $\frac{E}{I}=R$
• This non-resistive opposition to the flow of A.C. in a circuit is called the inductive reactance (X_L) of the circuit

$X_L=\omega L=2\pi FfL$  where f=frequency of A.C

• Unit of $X_L-Ohm$
• Inductive Reactance $X_L$ ∝ $f$ (Higher the frequency of A.C. higher is the inductive reactance offered by an inductor in an A.C. circuit.)
•  For d.c Circuit $f=0,X_L=X_L=2\pi FfL=0$
• Hence inductor offers no opposition to the flow of d.c whereas a resistive path to a.c

4.0AC Voltage Across Series LCR

• A circuit containing a series combination of an resistance R, a coil of inductance L and a capacitor of capacitance C, connected with a source of alternating e.m.f. of peak value $E_0$.

5.0Phasor Diagram for Series LCR Circuit

• Let in series LCR circuit applied alternating emf is $E=E_{0}sin\omega t$. As L, C and R are joined in series,as a result ,the current flowing through the three elements at any given moment has the same amplitude and phase.
• However, voltage across each element bears a different phase relationship with the current
• Potential difference across L,C and R 

$V_L=I{X}_L$ $V_C=I{X}_C$ $V_R=IR$

• Now, V_R is in phase with current I but V_L leads I by 90°,While V_C lags behind I by 90°.
• The vector OP represents V_R (which is in phase with I) , the vector OQ represents V_L (which leads I by 90°) and the vector OS represents V_C (which legs behind I by 90°) V_L and V_C are opposite to each other.

• If V_L > V_C (as shown in figure) then their resultant will be (V_L – V_C) which is represented by OT.  Finally, the vector OK represents the resultant of V_R and (V_L – V_C), that is, the resultant of all the three applied e.m.f

$E=\sqrt{V_R^2+{\left(V_L-V_C\right)}^2}=I\sqrt{R^2+{\left(X_L-X_C\right)}^2}$

$I=\frac{E}{\sqrt{R^2+{\left(X_L-X_C\right)}^2}}$

$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}=\sqrt{R^2+{\left(\omega L-\frac{1}{\omega C}\right)}^2}$

Phasor diagram also shown that in LCR circuit the applied e.m.f leads the current I by a phase angle $\Phi$

$\mathrm{Tan}\varphi =\frac{X_L-X_C}{R}$

Power Associated with series LCR Circuit 

$P_{\text{avg }}=V_{rms}{I}_{rms}\mathrm{Cos}\varphi =0$

6.0Sample Questions on AC Voltage

Q-1.An ideal inductor consumes no electric power in an a.c circuit .Explain?

Solution:

$P_{\text{avg }}=V_{rms}{I}_{rms}\mathrm{Cos}\varphi =0$

Average power is zero ,in case of inductor the phase angle among current and voltage is $\frac{\pi }{2}$

Q-2. What is the value of impedance of the LR circuit?

Solution: The effective opposition created due to the inductor and resistor is called impedance of the LR circuit,

$Z=\sqrt{R^2+X_L^2}=\sqrt{R^2+{\omega }^2{L}^2}$

Q-3. The frequency of an a.c source is doubled. What will be the new reactance of an inductor?

Solution:

$X_L=\omega L=2\pi fL$

$X_L\propto f$

When f is doubled $X_L$ becomes two times.