Armstrong FM transmitter numerical (Δf1 = 24 Hz)

Given: $f_1 = 200$ kHz, $\Delta f_1 = 24$ Hz, $n_1 = 65$, $f_{LO} = 10.8$ MHz, $n_2 = 50$.

Rules: a multiplier multiplies both carrier and deviation; a mixer changes only the carrier (deviation unchanged). The mixer output is taken as the difference frequency (normal Armstrong design, to bring the carrier down before the second multiplier).

Step 1: After the first multiplier ($\times 65$)

$$\begin{aligned} f_2 &= n_1 f_1 = 65 \times 200\ \text{kHz} = 13\ \text{MHz} \\ \Delta f_2 &= n_1 \Delta f_1 = 65 \times 24 = 1560\ \text{Hz} \end{aligned}$$

Step 2: After the mixer

$$f_3 = f_2 - f_{LO} = 13 - 10.8 = 2.2\ \text{MHz}, \qquad \Delta f_3 = 1560\ \text{Hz}$$

Step 3: After the second multiplier ($\times 50$)

$$\begin{aligned} f_c &= n_2 f_3 = 50 \times 2.2\ \text{MHz} = 110\ \text{MHz} \\ \Delta f_c &= n_2 \Delta f_3 = 50 \times 1560 = 78\,000\ \text{Hz} = 78\ \text{kHz} \end{aligned}$$

Check: $\Delta f_c = n_1 n_2 \Delta f_1 = 3250 \times 24 = 78$ kHz.

(If the sum frequency 23.8 MHz were selected, $f_c$ would be 1190 MHz with the same deviation, which is not a practical FM carrier.)

Answer: maximum frequency deviation $\Delta f_c = 78$ kHz, carrier frequency $f_c = 110$ MHz.