A 100 MHz un-modulated carrier delivers 100 Watts of power to a load. The carrier is frequency modulated by a 2 kHz modulating signal causing a maximum frequency deviation of 8 kHz... A 100 MHz un-modulated carrier delivers 100 Watts of power to a load. The carrier is frequency modulated by a 2 kHz modulating signal causing a maximum frequency deviation of 8 kHz. This FM signal is coupled to a load through an ideal Band Pass filter with 100MHz as center frequency and a variable bandwidth. Calculate the modulation index. Determine the number of sets of significant side frequencies. Calculate their powers of each significant side frequencies. Determine the bandwidth of the FM signal if the bandwidth is less than 15 kHz. Read more

Steps to Solve

1. Identify Given Values

   • Carrier frequency, ( f_c = 100 , \text{MHz} = 100 \times 10^6 , \text{Hz} )
   • Modulating frequency, ( f_m = 2 , \text{kHz} = 2000 , \text{Hz} )
   • Maximum frequency deviation, ( \Delta f = 8 , \text{kHz} = 8000 , \text{Hz} )

2. Calculate Modulation Index

   • The modulation index ( m ) is calculated using the formula: $$ m = \frac{\Delta f}{f_m} $$
   • Substituting the values: $$ m = \frac{8000 , \text{Hz}}{2000 , \text{Hz}} = 4 $$

3. Determine Number of Significant Side Frequencies

   • The number of significant side frequencies can be determined using: $$ n = 2m + 1 $$
   • For ( m = 4 ): $$ n = 2(4) + 1 = 9 $$

4. Calculate Powers of Significant Side Frequencies

   • The power of each side frequency ( P_n ) is given by: $$ P_n = \frac{P_c}{(n^2)} $$
   • Where ( P_c = 100 , \text{Watts} ):
     • For significant frequencies, we take ( n = 1 ) to ( n = 4). Calculate:
       • ( n = 1: P_1 = \frac{100}{1^2} = 100 , \text{W} )
       • ( n = 2: P_2 = \frac{100}{2^2} = 25 , \text{W} )
       • ( n = 3: P_3 = \frac{100}{3^2} = 11.11 , \text{W} )
       • ( n = 4: P_4 = \frac{100}{4^2} = 6.25 , \text{W} )

5. Determine Bandwidth of the FM Signal

   • The bandwidth ( BW ) can be calculated with Carson's Rule: $$ BW = 2(\Delta f + f_m) $$
   • Substituting the values: $$ BW = 2(8000 , \text{Hz} + 2000 , \text{Hz}) = 2(10000, \text{Hz}) = 20000 , \text{Hz} = 20 , \text{kHz} $$