--- title: "Angle modulation; Armstrong method of FM" subject: "Communication Systems" chapter: 3 key: "semester_6/communication_systems:angle-modulation-armstrong" source: https://www.bctengineer.com/q/semester_6/communication_systems/angle-modulation-armstrong site: BCT Engineering --- # Angle modulation; Armstrong method of FM Communication Systems · Chapter 3: Angle Modulation - Asked in: 2073 Magh (CS I), 4+6 marks - Video: [FM Generation by Indirect Method of Armstrong (Basics, Block Diagram, Circuit & Working) Explained](https://www.youtube.com/watch?v=hhto6WRgl1g) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/angle-modulation-armstrong - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3 ## Question What is angle modulation? Explain with the help of equation and block diagram, the Armstrong method of generating FM signal. ## Short answer Angle modulation varies the carrier angle: PM, θ_i = ω_c t + k_p m(t); FM, θ_i = ω_c t + 2π k_f ∫ m dt. The Armstrong (indirect) method first generates NBFM (β << 1) using integrator, –90° phase shift, balanced modulator and adder, then uses frequency multipliers and a mixer to produce WBFM with desired deviation and carrier frequency. ## Answer #### 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. ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What approximation yields NBFM for small beta? A. cos(beta sin wm t) is about 1 and sin(beta sin wm t) is about beta sin wm t B. cos is about 0 and sin is about 1 C. Both are about beta D. Both are about 1 Answer: A. This gives s(t) = Ac cos wc t - beta Ac sin wm t sin wc t. (Key point: NBFM approximation) 2. How is NBFM realised? A. Differentiate m(t) and multiply by the carrier in an envelope detector B. Integrate m(t), multiply by a -90 degree carrier in a balanced modulator, and add the carrier C. Square the message and filter D. Pass the message through a limiter Answer: B. The integrator makes PM look like FM; adding the carrier gives the NBFM wave. (Key point: NBFM generation) 3. What happens in the frequency multiplier? A. Only carrier multiplies B. Deviation falls by n C. Carrier and deviation both multiply by n D. Beta stays the same Answer: C. Output is cos(n wc t + n beta sin wm t). (Key point: Multiplier effect) 4. Which is a drawback of Armstrong method? A. Poor carrier stability B. Cannot generate FM at all C. Needs no filters D. Noise and phase errors are multiplied by the many multiplier stages Answer: D. Multiplication also scales noise. 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