--- title: "NBFM and WBFM; Armstrong method" subject: "Communication Systems" chapter: 3 key: "semester_6/communication_systems:nbfm-wbfm-armstrong" source: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-wbfm-armstrong site: BCT Engineering --- # NBFM and WBFM; Armstrong method Communication Systems · Chapter 3: Angle Modulation - Asked in: 2076 Baisakh (CS I), 2+6 marks - Video: [ARMSTRONG METHOD || GENERATION OF FM WAVE || ADC || Electronics and Communication Engineering](https://www.youtube.com/watch?v=pGzV1bsXoMk) — Friend's Explanation - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/nbfm-wbfm-armstrong - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3 ## Question Define narrow band and wide band FM. Explain the process of generation of wide band FM wave using Armstrong method. ## Short answer NBFM has β ≪ 1, bandwidth ≈ 2f_m; WBFM has β ≫ 1, bandwidth ≈ 2(Δf+f_m). Armstrong method: integrate the message, phase-modulate a crystal oscillator to produce NBFM with small deviation, then multiply frequency and mix to reach desired large deviation and carrier frequency. ## Answer #### 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. ```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] ``` **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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. When is an FM wave called narrow band? A. When beta << 1 (below about 0.3), giving bandwidth about 2fm B. When beta >> 1 C. When Delta f = 75 kHz D. When the carrier is below 1 MHz Answer: A. NBFM has small beta and only a carrier plus one sideband pair. (Key point: NBFM definition) 2. Why is the indirect Armstrong method used instead of direct LC-oscillator FM? A. It needs no multipliers B. It uses a crystal oscillator, giving a much more stable carrier C. It gives larger deviation directly D. It avoids the need for an integrator Answer: B. Direct LC oscillator FM drifts; the crystal gives frequency stability. (Key point: Why Armstrong method) 3. What is the purpose of the mixer between the two multipliers? A. Increase the deviation B. Remove noise C. Shift the carrier to a suitable value without changing the deviation D. Integrate the message Answer: C. It lets the final carrier be set independently of the deviation. (Key point: Role of mixer) 4. Which relations hold at the output? A. fc = n1 n2 fc1 and Delta f = n2 Delta f1 B. fc = n1 n2 (fc1 - fc2) and Delta f = Delta f1 C. Delta f = (n1 + n2) Delta f1 D. fc = n2(n1 fc1 - fc2) and Delta f = n1 n2 Delta f1 Answer: D. The mixer subtracts fc2 only from the carrier; deviation is multiplied by both n1 and n2. (Key point: Final formulas) ## Related in Communication Systems - [NBFM spectrum vs AM; Armstrong NBFM generation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-spectrum-vs-am-armstrong) — Communication Systems, chapter 3 - [Armstrong FM modulator: find n1 and n2](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aarmstrong-n1-n2-numerical) — Communication Systems, chapter 3 - [Armstrong (indirect) FM generation; pre-emphasis](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aarmstrong-indirect-pre-emphasis) — Communication Systems, chapter 3 - [Armstrong FM transmitter numerical](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aarmstrong-numerical-200khz-25hz) — Communication Systems, chapter 3 - [Armstrong FM transmitter numerical (Δf1 = 24 Hz)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aarmstrong-numerical-24hz) — Communication Systems, chapter 3 - [NBFM time domain expression; NBFM to WBFM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anbfm-time-domain-wbfm) — Communication Systems, chapter 3 ## Related in other subjects - [Astable multivibrator for square wave generation](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Aastable-multivibrator-square-wave) — Electronics Devices and Circuit, chapter 5