physics_ac_currentAlternating Current
 AC Voltage applied to registor V=Vm sin 𝜔t $V={V}_{m}\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}𝜔t\phantom{\rule{0.22em}{0ex}}$ Current =i=VmR sin 𝜔t $Current\phantom{\rule{0.22em}{0ex}}=i=\frac{{V}_{m}}{R}\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}𝜔t\phantom{\rule{0.22em}{0ex}}$ Im=VmR ${I}_{m}=\frac{{V}_{m}}{R}\phantom{\rule{0.22em}{0ex}}$ AC Voltage and current are in same phase with each other Instantaneous power=P=i2m R sin2𝜔t$Instantaneous\phantom{\rule{0.22em}{0ex}}power=P={i}_{m}^{2}\phantom{\rule{0.22em}{0ex}}R\phantom{\rule{0.22em}{0ex}}si{n}^{2}𝜔t$ Average Power=⏨⏨ P =12=I2m R$Average\phantom{\rule{0.22em}{0ex}}Power=\overline{\phantom{\rule{0.22em}{0ex}}P\phantom{\rule{0.22em}{0ex}}}=\frac{1}{2}={I}_{m}^{2}\phantom{\rule{0.22em}{0ex}}R$ Irms=Im2 =0.707 Im ${I}_{rms}=\frac{{I}_{m}}{\sqrt{2}}\phantom{\rule{0.22em}{0ex}}=0.707\phantom{\rule{0.22em}{0ex}}{I}_{m}\phantom{\rule{0.22em}{0ex}}$ Vm=2 Vrms${V}_{m}=\sqrt{2}\phantom{\rule{0.22em}{0ex}}{V}_{rms}$
 AC Voltage applied to Inductor EquationaV-Ldidt=0 didt=VmL sin𝜔t$\begin{array}{l}V-L\frac{di}{dt}=0\phantom{\rule{0.22em}{0ex}}\\ \frac{di}{dt}=\frac{{V}_{m}}{L}\phantom{\rule{0.22em}{0ex}}sin𝜔t\end{array}$ i=imsin(𝜔t-𝜋2) $i={i}_{m}sin\left(𝜔t-\frac{𝜋}{2}\right)\phantom{\rule{0.22em}{0ex}}$ im=Vm𝜔L ${i}_{m}=\frac{{V}_{m}}{𝜔L}\phantom{\rule{0.22em}{0ex}}$ inductive reactance=XL=𝜔L $\phantom{\rule{0.22em}{0ex}}inductive\phantom{\rule{0.22em}{0ex}}reactance={X}_{L}=𝜔L\phantom{\rule{0.22em}{0ex}}$ im=VmXL ${i}_{m}=\frac{{V}_{m}}{{X}_{L}}\phantom{\rule{0.22em}{0ex}}$ Instantaneous Power=PL=-imVm2 sin2𝜔t $Instantaneous\phantom{\rule{0.22em}{0ex}}Power={P}_{L}=-\frac{{i}_{m}{V}_{m}}{2}\phantom{\rule{0.22em}{0ex}}sin2𝜔t\phantom{\rule{0.22em}{0ex}}$ The average power supplied to an inductor over one complete cycle is zero.
 AC Voltage applied to Capacitor EquationaV=qC Vmsin𝜔t=qC $\begin{array}{l}V=\frac{q}{C}\phantom{\rule{0.22em}{0ex}}\\ {V}_{m}sin𝜔t=\frac{q}{C}\phantom{\rule{0.22em}{0ex}}\end{array}$ q=VmCsin𝜔t $q={V}_{m}Csin𝜔t\phantom{\rule{0.22em}{0ex}}$ i=imsin(𝜔t+𝜋2) $i={i}_{m}sin\left(𝜔t+\frac{𝜋}{2}\right)\phantom{\rule{0.22em}{0ex}}$ im=Vm1𝜔C ${i}_{m}=\frac{{V}_{m}}{\frac{1}{𝜔C}}\phantom{\rule{0.22em}{0ex}}$ Capacitive reactanceXC=1𝜔C ${X}_{C}=\frac{1}{𝜔C}\phantom{\rule{0.22em}{0ex}}$ im=VmXC ${i}_{m}=\frac{{V}_{m}}{{X}_{C}}\phantom{\rule{0.22em}{0ex}}$ Instantaneous Power=PL=imVm2 sin2𝜔t $Instantaneous\phantom{\rule{0.22em}{0ex}}Power={P}_{L}=\frac{{i}_{m}{V}_{m}}{2}\phantom{\rule{0.22em}{0ex}}sin2𝜔t\phantom{\rule{0.22em}{0ex}}$ The average power supplied to a capacitor over one complete cycle is zero.
 AC Voltage applied to LCR Circuit EquationaLdidt+Ri+qC=v=Vmsin𝜔t Ld2qdt2+Rdqdt+qC=Vmsin𝜔t$\begin{array}{l}L\frac{di}{dt}+Ri+\frac{q}{C}=v={V}_{m}sin𝜔t\\ \\ L\frac{{d}^{2}q}{d{t}^{2}}+R\frac{dq}{dt}+\frac{q}{C}={V}_{m}sin𝜔t\end{array}$ V=Vm sin 𝜔t $V={V}_{m}\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}𝜔t\phantom{\rule{0.22em}{0ex}}$ i=imsin(𝜔t+𝜙) $i={i}_{m}sin\left(𝜔t+𝜙\right)\phantom{\rule{0.22em}{0ex}}$ Peak Voltage VRm=imR${V}_{Rm}={i}_{m}R$VCm=imXC${V}_{Cm}={i}_{m}{X}_{C}$VLm=imXL${V}_{Lm}={i}_{m}{X}_{L}$ Input voltage peak value = Vm${V}_{m}$ V2m=V2Rm+(VCm-VLm)2${V}_{m}^{2}={V}_{Rm}^{2}+\left({V}_{Cm}^{}-{V}_{Lm}{\right)}^{2}$ im=VmR2+(XC-XL)2 ${i}_{m}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left({X}_{C}^{}-{X}_{L}^{}{\right)}^{2}}}\phantom{\rule{0.22em}{0ex}}$ Impedance=Z=R2+(XC-XL)2$Impedance=Z=\sqrt{{R}^{2}+\left({X}_{C}^{}-{X}_{L}^{}{\right)}^{2}}$ im=VmZ ${i}_{m}=\frac{{V}_{m}}{Z}\phantom{\rule{0.22em}{0ex}}$ i=imsin(𝜔t+𝜙) $i={i}_{m}sin\left(𝜔t+𝜙\right)\phantom{\rule{0.22em}{0ex}}$ tan𝜙=XC-XLR$tan𝜙=\frac{{X}_{C}-{X}_{L}}{R}$ The current phasor ( I ) is always parallel to VR${V}_{R}$ 𝜙 $𝜙\phantom{\rule{0.22em}{0ex}}$ is angle between VR${V}_{R}$ and V If XC>XL${X}_{C}^{}>{X}_{L}^{}$ then 𝜙$𝜙$ is positive and the circuit is predominantly capacitive.So in this case the current in the circuit leads the source voltage. If XC
 Resonance Frequencies at which the response of the amplitude is maximum is known as resonant frequencies.Current amplitude is maximum at resonant frequencies. In RLC circuitim=VmZ =VmR2+(XC-XL)2 ${i}_{m}=\frac{{V}_{m}}{Z}\phantom{\rule{0.22em}{0ex}}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left({X}_{C}^{}-{X}_{L}^{}{\right)}^{2}}}\phantom{\rule{0.22em}{0ex}}$ Current amplitude is maximum when impedance (Z) is minium Z is minumim when XC=XL $\phantom{\rule{0.22em}{0ex}}{X}_{C}={X}_{L}\phantom{\rule{0.22em}{0ex}}$ The frequency at which impedance (Z) is minimum is called resonant frequency. Resonant frequency=𝜔0=1LC $Resonant\phantom{\rule{0.22em}{0ex}}frequency={𝜔}_{0}=\frac{1}{\sqrt{LC}}\phantom{\rule{0.22em}{0ex}}$ Resonant CircuitIn AC circuit with inductor and capacitor: When frequency of AC is changed at certain frequency so that impedance becomes maximum or minimum, then that condition is called resonance and such a circuit is called resonant circuit Series Resonance Parallel Resonance
 Sharpness of resonance im=VmR2+(XL-XC)2 =VmR2+(𝜔L-1𝜔C)2${i}_{m}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left({X}_{L}^{}-{X}_{C}^{}{\right)}^{2}}}\phantom{\rule{0.22em}{0ex}}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left(𝜔L-\frac{1}{𝜔C}{\right)}^{2}}}$ im is maximum when 𝜔=𝜔0=${i}_{m}\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}maximum\phantom{\rule{0.22em}{0ex}}when\phantom{\rule{0.22em}{0ex}}𝜔={𝜔}_{0}=$1LC$\frac{1}{\sqrt{LC}}$ (im)max=VmR $\left({i}_{m}{\right)}_{max}=\frac{{V}_{m}}{R}\phantom{\rule{0.22em}{0ex}}$ We choose 𝜔$𝜔$ where current amplitude is 12$\frac{1}{\sqrt{2}}$ times its maximum value. a𝜔1=𝜔0+𝛥𝜔 𝜔2=𝜔0-𝛥𝜔 $\begin{array}{l}{𝜔}_{1}={𝜔}_{0}+𝛥𝜔\phantom{\rule{0.22em}{0ex}}\\ {𝜔}_{2}={𝜔}_{0}-𝛥𝜔\phantom{\rule{0.22em}{0ex}}\\ \end{array}$𝜔1-𝜔2=2𝛥𝜔=bandwidth of circuit ${𝜔}_{1}-{𝜔}_{2}=2𝛥𝜔=bandwidth\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}circuit\phantom{\rule{0.22em}{0ex}}$ Sharpness of resonance=𝜔02𝛥𝜔$Sharpness\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}resonance=\frac{{𝜔}_{0}}{2𝛥𝜔}$ In order to determine 𝛥𝜔$𝛥𝜔$ we consider im=i maxm2${i}_{m}=\frac{{i}_{m}^{\phantom{\rule{0.22em}{0ex}}max}}{\sqrt{2}}$ At 12$\frac{1}{\sqrt{2}}$ times the current amplitude , power dissipated by he circuit becomes half. Sharpness of resonance=𝜔0LR$Sharpness\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}resonance=\frac{{𝜔}_{0}L}{R}$ Sharpness of resonance = Quality factorQ=𝜔0LR$Q=\frac{{𝜔}_{0}L}{R}$
 Power in AC Circuit (RLC circuit) im=VmZ${i}_{m}=\frac{{V}_{m}}{Z}$ 𝜙=tan-1 XC-XLR$𝜙=ta{n}^{-1}\phantom{\rule{0.22em}{0ex}}\frac{{X}_{C}-{X}_{L}}{R}$ Inatantenous power p suplied by sourceap=VI=(Vmsin𝜔t)×(im sin(𝜔t+𝜙) ) p=Vmim2 [cos𝜙-cos(2𝜔t+𝜙)] Average Power=P=Vmim2 cos𝜙P=Vm2 im2 cos𝜙 P=VI cos𝜙 P=I2 Z cos𝜙 $\begin{array}{l}p=VI=\left({V}_{m}sin𝜔t\right)×\left({i}_{m}\phantom{\rule{0.22em}{0ex}}sin\left(𝜔t+𝜙\right)\phantom{\rule{0.22em}{0ex}}\right)\phantom{\rule{0.22em}{0ex}}\\ \\ p=\frac{{V}_{m}{i}_{m}}{2}\phantom{\rule{0.22em}{0ex}}\left[cos𝜙-cos\left(2𝜔t+𝜙\right)\right]\\ \\ Average\phantom{\rule{0.22em}{0ex}}Power=P=\frac{{V}_{m}{i}_{m}}{2}\phantom{\rule{0.22em}{0ex}}cos𝜙\\ P=\frac{{V}_{m}}{\sqrt{2}}\phantom{\rule{0.22em}{0ex}}\frac{{i}_{m}}{\sqrt{2}}\phantom{\rule{0.22em}{0ex}}cos𝜙\phantom{\rule{0.22em}{0ex}}\\ \\ P=VI\phantom{\rule{0.22em}{0ex}}cos𝜙\\ \\ P={I}^{2}\phantom{\rule{0.22em}{0ex}}Z\phantom{\rule{0.22em}{0ex}}cos𝜙\\ \end{array}$ Power Factor=cos𝜙$Power\phantom{\rule{0.22em}{0ex}}Factor=cos𝜙$
 LC Oscillation Capacitor is charged and connected to a conductor. aqC-Ldidt=0 i=-dqdt d2qdt2+1LCq=0 q=qm cos(𝜔0t+𝜙) 𝜔0=1LC and 𝜙=0 q=qm cos(𝜔0t) $\begin{array}{l}\frac{q}{C}-L\frac{di}{dt}=0\\ \\ i=-\frac{dq}{dt}\\ \\ \frac{{d}^{2}q}{d{t}^{2}}+\frac{1}{LC}q=0\phantom{\rule{0.22em}{0ex}}\\ \\ q={q}_{m}\phantom{\rule{0.22em}{0ex}}cos\left({𝜔}_{0}t+𝜙\right)\\ \\ {𝜔}_{0}=\frac{1}{\sqrt{LC}}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}𝜙=0\phantom{\rule{0.22em}{0ex}}\\ \\ q={q}_{m}\phantom{\rule{0.22em}{0ex}}cos\left({𝜔}_{0}t\right)\phantom{\rule{0.22em}{0ex}}\end{array}$
 Transformer Transformer works using principal of mutual induction. Primary Coil= It is input coil.Number of turns=Np$={N}_{p}$ Secondary Coil= It is output coil.Number of turns=Ns$={N}_{s}$ Induced emf in secondary coilVS=-NSd𝜙dt${V}_{S}=-{N}_{S}\frac{d𝜙}{dt}$ Back emf in primary coilVP=-NPd𝜙dt${V}_{P}=-{N}_{P}\frac{d𝜙}{dt}$ Transformer equation VSVP=NSNP$\frac{{V}_{S}}{{V}_{P}}=\frac{{N}_{S}}{{N}_{P}}$ Assumption in transformer equationa. primary resistance and current are smallb. same flux links both primary and secondaryc. the secondary current is small. For 100% efficiencyiPVP=iSVs ${i}_{P}{V}_{P}={i}_{S}{V}_{s}\phantom{\rule{0.22em}{0ex}}$iPiS=VSVP=NSNP$\frac{{i}_{P}}{{i}_{S}}=\frac{{V}_{S}}{{V}_{P}}=\frac{{N}_{S}}{{N}_{P}}$ Energy losses happens in transformer due to followinga. Flux Leakage: There is always some flux leakage; that is, not all ofthe flux due to primary passes through the secondary due to poordesign of the core or the air gaps in the core. It can be reduced bywinding the primary and secondary coils one over the other. b. Resistance of the windings: The wire used for the windings has someresistance and so, energy is lost due to heat produced in the wire (I2R)$\left({I}^{2}R\right)$ . In high current, low voltage windings, these are minimised byusing thick wire. c. Eddy currents: The alternating magnetic flux induces eddy currentsin the iron core and causes heating. The effect is reduced by having alaminated core. d. Hysteresis: The magnetisation of the core is repeatedly reversed bythe alternating magnetic field. The resulting expenditure of energy in the core appears as heat and is kept to a minimum by using a magnetic material which has a low hysteresis loss.
 Rectification is the process in which an alternating current is forced to only flow in one direction. Half-wave Rectification:a. Dalf wave rectifier is allows one half-cycle of an AC voltage waveform to pass, blocking the other half-cycle.b. half of the power is lost in half-wave rectification. Full-wave Rectificationa. Full wave rectifier converts both halves of each cycle of an alternating wave (AC signal) into pulsating DC signal.
 Questions Q.7.1A 100 Ω $100\phantom{\rule{0.22em}{0ex}}\Omega \phantom{\rule{0.22em}{0ex}}$ resistor is connected to a 220 V, 50 Hz ac supply.(a) What is the rms value of current in the circuit?(b) What is the net power consumed over a full cycle? A.7.1a(a) Irms=VrmsR=220100=2.2 A (b) This is AC voltage applied on resistor $\begin{array}{l}\left(a\right)\phantom{\rule{0.44em}{0ex}}{I}_{rms}=\frac{{V}_{rms}}{R}=\frac{220}{100}=2.2\phantom{\rule{0.22em}{0ex}}A\\ \\ \left(b\right)\phantom{\rule{0.22em}{0ex}}This\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}AC\phantom{\rule{0.22em}{0ex}}voltage\phantom{\rule{0.22em}{0ex}}applied\phantom{\rule{0.22em}{0ex}}on\phantom{\rule{0.22em}{0ex}}resistor\\ \end{array}$Average Power=P=Vrms×Irms$Average\phantom{\rule{0.22em}{0ex}}Power=P={V}_{rms}×{I}_{rms}$P=2.2×220=484 Watt $P=2.2×220=484\phantom{\rule{0.22em}{0ex}}Watt\phantom{\rule{0.22em}{0ex}}$ Q.7.7A charged 30 μF capacitor is connected to a 27 mH inductor. What is the angular frequency of free oscillations of the circuit? A.7.7𝜔0=1LC${𝜔}_{0}=\frac{1}{\sqrt{LC}}$a𝜔0=127×10-3×30×10-6=181×10-8 𝜔0=19×10-4=1.1×103 rad/sec$\begin{array}{l}{𝜔}_{0}=\frac{1}{\sqrt{27×{10}^{-3}×30×{10}^{-6}}}=\frac{1}{\sqrt{81×{10}^{-8}}}\\ \\ {𝜔}_{0}=\frac{1}{9×{10}^{-4}}=1.1×{10}^{3}\phantom{\rule{0.22em}{0ex}}rad/sec\end{array}$ Q.7.18 A circuit containing a 80 mH inductor and a 60 μF capacitor in series is connected to a 230 V, 50 Hz supply. The resistance of the circuit is negligible.(a) Obtain the current amplitude and rms values.(b) Obtain the rms values of potential drops across each element.(c) What is the average power transferred to the inductor?(d) What is the average power transferred to the capacitor?(e) What is the total average power absorbed by the circuit? [‘Average’ implies ‘averaged over one cycle’.] A.7.18a(a)Capacitance=C=60×10-6 FInductance=L=80×10-3 Ffrequency=50 Hz𝜔=2𝜋𝜈=2×3.14×50=314 capacitive reactance=XC=1𝜔CXC=1314×60×10-6 =53.1 inductive reactance=XL=𝜔LXL=314× 80×10-3 =25.12 Impedance=Z=R2+(XC-XL)2=27.98 Vrms=230VVm=2302Im=VmZ=230227.98=11.623 A Irms=VrmsZ=23027.98=8.22 A (b)RMS Potential drop across capacitance=V1V1=Irms×XC=8.22×53.1=436.482 V RMS Potential drop across capacitance=V2V2=Irms×XL=8.22×25.12=206.486 V $\begin{array}{l}\left(a\right)\\ Capacitance=C=60×{10}^{-6}\phantom{\rule{0.22em}{0ex}}F\\ Inductance=L=80×{10}^{-3}\phantom{\rule{0.22em}{0ex}}F\\ frequency=50\phantom{\rule{0.22em}{0ex}}Hz\\ 𝜔=2𝜋𝜈=2×3.14×50=314\phantom{\rule{0.22em}{0ex}}\\ capacitive\phantom{\rule{0.22em}{0ex}}reactance={X}_{C}^{}=\frac{1}{𝜔C}\\ {X}_{C}^{}=\frac{1}{314×60×{10}^{-6}\phantom{\rule{0.22em}{0ex}}}=53.1\phantom{\rule{0.22em}{0ex}}\\ \\ inductive\phantom{\rule{0.22em}{0ex}}reactance={X}_{L}^{}=𝜔L\\ {X}_{L}^{}=314×\phantom{\rule{0.22em}{0ex}}80×{10}^{-3}\phantom{\rule{0.22em}{0ex}}=25.12\phantom{\rule{0.22em}{0ex}}\\ \\ Impedance=Z=\sqrt{{R}^{2}+\left({X}_{C}^{}-{X}_{L}^{}{\right)}^{2}}=27.98\\ \\ {V}_{rms}=230V\\ {V}_{m}=230\sqrt{2}\\ {I}_{m}=\frac{{V}_{m}}{Z}=\frac{230\sqrt{2}}{27.98}=11.623\phantom{\rule{0.22em}{0ex}}A\\ \\ {I}_{rms}=\frac{{V}_{rms}}{Z}=\frac{230}{27.98}=8.22\phantom{\rule{0.22em}{0ex}}A\\ \\ \left(b\right)\\ RMS\phantom{\rule{0.22em}{0ex}}Potential\phantom{\rule{0.22em}{0ex}}drop\phantom{\rule{0.22em}{0ex}}across\phantom{\rule{0.22em}{0ex}}capacitance={V}_{1}\\ {V}_{1}={I}_{rms}×{X}_{C}=8.22×53.1=436.482\phantom{\rule{0.22em}{0ex}}V\\ \\ RMS\phantom{\rule{0.22em}{0ex}}Potential\phantom{\rule{0.22em}{0ex}}drop\phantom{\rule{0.22em}{0ex}}across\phantom{\rule{0.22em}{0ex}}capacitance={V}_{2}\\ {V}_{2}={I}_{rms}×{X}_{L}=8.22×25.12=206.486\phantom{\rule{0.22em}{0ex}}V\\ \end{array}$(c)The average power supplied to an inductor over one complete cycle is zero.aPaverage=-imVm2〔sin2𝜔t⦘ value of 〔sin2𝜔t⦘ is 0 in one cycle $\begin{array}{l}{P}_{average}=-\frac{{i}_{m}{V}_{m}}{2}〔sin2𝜔t⦘\phantom{\rule{0.22em}{0ex}}\\ value\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}〔sin2𝜔t⦘\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}0\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}one\phantom{\rule{0.22em}{0ex}}cycle\\ \end{array}$(d)The average power supplied to an capacitor over one complete cycle is zero.aPaverage=imVm2〔sin2𝜔t⦘ value of 〔sin2𝜔t⦘ is 0 in one cycle $\begin{array}{l}{P}_{average}=\frac{{i}_{m}{V}_{m}}{2}〔sin2𝜔t⦘\phantom{\rule{0.22em}{0ex}}\\ value\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}〔sin2𝜔t⦘\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}0\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}one\phantom{\rule{0.22em}{0ex}}cycle\\ \end{array}$(e)aAverage Power=I2Zcos𝜙 𝜙=tan-1 XC-XLR here R=0, 𝜙=90° so, cos𝜙=cos90°=0so, Average Power=I2Zcos𝜙=0 $\begin{array}{l}Average\phantom{\rule{0.22em}{0ex}}Power={I}^{2}Zcos𝜙\\ \\ 𝜙=ta{n}^{-1}\phantom{\rule{0.22em}{0ex}}\frac{{X}_{C}-{X}_{L}}{R}\\ \\ here\phantom{\rule{0.22em}{0ex}}R=0,\phantom{\rule{0.22em}{0ex}}𝜙={90}^{°}\\ \\ so,\phantom{\rule{0.22em}{0ex}}cos𝜙=cos{90}^{°}=0\\ so,\phantom{\rule{0.22em}{0ex}}Average\phantom{\rule{0.22em}{0ex}}Power={I}^{2}Zcos𝜙=0\\ \end{array}$ Q.7.20A series LCR circuit with L = 0.12 H, C = 480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.(a) What is the source frequency for which current amplitude is maximum. Obtain this maximum value.(b) What is the source frequency for which average power absorbed by the circuit is maximum. Obtain the value of this maximum power.(c) For which frequencies of the source is the power transferred to the circuit half the power at resonant frequency? What is the current amplitude at these frequencies?(d) What is the Q-factor of the given circuit? A.7.20(a)L = 0.12 HC = 480×10-9 F$480×{10}^{-9}\phantom{\rule{0.22em}{0ex}}F$ im=VmR2+(XL-XC)2 =VmR2+(𝜔L-1𝜔C)2${i}_{m}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left({X}_{L}^{}-{X}_{C}^{}{\right)}^{2}}}\phantom{\rule{0.22em}{0ex}}=\frac{{V}_{m}}{\sqrt{{R}^{2}+\left(𝜔L-\frac{1}{𝜔C}{\right)}^{2}}}$im is maximum when 𝜔=𝜔0=${i}_{m}\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}maximum\phantom{\rule{0.22em}{0ex}}when\phantom{\rule{0.22em}{0ex}}𝜔={𝜔}_{0}=$1LC$\frac{1}{\sqrt{LC}}$ a𝜔0=10.12×480×10-9 =112×12×4×10-10𝜔0=112×2×10-5=4167 rad/secfrequency=𝜈=𝜔2𝜋=41672×3.14=663 Hz i mmax=VmRVm=V2=2302i mmax=230223=14.14 A $\begin{array}{l}{𝜔}_{0}=\frac{1}{\sqrt{0.12×480×{10}^{-9}\phantom{\rule{0.22em}{0ex}}}}=\frac{1}{\sqrt{12×12×4×{10}^{-10}}}\\ {𝜔}_{0}=\frac{1}{12×2×{10}^{-5}}=4167\phantom{\rule{0.22em}{0ex}}rad/sec\\ frequency=𝜈=\frac{𝜔}{2𝜋}=\frac{4167}{2×3.14}=663\phantom{\rule{0.22em}{0ex}}Hz\\ \\ {{i}_{m}^{\phantom{\rule{0.22em}{0ex}}}}^{max}=\frac{{V}_{m}}{R}\\ {V}_{m}=V\sqrt{2}=230\sqrt{2}\\ {{i}_{m}^{\phantom{\rule{0.22em}{0ex}}}}^{max}=\frac{230\sqrt{2}}{23}=14.14\phantom{\rule{0.22em}{0ex}}A\phantom{\rule{0.22em}{0ex}}\end{array}$ (b)a Power=P=VIcos𝜙For maximum power, cos𝜙=1Maximum power will have when XL=XC and source frequency is 663 Hz V=230 VPmax=V2R=230×23023=2300 W$\begin{array}{l}\\ Power=P=VIcos𝜙\\ For\phantom{\rule{0.22em}{0ex}}maximum\phantom{\rule{0.22em}{0ex}}power,\phantom{\rule{0.22em}{0ex}}cos𝜙=1\\ Maximum\phantom{\rule{0.22em}{0ex}}power\phantom{\rule{0.22em}{0ex}}will\phantom{\rule{0.22em}{0ex}}have\phantom{\rule{0.22em}{0ex}}when\phantom{\rule{0.22em}{0ex}}{X}_{L}={X}_{C}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}source\phantom{\rule{0.22em}{0ex}}frequency\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}663\phantom{\rule{0.22em}{0ex}}Hz\phantom{\rule{0.22em}{0ex}}\\ \\ V=230\phantom{\rule{0.22em}{0ex}}V\\ {P}_{max}=\frac{{V}^{2}}{R}=\frac{230×230}{23}=2300\phantom{\rule{0.22em}{0ex}}W\end{array}$ (c)At 12$\frac{1}{\sqrt{2}}$ times the current amplitude , power dissipated by he circuit becomes half.So 𝜔$𝜔$ at which power will be dissipated becomes half will bea𝜔=𝜔0±𝛥𝜔𝛥𝜔=R2L=232×0.12=96 rad/sec 𝜔1=4167+96=4263 Hz 𝜔2=4167-96=4071 Hz 𝜈1=𝜔12𝜋=678 Hz𝜈2=𝜔22𝜋=647 Hz$\begin{array}{l}𝜔={𝜔}_{0}±𝛥𝜔\\ 𝛥𝜔=\frac{R}{2L}=\frac{23}{2×0.12}=96\phantom{\rule{0.22em}{0ex}}rad/sec\phantom{\rule{0.22em}{0ex}}\\ {𝜔}_{1}=4167+96=4263\phantom{\rule{0.22em}{0ex}}Hz\phantom{\rule{0.22em}{0ex}}\\ {𝜔}_{2}=4167-96=4071\phantom{\rule{0.22em}{0ex}}Hz\phantom{\rule{0.22em}{0ex}}\\ \\ {𝜈}_{1}=\frac{{𝜔}_{1}}{2𝜋}=678\phantom{\rule{0.22em}{0ex}}Hz\\ {𝜈}_{2}=\frac{{𝜔}_{2}}{2𝜋}=647\phantom{\rule{0.22em}{0ex}}Hz\end{array}$ (d)aQuality factor=Q=𝜔0LRQ=4167×0.1223=21.74 $\begin{array}{l}Quality\phantom{\rule{0.22em}{0ex}}factor=Q=\frac{{𝜔}_{0}L}{R}\\ Q=\frac{4167×0.12}{23}=21.74\phantom{\rule{0.22em}{0ex}}\end{array}$ Q.7.23A power transmission line feeds input power at 2300 V to a step-down transformer with its primary windings having 4000 turns. What should be the number of turns in the secondary in order to get output power at 230 V?A.7.23aVSVP=NSNP ⟹2302300=NS4000 ⟹NS=400 turns $\begin{array}{l}\frac{{V}_{S}}{{V}_{P}}=\frac{{N}_{S}}{{N}_{P}}\\ \\ ⟹\frac{230}{2300}=\frac{{N}_{S}}{4000}\\ \\ ⟹{N}_{S}=400\phantom{\rule{0.22em}{0ex}}turns\phantom{\rule{0.22em}{0ex}}\end{array}$