physics-class-12-unit-07-chapter-07-lcr-circuit-applications-l-7-10-zaosg9-l-ru

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> LCR Circuit </primary>
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- $ L \frac{d I}{d t}+I R+\frac{Q}{C}=V_m \sin \omega t $

- $ I= \frac{dQ}{d t} $

- $ L \frac{d^2 I}{d t^2} + R \frac{d I}{dt}+\frac{I}{C}=V_m \omega cos \omega t $

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

- $\varphi$ Amount of phase by which current leads 1hz voltage.

- $I_m = \frac{V_m}{Z}: Z = \sqrt {R^2 + (X_c - X_L)}$

- $ tan \varphi = \frac {X_c - X_L}{R}$

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<footer style = "font-size: 18px"> $\rightarrow$ LCR Circuit  Applications $\rightarrow$ <span style = "color:blue"> LCR Circuit </span> $\rightarrow$ Maximum Current $\rightarrow$ LCR Circuit </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Maximum Current </primary>
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- Maximum current $I_m$ has a frequency dependence.

- $I_m$ has a maximum value as a function of $\omega$

- $I_m$ has a maximum as a function of $\omega$ when $X_c - X_L$ $\rightarrow$ $\omega L = \frac {1}{\omega C}$

- $\omega = \omega_0 = \frac{1}{\sqrt{\omega c}} $ (Resonance)

- $I_m^{max} = \frac {V_m}{R}$

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<footer style = "font-size: 18px">LCR Circuit  Applications $\rightarrow$ LCR Circuit $\rightarrow$ <span style = "color:blue"> Maximum Current </span> $\rightarrow$ LCR Circuit $\rightarrow$ Full width half maximum </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Full width half maximum </primary>
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- At $\omega = \omega_0$, the circuit absorbs maximum power.

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

- $ I-m^{2} \propto \frac {1}{Z}$ maximum power

- $\rightarrow$ Z is minimum

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

- $ I_m =\frac {I_m^{max}}{\sqrt 2}$

- 1hz power absorbed is half the possible maximum $I_{m}^{max}$

- Full width at half max $2\Delta \omega$

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<footer style = "font-size: 18px">Maximum Current $\rightarrow$ LCR Circuit $\rightarrow$ <span style = "color:blue"> Full width half maximum </span> $\rightarrow$ Band Width $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Example </primary>
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- LCR circuit R = 5 $\Omega$, C= 20 $\mu$ F, L = 200 mH

- (i) Resonant frequency

- $\omega_0 = \frac{1}{\sqrt LC} = \frac {1}{\sqrt{ 2 \times 10^{-1} \times 2 \times 10^{-5}}}$

- = 500 rad/s (f $\approx$ 80 Hz)

- (ii) Value of $\omega$ for which half power maximum occurs.

- $\frac {I_m^{max}}{\sqrt 2} \Rightarrow (\omega L - \frac{1}{\omega c})^2 = R^2$

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<footer style = "font-size: 18px">Full width half maximum $\rightarrow$ Band Width $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
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- $ \omega L-\frac {1}{\omega C} = \pm R $

- $ \omega^2 L C \mp R \omega C-1=0 $

- $ \omega= \frac{ \pm R C+\sqrt{R^2 C^2+4 L C}}{2 L C} $

- $ \omega = 512.5 \mathrm{rad} / \mathrm{s} \text { or } 487.5 \mathrm{rad} / \mathrm{s}$

- $ \text { spread of } \pm 12.5 \mathrm{rad} / \mathrm{s} \text {. } $

- $ \text { BW }=2 \Delta W=25 \mathrm{rad} / \mathrm{s} $

- $ B W=2 \Delta W=\frac{R}{L}=\frac{5}{2 \times 10^{-1}}=25 \mathrm{rad} / \mathrm{s} $

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

<h3 id="success-stylefont-size25px-lcr-circuit-applications-l-7-success-1"><Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success></h3>
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<p>What in $\omega$ such that</p>
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<p>$\frac{V_{\text {out }}}{V^{\text {in }}}=\frac{1}{2} \quad \Rightarrow$ $V_out  = 5 \sin \omega t$.</p>
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<p>$ Z_{i n}=\sqrt{50^{\circ}+.04 \omega^2}$</p>
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<p>$I_m=\frac{10}{Z_{i n}}=\frac{10}{\sqrt{50^2+0.04 \omega^2}} $</p>
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<p>$ { Z_{out} }=\sqrt{10^2+0.04 \omega^2}$</p>
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<p>$V_{out }^{m}={I_m} \times Z_{out}$</p>
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<p>=$\frac{10}{\sqrt{50^2+0.04 \omega^2}} \times \sqrt{10^8+0.04 \omega^2} $</p>
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<p>$ X_{L}^{2} = 700 = \omega^2 \times 0.04$</p>
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<p>$\omega$ = 132 rad/s</p>
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<p>$f = 21 \mathrm{Hz}$</p>
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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ LR circuit $\rightarrow$ LR circuit </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
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- $I_m = \frac {V_in^{m}}{\sqrt {R^2 + L^2 \omega^2}} $

- $ V_{out} = \frac {V_in^{m}}{\sqrt{R^2 + L^2 \omega^2}} $ $\times L \omega$

- =$V_{in}^{m}$ .$\frac {L \omega} {\omega L (1 + \frac {R^2}{L^2 \omega^2})^{1/2}} $

- $\approx V_{in}^{m}[1+ \frac{1}{2} . \frac {R^2}{L^2 \omega^2}]$

- As $\omega$ increases. $V_{out}$ $\rightarrow$ $V_{in}$

- $ \omega \rightarrow$ 0 $V_{out}$ = 0

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> LR circuit </span> $\rightarrow$ LR circuit $\rightarrow$ Low Pass Filter </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
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- $V_{out} = \frac {V_{in} R}{\sqrt{R^2 + L^2 \omega^2}}$

- = $\frac {V_{in} R} {L \omega  (1 + \frac {R^2}{L^2 \omega^2})^{1/2}} $

- For large $\omega$ $V_{out}$ decreases.

- For $\omega \rightarrow 0$, $V_{out} \rightarrow V_{in}$

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<footer style = "font-size: 18px">Example $\rightarrow$ LR circuit $\rightarrow$ <span style = "color:blue"> LR circuit </span> $\rightarrow$ Low Pass Filter $\rightarrow$ Power Absorbed in AC circuit </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
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- $I_m = \frac {V_{in}^{m}}{Z} = \frac {10} {\sqrt{1o^4 + 0.04\omega^2}}$

- $V_{m}^{out} = \frac {10 \times 10^3} {\sqrt{1o^4 + 0.04\omega^2}}$

- $\omega$ = 500 rad/s : $V_{out}^m = 9.95 V$

- $\omega$ = 5000 rad/s : $V_{out}^m = 7.07 V$

- $\omega$ = 50000 rad/s : $V_{out}^m = 0.995 V$

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<footer style = "font-size: 18px">LR circuit $\rightarrow$ LR circuit $\rightarrow$ <span style = "color:blue"> Low Pass Filter </span> $\rightarrow$ Power Absorbed in AC circuit $\rightarrow$ Power Absorbed in AC circuit </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Power Absorbed in AC circuit </primary>
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- V(t) = $V_m sin \omega t$

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

- $I_m = \frac{V_m}{Z}$ : $\varphi = tan^{-1} \frac{X_c - X_L} {R} $

- P(t) = I(t) V(t)

- = $V_m I_m sin \omega t . sin (\omega t + \varphi)$

- =$V_m I_m [sin^2 \omega t cos \varphi +  sin \omega t + cos \omega t sin \varphi)]$

- $\<P(t)\> = \frac {V_m I_m}{2} . cos \varphi = \frac {I_m^2}{2} . cos \varphi = \frac {V_m^2}{2Z} cos \varphi.$

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<footer style = "font-size: 18px">LR circuit $\rightarrow$ Low Pass Filter $\rightarrow$ <span style = "color:blue"> Power Absorbed in AC circuit </span> $\rightarrow$ Power Absorbed in AC circuit $\rightarrow$ Resistive Circuit </footer>

<h3 id="success-stylefont-size25px-lcr-circuit-applications-l-7-success-2"><Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success></h3>
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<p>$ < P> = \frac {I_m^2  Z}{2} . Cos \phi$</p>
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<p>$ tan \phi = \frac {X_c - X_L}{R}$</p>
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<p>$cos \phi = {R} {\sqrt {R^2 + (X_c - X_L)^2}} = \frac {R}{Z}$</p>
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<p>$< P > = \frac {I_m^2 Z}{2} .\frac {R}{Z} = \frac {I_m^2 .R}{2} = I_{rms}^2. R$</p>
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<p>= $ \frac {V_{rms}^2}{R^2 + (X_c - X_L)^2} . R$</p>
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<footer style = "font-size: 18px">Low Pass Filter $\rightarrow$ Power Absorbed in AC circuit $\rightarrow$ <span style = "color:blue"> Power Absorbed in AC circuit </span> $\rightarrow$ Resistive Circuit $\rightarrow$ For Purely Capacitive or Inductive Circuit </footer>

<h3 id="success-stylefont-size25px-lcr-circuit-applications-l-7-success-3"><Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success></h3>
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<p>$ < P>$ is maximum at $X_L = X_c$</p>
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<p>$ < P> = \frac {V_{rms}^2}{R}$</p>
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<p>For a purely resistive circuit.</p>
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<p>$ cos \phi = \frac {R}{Z} \Rightarrow 1 : \phi = 0$</p>
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<footer style = "font-size: 18px">Power Absorbed in AC circuit $\rightarrow$ Power Absorbed in AC circuit $\rightarrow$ <span style = "color:blue"> Resistive Circuit </span> $\rightarrow$ For Purely Capacitive or Inductive Circuit $\rightarrow$ Maximum power </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> For Purely Capacitive or Inductive Circuit </primary>
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- $\phi = \pm \frac {\pi}{2} : cos \phi = 0$

- watt-loss circuits.

- L-C-R circuit

- $ tan \phi = \frac {\Chi_C - \Chi_L}{R} \rightarrow \phi \neq 0$

- Power dissiparon through resistances only

- Circuit at resonance $\Chi_L = \Chi_C : \phi = 0 $

- Maximum power dissipation (through resistance only)

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<footer style = "font-size: 18px">Power Absorbed in AC circuit $\rightarrow$ Resistive Circuit $\rightarrow$ <span style = "color:blue"> For Purely Capacitive or Inductive Circuit </span> $\rightarrow$ Maximum power $\rightarrow$ Quality factor </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Maximum power </primary>
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- $\<P\> = \frac {V_{rms}^2}{R^2 + (X_L - X_c)^2}$

- $\<P_{max}\> = \frac {V_{rms}^2}{R}$ ,

- When $X_L = X_c \rightarrow \omega = \omega_0$

- $\<P\> = \frac {V_{rms}^2}{2R} \rightarrow (X_L - X_c)^2 = R^2$

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<footer style = "font-size: 18px">Resistive Circuit $\rightarrow$ For Purely Capacitive or Inductive Circuit $\rightarrow$ <span style = "color:blue"> Maximum power </span> $\rightarrow$ Quality factor $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Lcr Circuit Applications L-7 </Success>
### <primary style="font-size:24px"> Quality factor </primary>
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- $ \omega= \pm \frac{R}{2 L} \pm \frac{1}{2} \sqrt{\frac{R^2}{L^2}+4 \omega_0^2} $

- $ \omega_1=\frac{R}{2 L}+\frac{1}{2} \sqrt{\frac{R^2}{L^2}+4 \omega_0^2} $

- $ \omega_2=-\frac{R}{2 L}+\frac{1}{2}\sqrt{ \frac {R^2}{L^2}+4 \omega_0^2}$

- $2 \Delta \omega=\omega_1-\omega_2=\frac{R}{L} $
 
- $ Q=\frac{\omega_0}{2 \Delta \omega}=\frac{\omega_0 L}{R}$

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<footer style = "font-size: 18px">For Purely Capacitive or Inductive Circuit $\rightarrow$ Maximum power $\rightarrow$ <span style = "color:blue"> Quality factor </span> $\rightarrow$ Thank You $\rightarrow$  </footer>