Alternating Current (AC) - Physics Past Paper Notes PDF

# Institute of Physics ## STD – XII (SCIENCE) PHYSICS CBSE & GSEB BOARD CONCEPT - 7 ## Alternating Current ### Alternating Current (AC) - An alternating current is that current whose magnitude changes continuously with time and direction reverses periodically. - We know that when a coil is rotated...

# Institute of Physics ## STD – XII (SCIENCE) PHYSICS CBSE & GSEB BOARD CONCEPT - 7 ## Alternating Current ### Alternating Current (AC) - An alternating current is that current whose magnitude changes continuously with time and direction reverses periodically. - We know that when a coil is rotated in a magnetic field, an alternating emf is induced which is given by the relation: - $E=E_o sinwt$ - From Ohm’s Law: - $I=E/R=E_osinwt/R$ - $∴I=I_osinwt$ - Where $I_o=E_o/R$ - peak or maximum value. - **Amplitude:** The Maximum value attained by an AC current in either direction is called amplitude ($I_o$). - **Time period:** The time taken by an AC to complete one cycle of its variation is called its time period and denoted by T. - $T=2π/ω$ - **Frequency:** the no. of cycles completed by an AC current is called frequency (f): - $f=ω/2π=1/T$ - Where $I=I_osinwt=I_osinωT/2πt=I_osin2πft/T$ ### Average Value of AC over one cycle - The AC current at any instant time t is given by: - $I=I_osinwt$ - Assuming the current remains constant for a small time dt, the amount of charge that flows through the circuit in time dt will be: - $dq=Idt=I_osinωtdt$ - The total charge that flows through the circuit in one complete cycle of AC is: - $q=\int dq=\int I_osinωtdt$ - $=I_o∫ sinωtdt$ - $=I_o [-cosωt/ω]_0^T$ - $=-I_o [cosωT/ω - cos0/ω]$ - $=-I_oT [cos2π - cos0]$ - $=-2I_oT[1-1]/2π=0$ - Therefore, the average value of AC over one complete cycle is: - $I_{av}=q/T=0$ ### Relation between Average Value and Peak Value - The value of AC at any instant t is given by: - $I=I_osinwt$ - Then the amount of charge that flows through the circuit in a small time dt is given by - $dg=Idt=I_osinωtdt$ - The total charge that flows through the circuit in the first half cycle, from t = 0 to t = T/2, is: - $q=\int dq=\int I_osinωtdt$ - $=I_o∫ sinωtdt$ - $=I_o [-cosωt/ω]_0^{T/2}$ - $=-I_o [cosωT/2/ω T/2 - cos0/ω]$ - $=-I_oT [cosπ - cos0]/2π$ - $=-2I_oT[1-1]/2π=2I_o/π$ - Therefore, the average value of AC over the first half cycle is: - $I_{av}=q/t=2I_o/π/T/2=2I_o/π/T=0.637I_o$ - $E_{av}=ε_o/π= 0.637 ε_o$ ### RMS Value of AC current/Voltage - It is defined as that value of direct current/or steady voltage which produces the same heating effect in a given resistor as is produced by AC current/emf when passed for the same time. It is denoted by $I_{rms}$, $I_u$ or $I_{eff}$. - $I_{rms}=\sqrt{\int_0^{T/2} I^2 dt}$ - RMS means Root mean square; the square root of the mean of the square of that quantity. - Considering E = ε_o sinwt: -$<&^2>=<ε_o^2sin^2ωt>$ - $=ε_o^2<1-cos2ωt>/2$ - $=ε_o^2/2-ε_o^2/2<cos2ωt>$ - $=ε_o^2/2 - ε_o^2/2 ∫_0^T{cos2ωt dt}/T$ - Since $∫_0^T{cos2ωt dt}/T=0$, then: - $<&^2>=ε_o^2/2$ - $E_{rms}=\sqrt{<&^2>}= ε_o/√2$ - Similarly, $I_{rms}=I_o/√2$ - Therefore, the rms value of AC is 1/√2 times its peak value. ### AC Circuit Containing only Resistor - Suppose a resistor of resistance R is connected to a source of alternating emf E: - E=E_o sinwt - Such a circuit is known as a pure resistive circuit. - Let I be the current flowing in the circuit. The potential drop across R will be IR: - $E_o sinwt=IR$ - $I=E_o sinwt/R$ - $I=I_o sinwt$ - From the relations between E and I, we can conclude they are a function of sinwt so E and I are in the same phase. This means that both E and I attain their zero, minimum and maximum values at the same time. ### AC Circuit with only Inductor - An inductor of inductance L is connected to a source of AC emf. The AC emf is given by: - $E=E_osinwt$ - According to Kirchhoff's Law, the following calculation can be done: - $E=L{dI}/{dt}$ - $dI={Eosinwt}/{L} dt$ - $∫dI=∫{Eosinwt}/{L} dt$ - $I=-{E_ocosωt}/{ωL} + C$ - $I=-{E_ocosωt}/{ωL} + C$ - $I={E_osin(ωt-π/2)}/{ωL}$ - $I=I_osin(ωt-π/2)$ - Where $I_o={E_o}/{ωL}$ - peak value ### Inductive reactance - Comparing the equation $I_o = {E_o}/{ωL}$ with $I_o={E_o}/{R}$, we find that ωL plays the same role here as resistance R in the resistive case. - The inductive reactance is a measure of the effective resistance offered by the inductor to the flow of AC current. - $XL=ωL=2πfL$ - For AC: $XL \propto f$ - For DC: $f=0$ so $XL=0$ - Therefore, an inductor allows DC flow through it easily but opposes the flow of AC. ### AC circuit with only capacitor - A pure capacitor C is connected across an AC emf. The AC emf is given by: - $E= E_o sinwt$ - It is a purely capacitive circuit. - $E=Q/C$ - $E_osinwt=Q/C$ - we know $I=dQ/dt$: - $I=d(ε_ocsinwt)/dt$ - $I=ωCε_ocoswt$ - $I=ωCε_osin(ωt+π/2)$ - $I=I_osin(ωt+π/2)$ - Where $I_0=ωCε_o=ε_o/ωC$ - current Amplitude. - We find that in a capacitive circuit, the current leads the voltage by π/2. ### Capacitive reactance - Comparing the relation $I_o=ε_o/{ωC}$ with the ohmic relation $I_o=ε_o/R$ we find that $1/{ωC}$ is the effective resistance offered by the capacitor to the flow of AC current. - It is called capacitive reactance: - $Xc=1/{ωC}=1/{2πfC}$ ### Sharpness of Resonance: Q-factor - **Resonance Condition of LCR circuit:** - When $XL=XC$ or $VL=VC$ - $Z=\sqrt{R^2+(XL-XC)^2}=R$ - The impedance becomes minimum. The circuit is purely resistive. The current and voltage are in the same phase. This resonance condition. The frequency at which the current amplitude $I_o$ attains a peak value is called the resonant frequency: - $ f_r=1/\sqrt{LC}$ ### Sharpness of resonance: - The Q-factor of a series resonant circuit is an indicator of sharpness of resonance. - It is defined as the ratio of the resonant frequency to the difference in two frequencies taken on both the sides of resonant frequency such that at each frequency, the current amplitude becomes 1/√2 times at resonant frequency. - $Q=ω_r/2Δω=ω_rL/R = 1/R\sqrt{L/C}=ω_rC/R$ - $ω_1$ and $ω_2$ are the frequencies at which the current falls to 1/√2 times its resonant value. - The frequency range ($ω_2-ω_1=2Δω$) is called bandwidth. The larger the Q-factor, the smaller the bandwidth and sharper the current peak. - **Expression for Q-Factor:** - Clearly at $ω_r$, the impedance is equal to R, while at $ω_1$ and $ω_2$, its value is √2R: - $Z=\sqrt{R^2+(ωL-1/ωC)^2}=\sqrt{2}R$ - $R^2+(ωL-1/ωC)^2=2R^2$ - $ωL-1/{ωC}=±R$ - We can write: $ω_1L-1/{ω_1C}=R$ and $ ω_2L-1/{ω_2C}=R$ - Adding the above two equations: - $(ω_1+ω_2)L-(1/ω_1+1/ω_2)C=0$ - $ω_1ω_2=1/LC$ -(1) - Subtracting equation (2) from (1), we get: - $(ω_2-ω_1)L+(1/ω_1+1/ω_2)C=2R$ - $(ω_2-ω_1)(L+1/ω_1ω_2C)=2R$ - $(ω_2-ω_1)(L+L)=2R $ - $ω_2-ω_1=R/L$ - $Q=ω_r/2Δω=ω_r/(ω_2-ω_1)=ω_rL/R$ - Since: $ω_r=1/\sqrt{LC}$ then $ω_r=1/{√LC}=√{1/LC}$ - $Q=1/R\sqrt{L/C}=ω_rC/R$ - If Q-factor is low or R is large, the bandwidth 2Δω is small. This means that resonance is sharp or a series resonant circuit is more selective. ### LC Oscillations - When charged capacitor is allowed to discharge through non-resistive inductor, electrical oscillations of constant amplitude and frequency are produced. These oscillations are called LC oscillations. - As shown in the figure, a capacitor with initial charge $q_o$ is connected to an ideal inductor. The electrical energy stored in capacitor is $U_E=1/2q_o^2/C$. As there is no current, the magnetic field energy stored in inductor is zero. - Now capacitor begins to discharge itself causing current E. As current increases, it builds up a magnetic field around the inductor. - A part of electrical energy of the capacitor gets stored in the inductor in the form of magnetic energy $U_B=1/2LI^2$. - At a later, the capacitor gets fully discharged, and the P.d across its plates becomes zero. Thus, the entire electrostatic energy of the capacitor has been converted into magnetic field energy of the inductor. - After the discharge of the capacitor is complete, the magnetic flux linked with the inductor decreases, inducing a current in the same direction as the earlier current and charges the capacitor in an opposite direction. The magnetic energy of the inductor begins to change into the electrostatic energy of the capacitor. This process continues till the capacitor is fully charged. but it is charged with a polarity opposite to that in its initial state. Therefore, the entire energy again is stored in the electric field of the capacitor. - The capacitor begins to discharge again, sending the current in the opposite direction. The energy is once again transferred to the magnetic field of the inductor. The process repeats in the opposite direction. - Therefore, the energy of the system is continuously surging back and forth between the electric field of the capacitor and the magnetic field of the inductor. This produces electrical oscillations of frequency $f_0$. These are called LC Oscillations. ### Series LCR Circuit - As shown in the figure, a resistor R, an inductor L and capacitor C are connected in series to a source of alternating emf E. - $E=E_osiniwt$ - Let I be the current in the circuit at any instant. Then: - VR=IR - Voltage across the resistance R will be in phase with I. So its amplitude is VR=IOR. - VL=XLI - Voltage across the inductor L is ahead of current I in phase by π/2 rad. Its amplitude is VL=I_oXL. - VC=X_CI - Voltage across the capacitor C lags behind the current I in phase by π/2 rad. Its amplitude is VC=I_oXC. - As VL and VC are in opposite directions, their resultant is (VL-VC). By parallelogram law, the resultant of VR and (VL-VC) must be equal to applied emf E. - Using the Pythagorean theorem: - $ε_o^2=(VR)^2+(VL-VC)^2$ - $ε_o^2=(I_oR)^2+(I_oXL-I_oXC)^2$ - $ε_o^2=I_o^2R^2+(XL-XC)^2I_o^2$ - $I_o=ε_o/\sqrt{R^2+(XL-XC)^2}$ - $√R^2+(XL-XC)^2$ is the effective resistance of the series LCR circuit which opposes the flow of current through it and is called its impedance. It’s denoted by Z and its unit is Ω: - $Z=\sqrt{R^2+(XL-XC)^2}=\sqrt{R^2+(ωL-1/ωC)^2}$ - It is important to note that: - $XL=ωL$ - inductive reactance - $XC=1/ωC$ - capacitive reactance ### Power in AC circuit - The rate at which electrical energy is consumed in a circuit is called its power. - In an AC circuit, we define instantaneous power as the product of instantaneous voltage and instantaneous current. - Suppose a voltage and current at any instant is given by: - $E=ε_osinωt$ and $I=I_osin(ωt-φ)$ - Where φ is the phase angle. - Therefore, the instantaneous power is: - $P=EI$ - $P=ε_oI_osinωtsin(ωt-φ)$ - $P=ε_oI_o/2(2sinωtsin(ωt-φ))$ - $P=ε_oI_o/2(cosφ-cos(2ωt-φ))$ - The average power is: - $P_{av}=ε_oI_o/2∫_0^T{cosφdt}/T+ε_oI_o/2∫_0^T{cos(2ωt-φ)dt}/T$ - Since $∫_0^T {cos(2ωt-φ)dt}=0$ and $∫_0^T{cosφdt}=Tcosφ$ then: - $P_{av}=ε_oI_ocosφ/2$ - $P_{av}=ε_{rms}I_{rms}cosφ$ ### Special cases: - **Circuit with only resistor** - For this case $φ=0$ - $P_{av}=ε_{rms}I_{rms}$ - **Circuit with only inductor** - For this phase difference $φ=π/2$ - $cosπ/2=0$ - $P=0$ - **Circuit with capacitor** - For this $φ=π/2$ - $cos(π/2)=0$ - $P=0$ ### Power factor - The average power of an AC circuit is given by: - $ P_{av} = ε_{rms} I_{rms} cos φ$ - The average power of an AC circuit does not give actual power. It gives actual power only when multiplied by factor $cos φ$. - The $cos φ $ is called the power factor. - Its value varies from 0 to 1. For series LCR, the power factor is defined as: - $cos φ=R/Z= R/\sqrt{R^2+(ωL-1/ωC)^2}$ ### Wattless Current - The current in an AC circuit is said to be wattless if the average power consumed in the circuit is zero. - $P_{av} = ε_{rms} I_{rms} cos φ =0$ - This happens in pure inductive or capacitive circuits in which voltage and current differ by π/2 so that $cos(π/2) = 0$. - Therefore, current in the circuit has no power. ### Average Power with Resistor - In the case of a pure resistor, the ε and I are in the same phase. So instantaneous power dissipated in the resistor is: - $P=I^2R=I_o^2sin^2wtR$ - The average value of power over one cycle; - $<P>=<I^2R>=<I_o^2sin^2wtR>$ - $=I_o^2R<sin^2wt>$ - $=I_o^2R<1-cos2wt>/2$ - $=I_o^2R/2<1-cos2wt>$ - $=I_o^2R/2(1-<cos2wt>)$ - $<cos2wt>=0$ - $<P>=I_o^2R/2=I_o^2R/2$ - $=I_o^2R/2=(I_o/√2)^2R$ - $<P>=I_o^2R/2=I_{rms} I_{rms}R= I_{rms}^2R $ ### Average Power with Inductor - In the case of an inductor, the current lags behind the voltage in phase by π/2 rad. - So the instantaneous power is: - $P_L=IE= I_osin(ωt-π/2)ε_osinωt$ - $P_L=-I_oε_ocosωtsinωt$ - $P_L=I_oε_o/2sin2ωt$ - The average power is: - $<P_L>=I_oε_o/2<sin2ωt>$ - $<sin2ωt>=0$, - $<P_L>= I_oε_o/2<sin2ωt>=0$ - The average power dissipated in the inductor is zero. ### Average Power with Capacitor - In the case of a capacitor, the current leads voltage by π/2 in phase. - The power supplied to the capacitor is: - $P_C=IE=I_osin(ωt+π/2)ε_osinωt$ - $P_C=I_oε_ocosωtsinωt$ - $P_C=I_oε_o/2sin2ωt$ - The average power is: - $<P_C>=I_oε_o/2<sin2ωt>$ - $<sin2ωt>=0$ - $<P_C>=I_oε_o/2<sin2ωt>=0$ - The average power dissipated in the capacitor is zero. ### Transformer - A transformer is an electrical device for converting AC current at low voltage into that at high voltage or vice versa. - If it increases the input voltage it is called a step-up transformer and if it decreases the input voltage, it is called a step-down transformer. - **Principle:** It works on the principle of mutual induction. - **Construction:** - A transformer consists of two coils of insulated copper wire having different numbers of turns and wound on the same soft iron core. - The coil to which electrical energy is supplied is called the primary and coil at which output is obtained is called the secondary. - To prevent the energy loss due to eddy currents, a laminated core is used, the entire magnetic flux due to current in the primary coil practically remains in the iron core, so it passes fully through the secondary. - Two types of arrangements are generally used for winding coils: (1) core type (2) shell type. - **Working:** - As current flows through the primary, it generates an alternating magnetic flux in the core, which also passes through the secondary. This set up an induced emf in the secondary. - If there is no leakage of magnetic flux, then the flux linked with each turn of the primary will be equal to that with each turn of the secondary - **Theory:** - Consider the terminals of the secondary are open. Let $N_1$ and $N_2$ be the no. of turns in the primary and secondary. - Then, the induced emf in the primary: - $ε_1=-N_1{dφ}/{dt}$ - The induced emf in the secondary: - $ε_2=-N_2{dφ}/{dt}$ - $∴ε_1/N_1=ε_2/N_2$ - By Lenz's law, the self-induced emf $ε_1$ opposes ε in the primary coil. - Resultant emf in the primary: - $E-ε1=I_1R$ - This emf sends current $I_1$ through the primary coil of resistance R. - But R is very small, so the term $I_1R$ can be neglected. - Then $ε=ε_1$ and $ε_1$ is the input emf. - $ε_2$ is the output emf: - $ε_2=ε_1N_2/N_1$ - $N_2/N_1$ is called the turns ratio or transformation ratio. - For step up, $N_2>N_1$, so $N_2/N_1>1$ and $ε_2>ε_1$. - For step down, $N_2<N_1$, so $N_2/N_1<1$ and $ε_2<ε_1$. - The ideal transformer is one so that there are no energy losses. - Input power = Output power. - $ε_1I_1=ε_2I_2$ - $ε_1/ε_2=N_1/N_2$ - $I_2/I_1=ε_1/ε_2=N_1/N_2$ - The efficiency of the transformer is: - $η=(output power/input power) \times 100$ ### Transformer losses - **Copper loss** - $(I^2R)$ losses - **Eddy current loss** - **Hysteresis loss** - **Flux leakage** ### Uses of Transformers: - Radio receivers - Loud speakers - Voltage regulators in TV, AC - Computer - Step-down transformers in induction furnaces - Electric welding - Step-up transformers used in X-ray production - Transmission lines ### AC Generator - An AC generator is a device which converts mechanical energy into electrical energy. - **Principle:** It is based on electromagnetic induction. When a coil is rotated in a uniform magnetic field, the magnetic field linked with the coil changes, and an induced emf is generated. - **Construction:** - **Field Magnet:** Is a permanent magnet (horse shoe). It produces a strong magnetic field in the region between its poles. - **Armature:** Is a large no. of copper wires turned around a soft iron cylindrical core. This can be rotated through a magnetic field about an axis perpendicular to B. - **Slip rings:** The two ends of armature coils are connected to two coaxial brass rings $S_1$ and $S_2$ called slip rings. The rings are fixed on the same shaft and insulated from each other, as well as from the shaft. The rings rotate along with the armature coil. - **Brushes:** Two graphite rods called brushes are tightly pressed against the two slip rings through brushes $B_1$ and $B_2$. The armature coil is rotated about its axis with help of a turbine. - **Working:** - As the armature coil rotates, the magnetic flux linked with it changes and a current is induced. - As the coil PQRS rotates, PQ moves downwards and SR moves upwards. - According to Fleming’s right-hand rule, the induced current flows from Q to P and R to $S_1$ so it flows as SRQP during the first half rotation of the coil, with B_1 acting as the positive terminal and brush B_2 as the negative terminal. - During the second half of the rotation, PQ side moves upward & SR moves downward, so the direction of induced current is reversed. It flows along PQRS, so brush B_2 acts as the positive terminal and B_1 as a negative terminal. - Therefore, the direction of current in the external circuit is reversed after every half-cycle. Hence, AC current is produced by the generator. - **Expression for induced emf:** - The magnetic flux at any time in the coil is: - $φ = NBACosθ= NBACosωt$ - By Faraday's Law, the induced emf is: - $ε=-dφ/dt=-d(NBACosωt)/dt$ - $=NBAωsinωt$ - $ε=ε_osinωt$, where $ε_o=NBAω$ - Then: - $I = ε/R= ε_osinωt/R=I_osinω t$