physics-class-11-unit-15-chapter-01-amplitude-procedure-to-generate-amplitude-modulated-wave-l-1-5-ozcxigsjfo0

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Equation of Amplitude Modulated Wave </primary>
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- Information Signal (message) $\rightarrow E_m = E_{mo} \sin(\omega_m t)$

- where:

- $E_{mo}$ : Amplitude of message

- $\omega_m$ : angular frequency of message

- Carrier wave $\rightarrow E_c = E_{co} \sin ({\omega_c t})$

- Amplitude modulated $\rightarrow E_{am} = (E_{co} + E_m ) \sin ({\omega_c t)}$

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<footer style = "font-size: 18px"> $\rightarrow$ Amplitude and Phase Frequency Modulation, Procedure to Generate Amplitude Modulated Waves $\rightarrow$ <span style = "color:blue"> Equation of Amplitude Modulated Wave </span> $\rightarrow$ Amplitude Modulation Index $\rightarrow$ Amplitude Modulation Index </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Amplitude Modulation Index </primary>
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- **Amplitude modulated wave has 3 frequencies**

- $\omega_c$ - $\omega_m$ : Lower side band

- $\omega_c$ : carrier frequency

- $\omega_c$ + $\omega_m$ : upper side band

- Amplitude modulation index ($\mu$) = $\frac {A_m}{A_c}$  = $\frac{E_{mo}}{E_{co}}$ = $\frac {\text {Amplitude of modulately wave (message)}}{\text{Amplitude of Carrier wave}}$

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<footer style = "font-size: 18px">Amplitude and Phase Frequency Modulation, Procedure to Generate Amplitude Modulated Waves $\rightarrow$ Equation of Amplitude Modulated Wave $\rightarrow$ <span style = "color:blue"> Amplitude Modulation Index </span> $\rightarrow$ Amplitude Modulation Index $\rightarrow$ Amplitude Modulation Index </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Amplitude Modulation Index </primary>
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- $\mu$ = $\frac{E_{mo}}{E_{co}}$ = $\frac{2 E_{mo}}{2 E_{co}}$ = $\frac {2 E_{mo} + E_{co} - E_{co}}{2 E_{co} + E_{mo} - E_{mo}}$

- = $\frac {(E_{co} + E_{mo}) - (E_{co} - E_{mo})}{ (E_{co} + E_{mo}) + (E_{co} - E_{mo})}$

- $\mu$ = $\frac{(E_{a m})^{max} - (E_{a m})^{min }} {(E_{a m})^{max } + (E_{a m})^{min }}$

- $E_{a_m}=[E_{c_0}+E_{mo} \sin (\omega_{m} t)] \sin (\omega_c t)$

- Maximum of amplitude modulated Electric field = $E_{c_0}+E_{m 0}$

- Minimum of amplitude modulated field =$E_{c_0}-E_{mo}$.

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<footer style = "font-size: 18px">Equation of Amplitude Modulated Wave $\rightarrow$ Amplitude Modulation Index $\rightarrow$ <span style = "color:blue"> Amplitude Modulation Index </span> $\rightarrow$ Amplitude Modulation Index $\rightarrow$ Amplitude Modulated Wave </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Amplitude Modulation Index </primary>
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- The envelope of amplitude modulated wave represents the original modulating wave.

- $\mu = 1$, efficient amplitude modulated transfer.

- <span style="color:blue">**Generation of Amplitude modulated wave**</span>

- <span style="color:green">After adder: </span> x(t) = m(t) + $m_c (t)$

- <span style="color:green">After square law device: </span> y(t) = B x (t) + C $x^2 (t)$

- $E_{mod}$ = m(t) = $E_{mo} \sin(\omega_m t)$

- $E_c (t)$ = $m_c (t) $= $E_{co} \sin (\omega_c t)$

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<footer style = "font-size: 18px">Amplitude Modulation Index $\rightarrow$ Amplitude Modulation Index $\rightarrow$ <span style = "color:blue"> Amplitude Modulation Index </span> $\rightarrow$ Amplitude Modulated Wave $\rightarrow$ Amplitude Modulated Wave </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Amplitude Modulated Wave </primary>
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- <span style="color:green">After adder:</span>

- $x(t)=E_{mo} \sin (\omega_m t)+ E_{co} \sin (\omega_c t)$

- <span style="color:green">After square law device: </span>

- $y(t)=  B x(t)+ C x^2 (t) $

- =  $B[E_{mo} \sin (\omega_m t)+E_{co} \sin (\omega_c t)] + c [E_{mo} \sin (\omega_m t)+E_{co} \sin (\omega_c t)]^2$

- y(t)=$B E_{mo} \sin (\omega_m t)+B E_{co} \sin (\omega_c t)$ + $c[E_{mo}^2 \sin ^2(\omega_{m} t)+E_{co}^2 \sin ^2 (\omega_c t)+2 E_{mo} E_{co} \sin (\omega_m t) \times \sin \ (\omega_c t)]$

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<footer style = "font-size: 18px">Amplitude Modulation Index $\rightarrow$ Amplitude Modulation Index $\rightarrow$ <span style = "color:blue"> Amplitude Modulated Wave </span> $\rightarrow$ Amplitude Modulated Wave $\rightarrow$ Band-Pass Filter </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Amplitude Modulated Wave </primary>
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- $\sin^2 (\omega t)  = \frac {1 - \cos (2 \omega t)}{2}$ ;

- $\sin (x) \sin(y) = \frac{1}{2} \times [ -\cos (x+y) + cos (x-y)]$

- $y(t)=B E_{mo} \sin (\omega_{m} t)+B E_{co} \sin (\omega_c t) $ + $ c [E_{mo}^2(\frac{1-\cos (2 \omega_m t)}{2})$ +$ E_{co}^2(\frac{1-\cos (2 \omega_c t)}{2})$ + $E_{mo} E_{co} \cos {(\omega_c-\omega_m) t}$
- $-E_{mo} E_{co} \cos {(\omega_c+\omega_m) t}]$

- = $B E_{mo} \sin ({\omega_m t}) t)$+$B E_{co} \sin (\omega_c t)$ -$ \frac{C E_{mo}^2}{2} \cos (2 \omega_{w} t)$ - $\frac{c E_{co}^2}{2} \cos (2 \omega_c t ) $ + $c E_{mo} E_{c_0} \ cos {(\omega_c-\omega_{m}) t} - c$ $E_{mo} E_{co} \cos \{(\omega_c+\omega_{m}) t} $ + $ \frac{c (E_{mo}^2 + E_{co}^2)}{2}$

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<footer style = "font-size: 18px">Amplitude Modulation Index $\rightarrow$ Amplitude Modulated Wave $\rightarrow$ <span style = "color:blue"> Amplitude Modulated Wave </span> $\rightarrow$ Band-Pass Filter $\rightarrow$ Band-Pass Filter </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Band-Pass Filter </primary>
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- We saw that after passing through the square law device, the  resultant wave has 6 frequencies.

- 1. $\omega_m$
- 2. $\omega_c$
- 3. 2$\omega_m$
- 4. 2$\omega_c$
- 5. $(\omega_m-\omega_c)$
- 6. $(\omega_c-\omega_m)$
 
- $\omega_c, \omega_c + \omega_m$

- $\omega_c$- $\omega_m$

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<footer style = "font-size: 18px">Amplitude Modulated Wave $\rightarrow$ Amplitude Modulated Wave $\rightarrow$ <span style = "color:blue"> Band-Pass Filter </span> $\rightarrow$ Band-Pass Filter $\rightarrow$ Frequency Modulation </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Band-Pass Filter </primary>
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- <span style="color:green">At output of band-pass filter</span>

- y(t) = $B E_{co} \sin (\omega_c t)$ + $c E_{mo} E_{co} \cos [(\omega_c - \omega_m)t]$ - $c E_{mo} E_{co} \cos[(\omega_c + \omega_m)t]$

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<footer style = "font-size: 18px">Amplitude Modulated Wave $\rightarrow$ Band-Pass Filter $\rightarrow$ <span style = "color:blue"> Band-Pass Filter </span> $\rightarrow$ Frequency Modulation $\rightarrow$ Frequency Modulation Index </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Frequency Modulation </primary>
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- In frequency modulation the frequency of the carrier wave is charged in acceleration with the modulating(message) wave.

- Carrier wave $E_c$ = $E_{co} \sin (\omega_c t)$

- Message wave $E_{mod}$ = $E_{mo} \sin (\omega_m t)$

- $E_{fm}$ = $E_{co} \sin ( \omega_c t + \theta)$ = $E{co} \sin (\omega_c t + \theta)$

- $\theta \rightarrow$ in accordance with message wave.

- $E_{fm} = E_{co} \sin (\omega_c t + \theta)$

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<footer style = "font-size: 18px">Band-Pass Filter $\rightarrow$ Band-Pass Filter $\rightarrow$ <span style = "color:blue"> Frequency Modulation </span> $\rightarrow$ Frequency Modulation Index $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Amplitude Procedure To Generate Amplitude Modulated Wave L-1 </Success>
### <primary style="font-size:24px"> Frequency Modulation Index </primary>
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- If $\omega_i$ is the instantaneous frequency of the frequency modulated wave.

- $\omega_i = \omega_c + \Delta \omega \sin (\omega_m t )$

- <span style="color:blue">Angular frequency</span> $\omega = 2 \pi \nu = \frac {d \theta }{dt}$

- d$\theta$  = $\omega$ dt $\Rightarrow \theta  = \int{d\theta} = \int{\omega d t }$ =$\int {\omega_i dt} = \int{[\omega_c + \Delta \omega sin (\omega_m t) ]}dt$

- <span style="color:green">Equation of frequency modulated wave</span>

- $E_{fm} = E_{co} \sin [\omega_c t - \frac {\Delta \omega}{\omega_m} \cos (\omega_m t)]$

- <span style="color:GREEN">Frequency modulation index</span> = $\frac {\Delta \omega}{\omega_m}$ = $\frac{\Delta f }{f_m}$

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<footer style = "font-size: 18px">Band-Pass Filter $\rightarrow$ Frequency Modulation $\rightarrow$ <span style = "color:blue"> Frequency Modulation Index </span> $\rightarrow$ Thank You $\rightarrow$  </footer>