Alternating Current

Alternating Current is a type of electrical current in which the flow of electric charge periodically reverses direction. Unlike Direct Current (DC), where the flow of electrons is unidirectional, AC alternates back and forth, typically in the form of a sine wave.In AC, the voltage varies sinusoidally over time, changing its polarity from positive to negative in a regular pattern. This is typically produced by an AC generator or an alternator.

1.0Definition of Alternating Current

An alternating quantity (such as current or voltage) is one whose magnitude continuously varies with time between zero and a maximum value, while its direction periodically reverses.

2.0Comparison of AC and DC

Alternating Current AC	Direct Current DC
Changes direction periodically	Flows only in one direction
Can be Generated by using AC Generator.	Can be Generated by using DC Generator, Battery, solar panels.
Inverter converts AC into DC.	The rectifier converts AC into DC.
Can be controlled using a Transformer.	Cannot be controlled using a Transformer.

3.0Advantage of AC

• A.C. is cheaper than D.C
• It can be transmitted over long distances at low power loss.
• It can be stepped up or stepped down with the help of a transformer.
• It can be controlled easily (by choke coil).

4.0Equation of Voltage and Current for AC

Alternating current or voltage varying as sine or cosine function.

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

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

5.0Peak Value or Maximum Value

• The peak value is defined as the maximum value of an alternating quantity (such as current or voltage.

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

• Peak Value=$I_0$  or Amplitude = $I_0$

6.0Average Value

The mean value of A.C. over any half cycle (either positive or negative) is that value of DC which would send the same amount of charge through a circuit as is sent by the AC through the same circuit at the same time.

$I_{avg}=<I>=\frac{{\int }_{t_1}^{t_2}Idt}{{\int }_{t_1}^{t_2}Idt}=\frac{AreaUnderI-TGraph}{TimeInterval}$

Physical Significance of Average Current:

$I_{avg}=\frac{{\int }_0^{t}idt}{{\int }_0^{t}dt}$

$i_{av}\times \Delta t={\int }_0^{t}idt$

Charge flown by DC of value $i_{av}$ = Charge flown by AC of value i

7.0Root Mean Square (R.M.S) Value

It is the equivalent value of DC that would generate the same amount of heat in a given resistance over a specific time period as the alternating current does when passed through the same resistance for the same duration."

$I_{rms}=\sqrt{\frac{{\int }_{t_1}^{t_2}{I}^{2}dt}{{\int }_{t_1}^{t_2}dt}}$

$I_{rms}=\frac{I_0}{\sqrt{2}}$

$I_{rms}=0.707I_0$

Significance of RMS Value

$i_{rms}^2\times R\times \Delta T={\int }_0^T{i}^{2}Rdt$

Heat produced by DC=Heat produced by AC

8.0Difference Between DC Meter and AC Meter

9.0Phasor Diagram

• Physical quantity which represents both the instantaneous value and direction of alternating quantity at any instant is called its phase.
• Its SI unit is radian. It is a dimensionless quantity.

$I=I_{0}Sin(\omega t+\varphi )$ The argument of sine is called its phase.

$\omega t$= instantaneous phase (changes with time)

$\varphi$ = initial phase or phase constant(constant w.r.t. time)

• Phase difference : The phase difference refers to the difference in phase between the current and voltage.

$V=V_{0}Sin(\omega t+{\varphi }_1)$

$i=i_{0}Sin(\omega t+{\varphi }_2)$

Phase Difference $\Delta \varphi ={\varphi }_2-{\varphi }_1$ (relative to current) or (relative to voltage)

• Time difference : If phase difference between alternating current and voltage is  then time difference between them is given as, $\frac{\Delta \varphi }{2\pi }=\frac{\Delta t}{T}$

10.0AC Circuit Containing 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,

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

$I=\frac{E_0}{R}Sin\omega t=I_{0}Sin\omega t$ ($I_0=\frac{E_0}{R}$=Peak or Maximum Value of Current

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

11.0AC Circuit Containing 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,

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

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

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

Hence $I=I_{0}Sin(\omega t-\frac{\pi }{2})$

• 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

12.0AC Circuit Containing 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,

$E=\frac{q}{C}$

$q=CE\Rightarrow q=C{E}_{0}Sin\omega t$

The instantaneous value of current

$I=\frac{dq}{dt}=\frac{d}{dt}(C{E}_{0}Sin\omega t)=C{E}_0\omega Cos\omega t$

$\Rightarrow I_{0}Sin(\omega t+\frac{\pi }{2})where{I}_0=\omega C{E}_0$

• 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 $X_c$. $X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}$
• Unit of $X_C$:Ohm
• It is inversely proportional to frequency of A.C 
• $X_C$ decreases as the frequency increases.                  

• For d.c circuit f=0

$X_C=\frac{1}{2\pi fC}$=∞ 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).

13.0Key Formula for AC Circuit containing R,L,C

14.0AC Circuit Containing RL,RC,LC Circuit

15.0AC Circuit Containing Series LCR Circuit

• 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 $E_0$.

16.0Phasor 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

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

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

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

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

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

Power Associated with series LCR Circuit  $P_{avg}=V_{rms}{I}_{rms}Cos\varphi$

Half -Power Frequency: There are two such frequencies of applied ac-source in LCR-series circuit where power consumption is exactly half of the maximum at resonance; such frequencies are called half power frequencies.

${\omega }_1$and ${\omega }_2$ are half power frequency

${\omega }_1={\omega }_0-\Delta \omega$

${\omega }_2={\omega }_0+\Delta \omega$ 

Bandwidth= $2\Delta ={\omega }_2-{\omega }_1$

At ${\omega }_1$and ${\omega }_2$$\Rightarrow P=\frac{P_{max}}{2}andI=\frac{I_{max}}{2}$

${\omega }_1{\omega }_2=\frac{1}{LC}\Rightarrow {\omega }_0=\sqrt{{\omega }_1{\omega }_2}$

${\omega }_2-{\omega }_1=\frac{R}{L}\Rightarrow 2\Delta \omega =\frac{R}{L}$

17.0Quality Factor

This factor gives relative information about stored energy and lost energy per cycle.

$Q=\frac{{\omega }_0}{Bandwidth}=\frac{{\omega }_0}{2\Delta \omega }$

$Q=\frac{{\omega }_{0}L}{R}=\frac{1}{{\omega }_{0}RC}$

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

18.0Power In AC

The rate of doing work or the amount of energy transferred by a circuit per unit time is known as power in AC circuits. It is used to calculate the total power required to supply a load.

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

$Cos\varphi =\frac{R}{Z}$$=PowerFactorofaccircuit$

• For Capacitive Circuit,$P_{avg}=0$
• For Inductive Circuit, $P_{avg}=0$
• RMS Power, $P_{rms}=V_{rms}{I}_{rms}$

19.0Wattless Current

• That component of current in ac-circuit which is not active, the component is I Sin
• Wattful Power: Average power is also known as watt full power

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

• Wattless Power: That component of current in ac-circuit which is not active. 

$P_{avg}=V_{rms}{I}_{rms}Sin\varphi$                           

20.0Power Factor

• Average power $P_{avg}=V_{rms}{I}_{rms}Cos\varphi$
• Power Factor($Cos\varphi$)=$\frac{AveragePower}{RMSPower}$

$Cos\varphi =\frac{R}{Z}$

21.0Sample Questions on Alternating Current

Q-1. Define capacitive reactance. Write its SI Units?

Solution: This non-resistive opposition to the flow of A.C in a pure capacitive circuit is known as capacitive reactance $X_C$.

$X_C=\frac{1}{\omega C}=\frac{1}{2\pi fC}$

Unit of $X_C$ :Ohm

Q-2. Define Quality factor of resonance in series LCR Circuit and write its SI Units?

Solution: It is the ratio of the resonant frequency to frequency bandwidth of the resonance curve.

$Q=\frac{{\omega }_0}{Bandwidth}=\frac{{\omega }_0}{2\Delta \omega }$

$Q=\frac{{\omega }_{0}L}{R}=\frac{1}{{\omega }_{0}RC}$

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

It has no units; it determines the nature of sharpness of resonance.

Q-3. Why capacitors block Direct Current(DC)?

Solution: For d.c circuit f=0

$X_C=\frac{1}{2\pi fC}$=∞ 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)

Q-4.The power factor of an ac circuit is 0.5.What is the phase difference between voltage and current in the circuit?

Solution: Power factor between voltage and current is given by Cos $\varphi$ where $\varphi$ is the phase difference

$Cos\varphi =0.5=\frac{1}{2}\Rightarrow Co{s}^{-1}(\frac{1}{2})=\frac{\pi }{3}$

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$