NBFM and WBFM; Armstrong method

Narrow band and wide band FM

• Narrow band FM (NBFM): FM with a small modulation index, $\beta = \Delta f/f_m \ll 1$ (practically $\beta < 0.3$). Its bandwidth is about $2f_m$, like AM, with only the carrier and one pair of sidebands:

$$s(t) \approx A_c\cos\omega_ct - \beta A_c\sin\omega_mt\,\sin\omega_ct$$

• Wide band FM (WBFM): FM with $\beta \gg 1$. It has many significant sidebands and bandwidth $B_T \approx 2(\Delta f + f_m)$ (Carson's rule). Example: FM broadcast with $\Delta f = 75$ kHz, $W = 15$ kHz, $\beta = 5$.

Generation of WBFM by Armstrong (indirect) method

Direct FM using an LC oscillator is not stable enough. Armstrong's method first makes a stable NBFM wave with a crystal oscillator, then uses frequency multiplication to obtain the required large deviation.

 m(t)->[Integrator]->[Balanced]--( Σ )-> NBFM
                     [modulator] ^ -
                         ^       | +
          +--[-90 deg]---+       |
 [Crystal osc fc1]---------------+
          (NB phase modulator)

 NBFM ->[ x n1 ]->[ Mixer ]->[ x n2 ]->[PA]-> WBFM
                      ^
                 [Osc fc2]

Step 1: NBFM by phase modulation of the integrated message. The message is integrated (so that phase modulation of $\int m\,dt$ gives FM) and fed to a balanced modulator with a carrier shifted by −90°. Adding the carrier gives

$$s_1(t) = A_c\cos\omega_{c1}t - A_c\beta_1\sin\omega_mt\,\sin\omega_{c1}t \approx A_c\cos(\omega_{c1}t + \beta_1\sin\omega_mt)$$

with $\beta_1$ kept small (< 0.5) to limit distortion. Typical values: $f_{c1} = 100$–200 kHz, $\Delta f_1 \approx 25$ Hz.

Step 2: frequency multiplier (×n₁). A non-linear device followed by a BPF tuned to the n-th harmonic. The output is $\cos(n_1\omega_{c1}t + n_1\beta_1\sin\omega_mt)$: both carrier and deviation are multiplied by $n_1$.

Step 3: mixer (frequency converter). Mixing with a crystal oscillator $f_{c2}$ shifts the carrier to $n_1f_{c1} - f_{c2}$ while the deviation stays $n_1\Delta f_1$. This lets the final carrier be set independently of the deviation.

Step 4: frequency multiplier (×n₂) to reach the final values:

$$f_c = n_2(n_1f_{c1} - f_{c2}), \qquad \Delta f = n_1n_2\Delta f_1$$

Step 5: power amplifier and antenna.

Example: $\Delta f_1 = 25$ Hz needs $n_1n_2 = 3000$ for $\Delta f = 75$ kHz.

Advantages: high carrier stability from crystal oscillators; good linearity. Disadvantages: many multiplier stages; noise is multiplied too; more complex.