Armstrong FM transmitter numerical

In the Armstrong method, frequency multipliers multiply both carrier frequency and deviation; the mixer shifts the carrier frequency but leaves the deviation unchanged.

Diagram

 NBFM        x65          Mixer       x50
 200 kHz -->[n1]-->13 MHz-->[X]-->2.2M-->[n2]--> WBFM
 df=25 Hz       df=1625 Hz   ^   df=1625    110 MHz
                             |              df=81.25k
                      [Osc 10.8 MHz]

Step 1: after first multiplier (n₁ = 65)

$$\begin{aligned} f_1 &= 65 \times 200\ \text{kHz} = 13\ \text{MHz} \\ \Delta f_1 &= 65 \times 25 = 1625\ \text{Hz} \end{aligned}$$

Step 2: mixer with 10.8 MHz

The mixer gives $13 \pm 10.8$ MHz, i.e. 2.2 MHz (difference) or 23.8 MHz (sum). Deviation stays 1625 Hz.

Step 3: second multiplier (n₂ = 50)

$$\begin{aligned} \Delta f &= 50 \times 1625 = 81250\ \text{Hz} = 81.25\ \text{kHz} \\ f_c &= 50 \times 2.2 = 110\ \text{MHz} \quad (\text{difference}) \\ f_c &= 50 \times 23.8 = 1190\ \text{MHz} \quad (\text{sum}) \end{aligned}$$

Valid carrier frequency

The sum term gives 1190 MHz (UHF), which is far outside the VHF FM broadcast range and is rejected by filtering after the mixer. The difference term is the usable one, giving

Answer: maximum frequency deviation Δf = 81.25 kHz; valid WBFM carrier f_c = 110 MHz (from the 2.2 MHz difference output).

Note: 110 MHz lies just above the 88–108 MHz broadcast band and 81.25 kHz is slightly above the 75 kHz standard, so in practice the oscillator frequency or multiplier would be adjusted slightly; with the given data, 110 MHz is the only practical (VHF) choice.