physics-class-12-unit-07-chapter-06-lcr-circuits-analytical-solution-resonance-l-6-10-faioem7qvg4

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Impedance </primary>
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- $Z=\sqrt{R^2+(\Chi_C-\Chi_L)^2}$

- $\Chi_C=\frac{1}{\omega C};\Chi_L=\omega L$

- $\omega C \rightarrow$ capacitive Reactance

- $\Chi_L \rightarrow$ Inductive Reactance

- $I(t)=I_m \sin (\omega t + \phi)$

- $\phi \rightarrow$ phase by which current leads the voltage

- $I_m = \frac{V_m}{2}$

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<footer style = "font-size: 18px"> $\rightarrow$ Circuits Analytical Solution Resonance $\rightarrow$ <span style = "color:blue"> Impedance </span> $\rightarrow$ Capacitive circuit $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $100 \mu F$ capacitor is series with a $400\Omega$ resistance, connected to a $110V (rms)/60Hz$ supply.

- What is the time lag between current maximum and voltage maximum?

- $60Hz \rightarrow \omega=2 \pi \times 60 \approx 377 rad/s$

- Capacitive reactance

- $\frac{1}{\omega C}=\frac{1}{377 \times 10^{-4}}=26.5 \Omega$

- $Z=\sqrt{40^2 (26.5)^2} \approx 48 \Omega$

- $I_{vms} = \frac{110}{48}=2.29A \rightarrow I_m = 2.29 \sqrt{2} =3.24 A$
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<footer style = "font-size: 18px">Impedance $\rightarrow$ Capacitive circuit $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Analytical Solution </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Example </primary>
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- **Angle by which Current Leads voltage**

- $\frac{\phi}{\omega}=1.55 ms$

- $f=1KHZ \rightarrow \omega = 6263 rad/s$

- $\omega C=6263 \times 10^{-4} \approx 0.63 \Omega^{-1}$

- $\Chi_C = \frac{1}{\omega C}=1.59 \Omega$

- $Z=\sqrt{40^2+(1.59)^2}=40.03 \Omega$

- $\frac{\Chi_C}{R}=\frac{1.53}{40}=0.039 \rightarrow \phi = \tan^{-1} \frac{\Chi_C}{R}\approx 0.039 rad$

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<footer style = "font-size: 18px">Capacitive circuit $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Analytical Solution $\rightarrow$ Analytical Solution </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Analytical Solution </primary>
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- $\frac{dI}{dl}+IR+\frac{\phi}{C}=V_m \sin \omega t$

- $I=\frac{d \phi}{dt}$

- $\frac{d^2I}{dt^2}+R\frac{dI}{dt}+\frac{1}{C}.I=V_m \omega \cos (\omega t)$

- $I=I_m \sin (\omega t +\phi)$

- $\frac{dI}{dt}=I_m \omega \cos (\omega t + \phi)$

- $\frac{d^2 I}{dt^2}=-I_m \omega^2 \sin (\omega t + \phi)$

- $I_m \omega [(-L \omega + \frac{1}{\omega C}) \sin (\omega t + \phi)+R \cos (\omega t + \phi )]$

- $=V_m \omega \cos \omega t$

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Analytical Solution </span> $\rightarrow$ Analytical Solution $\rightarrow$ Resonance </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Analytical Solution </primary>
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- $I_m \omega [(-L \omega + \frac{1}{\omega C})\sin (\omega t + \phi)+R \cos (\omega t + \phi)]$

- $=V_m \omega \cos \omega t$

- $R=A \cos \theta$

- $\Chi_C - \Chi_L = A \sin \theta $

- $A^2 = R^2 + (\Chi_C - \Chi_L)^2 = Z^2 \rightarrow A = z$

- $tan \theta = \frac{\Chi_C - \Chi_L}{R}$

- $I_m Z \cos (\omega t + \phi - \theta)=V_m \cos (\omega t)$

- $I_m z = V_m \rightarrow I_m =\frac{V_m}{z}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Analytical Solution $\rightarrow$ <span style = "color:blue"> Analytical Solution </span> $\rightarrow$ Resonance $\rightarrow$ Resonance </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Resonance </primary>
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- $I_m =\frac{V_m}{\sqrt{R^2 +(\Chi_C - \Chi_L)^2}}$

- $\Chi_C = \frac{1}{\omega C}; \Chi_L= \omega L$

- Z is minimum when $\frac{1}{\omega_C}= \Delta \omega L$

- $\rightarrow \omega = \omega_0 = \sqrt{\frac{1}{LC}}$

- $ \omega_0 $ = Resonant frequency

- At resonance $I_m=\frac{V_m}{R}; $ phase

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<footer style = "font-size: 18px">Analytical Solution $\rightarrow$ Analytical Solution $\rightarrow$ <span style = "color:blue"> Resonance </span> $\rightarrow$ Resonance $\rightarrow$ Bandwidth </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Resonance </primary>
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- Both L and C must be Present

- If $\omega^2 > \omega_0^2$

- $\omega^2 > \frac{1}{LC}$

- $I \propto \frac{1}{Z}$

- $I^2 Z$ is maximum at $\omega = \omega_0$

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<footer style = "font-size: 18px">Analytical Solution $\rightarrow$ Resonance $\rightarrow$ <span style = "color:blue"> Resonance </span> $\rightarrow$ Bandwidth $\rightarrow$ Bandwidth </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Bandwidth </primary>
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- Increase or decrease frequency (standing walk $\omega_0$) till power absorbed is half the maximum(at $\omega_0)$

- $20 \omega =(\omega_1 - \omega_2)$

- Bandwitdh (BW)

- $P=0.5 I^3 R = (\frac{I}{\sqrt{2})^2}.R \rightarrow$ At half power point

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<footer style = "font-size: 18px">Resonance $\rightarrow$ Resonance $\rightarrow$ <span style = "color:blue"> Bandwidth </span> $\rightarrow$ Bandwidth $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Bandwidth </primary>
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- Let, $\omega_1 = \omega_0 + \Delta \omega$

- $\omega_2 = \omega_0 - \Delta \omega$

- $2 \Delta \omega = $ Band width

- Sharper resonance $\rightarrow$ Lower bandwidth

- At, $\omega = \omega_1 \rightarrow I_m $

- $= \frac {V_ m}{\sqrt{R^2 +(\omega_ 1 L - \frac{1}{\omega_ 1 c^2})}} \equiv \frac{I_ m^{max}}{\sqrt{2}}=\frac{V_m}{R.\sqrt{2}}$

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<footer style = "font-size: 18px">Resonance $\rightarrow$ Bandwidth $\rightarrow$ <span style = "color:blue"> Bandwidth </span> $\rightarrow$ Example $\rightarrow$ Quality Factor </footer>

### <Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $R^2 (\omega_1 L - \frac{1}{\omega_1 C})^2 = 2R^2$

- $\omega_1 L - \frac{1}{\omega_1 C}=R$

- $(\omega_0 + \Delta \omega)L-\frac{1}{\omega_0 + \Delta \omega)C}=R$

- $\omega_0 L[1+\frac{\Delta \omega}{\omega_0}]-\frac{1}{\omega_0 C}[1-\frac{\Delta \omega}{\omega_0}]=R$

- $L. \Delta \omega +\frac{1}{\omega_0 C}.\frac{\Delta \omega}{\omega_0}=R$

- $(L).2 \Delta \omega = R$

- $\Delta \omega =\frac{R}{2 L}$
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<footer style = "font-size: 18px">Bandwidth $\rightarrow$ Bandwidth $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Quality Factor $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-lcr-circuits-analytical-solution-resonance-l-6-success-1"><Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success></h3>
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<p>$V(t)=(240 V) \sin \omega t$</p>
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<p>$L=10mH$</p>
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<p>$C=1 \mu F$</p>
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<p>$R=40 \Omega$</p>
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<p>(a) Resonance Frequency</p>
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<p>$\omega_0 = \frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{10^{-3} \times 10^{-6}}}$</p>
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<p>$=10^{-4} rad/s$</p>
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<p><img alt="image" src="https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-12-unit-07-chapter-06-lcr-circuits-analytical-solution-resonance-l-6_10-faioem7qvg4-241-2752.8.jpg"></p>
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<footer style = "font-size: 18px">Quality Factor $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

<h3 id="success-stylefont-size25px-lcr-circuits-analytical-solution-resonance-l-6-success-2"><Success style="font-size:25px"> Lcr Circuits Analytical Solution Resonance L-6 </Success></h3>
<h3 id="primary-stylefont-size24px-example-primary-2"><primary style="font-size:24px"> Example </primary></h3>
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<p>(b) What is the amplitude of current at resonance></p>
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<p>$I_m^{max} =\frac{V_m}{R} = \frac{240}{R}=6A.,R=40 \Omega$</p>
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<p>(c) $Q=\frac{\omega_0 L}{R}=\frac{10^{4} \times 10^{-2}}{40}=2.5$</p>
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<p>(d) Voltage across inductor at resonance</p>
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<p>$V_L^{max} = I_m^{max}.\omega_0L$</p>
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<p>$=6 \times [10^4 \times 10^{-2}]$</p>
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<footer style = "font-size: 18px">Quality Factor $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Thank You </footer>