Angle modulation; Armstrong method of FM

Angle modulation

Angle modulation varies the angle of the carrier according to the message, with constant amplitude: $s(t) = A_c\cos\theta_i(t)$.

• PM: $\theta_i(t) = \omega_ct + k_pm(t)$
• FM: $\theta_i(t) = \omega_ct + 2\pi k_f\int_0^tm(\tau)d\tau$, so $f_i = f_c + k_fm(t)$

For a tone $m = A_m\cos\omega_mt$, FM gives $s(t) = A_c\cos(\omega_ct + \beta\sin\omega_mt)$, $\beta = k_fA_m/f_m$.

Armstrong method of generating FM

Armstrong's (indirect) method generates a stable narrow-band FM wave with a crystal oscillator and then converts it to wide-band FM by frequency multiplication.

 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]

1. NBFM generation. For small $\beta$:

$$\begin{aligned} s(t) &= A_c\cos(\omega_ct + \beta\sin\omega_mt) \\ &= A_c\cos\omega_ct\cos(\beta\sin\omega_mt) - A_c\sin\omega_ct\sin(\beta\sin\omega_mt) \\ &\approx A_c\cos\omega_ct - \beta A_c\sin\omega_mt\,\sin\omega_ct \end{aligned}$$

using $\cos x \approx 1$, $\sin x \approx x$ for small $x$. This is realised by:

• integrating $m(t)$ (so that phase modulation gives FM),
• multiplying it in a balanced modulator with the carrier shifted by −90° ($\sin\omega_ct$),
• adding the carrier $A_c\cos\omega_ct$ back.

Typical: $f_{c1} = 200$ kHz, $\beta < 0.5$, $\Delta f_1 \approx 25$ Hz.

2. Frequency multiplier ×n₁. A non-linear stage and filter; output $\cos(n_1\omega_{c1}t + n_1\beta\sin\omega_mt)$. Carrier and deviation both multiply by $n_1$.

3. Mixer. Mixing with a crystal oscillator $f_{c2}$ moves the carrier to $n_1f_{c1} - f_{c2}$ without changing deviation. This keeps the final carrier at the wanted value.

4. Frequency multiplier ×n₂ and power amplifier:

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

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

Advantages: crystal-stable carrier, good linearity. Disadvantages: many multiplier stages, multiplied noise and phase errors, complex circuit.