---
title: "General FM expression; FM using phase modulator"
subject: "Communication Systems"
chapter: 3
key: "semester_6/communication_systems:fm-general-expression-phase-modulator"
source: https://www.bctengineer.com/q/semester_6/communication_systems/fm-general-expression-phase-modulator
site: BCT Engineering
---

# General FM expression; FM using phase modulator

Communication Systems · Chapter 3: Angle Modulation

- Asked in: 2071 Magh (old course), 3+5 marks
- Video: [FREQUENCY MODULATION (FM) - DERIVATIONS.](https://www.youtube.com/watch?v=Ra0rbY_1qaE) — Prof. Barapate's Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/fm-general-expression-phase-modulator
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-3

## Question

Derive the general expression for frequency modulation. Explain with the block diagram how can you generate FM using phase modulator.

## Short answer

The FM wave is s(t)=A_c cos[2πf_c t + 2πk_f ∫ m(τ)dτ]. For tone m(t)=A_m cos2πf_m t, it becomes s(t)=A_c cos[2πf_c t + β sin2πf_m t] with β=Δf/f_m. FM can be generated using a phase modulator preceded by an integrator: the integrated message phase-modulates the carrier, producing indirect FM.

## Answer

#### General expression for FM

Let the carrier be $c(t) = A_c\cos 2\pi f_c t$. A general angle-modulated wave is

$$
s(t) = A_c\cos\theta_i(t)
$$

The **instantaneous frequency** is $f_i(t) = \frac{1}{2\pi}\frac{d\theta_i}{dt}$.

In FM, the instantaneous frequency varies linearly with the message $m(t)$:

$$
f_i(t) = f_c + k_f\, m(t)
$$

where $k_f$ is the frequency sensitivity (Hz/V). Integrating,

$$
\theta_i(t) = 2\pi\int_0^t f_i(\tau)\,d\tau = 2\pi f_c t + 2\pi k_f \int_0^t m(\tau)\,d\tau
$$

Hence the **general FM expression**:

$$
s_{FM}(t) = A_c\cos\left[2\pi f_c t + 2\pi k_f\int_0^t m(\tau)\,d\tau\right]
$$

**Single-tone case:** for $m(t) = A_m\cos 2\pi f_m t$,

$$
\theta_i(t) = 2\pi f_c t + \frac{k_f A_m}{f_m}\sin 2\pi f_m t
$$

$$
s(t) = A_c\cos\left[2\pi f_c t + \beta\sin 2\pi f_m t\right], \qquad \beta = \frac{k_f A_m}{f_m} = \frac{\Delta f}{f_m}
$$

#### FM generation using a phase modulator

A PM wave is $A_c\cos[2\pi f_c t + k_p\, x(t)]$: its phase follows the input $x(t)$. Comparing with the FM expression, if the input to a phase modulator is the **integral of the message**, the output is FM.

```text
 m(t)  +------------+ x(t)=int m  +-----------+  FM wave
 ----->| Integrator |------------>|  Phase    |---------->
       +------------+             | modulator |
                                  +-----------+
                                        ^
                                  carrier Ac cos wc t
```

With $x(t) = \int_0^t m(\tau)d\tau$:

$$
s(t) = A_c\cos\left[2\pi f_c t + k_p\int_0^t m(\tau)\,d\tau\right]
$$

This is FM with $k_p = 2\pi k_f$. This is the basis of the **indirect (Armstrong) method**: the phase modulator is built from a balanced modulator and a 90° phase shift of a crystal oscillator, giving NBFM with high carrier stability. Frequency multipliers and a mixer then convert it to wideband FM. (Conversely, a differentiator followed by a frequency modulator gives PM.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In FM, which quantity varies linearly with the message m(t)?
   A. Carrier amplitude
   B. Instantaneous frequency, fi = fc + kf m(t)
   C. Instantaneous phase
   D. Carrier power
   Answer: B. FM is defined by fi(t) = fc + kf m(t); the phase is then the integral of this. (Key point: FM definition)

2. Which is the general FM expression?
   A. Ac cos[2π fc t + kf m(t)]
   B. Ac cos[2π fc t + 2π kf d m/dt]
   C. Ac cos[2π fc t + 2π kf integral of m(t) dt]
   D. Ac [1 + kf m(t)] cos 2π fc t
   Answer: C. Integrating fi gives theta = 2π fc t + 2π kf integral m, so the phase contains the integral of the message. (Key point: General FM expression)

3. For m(t) = Am cos 2π fm t, what is the modulation index?
   A. beta = kf Am fm
   B. beta = Am/fm
   C. beta = kf fm/Am
   D. beta = kf Am / fm = delta f / fm
   Answer: D. Integrating the cosine gives (kf Am/fm) sin 2π fm t, so beta = kf Am/fm = delta f/fm. (Key point: Single-tone index)

4. How can a phase modulator generate FM?
   A. Feed the integral of the message into the phase modulator
   B. Feed the derivative of the message into it
   C. Feed the message directly into it
   D. Square the message first
   Answer: A. A phase modulator makes phase proportional to its input; if the input is the integral of m(t), the phase matches the FM expression. This is the basis of the Armstrong method. (Key point: FM from a phase modulator)

## Related in Communication Systems

- [Angle modulation; single tone FM time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-single-tone-fm-time-freq) — 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
- [Stereo FM encoder and decoder block diagram](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Astereo-fm-encoder-decoder-blocks) — Communication Systems, chapter 3
- [Analog communication block diagram; modulation and its reasons](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aanalog-comm-modulation-reasons) — Communication Systems, chapter 1, asked 3×
- [Modulation; analog communication system block diagram](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-analog-block-diagram) — Communication Systems, chapter 1, asked 3×
- [Reasons for modulation; digital communication block diagram](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-reasons-digital-adv-duplex) — Communication Systems, chapter 1

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- [Frequency counter operation with block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Afrequency-counter) — Digital Logic, chapter 11, asked 14×
- [Frequency modulation](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afm-explain) — Data Communication, chapter 5
- [Modulation; AM and FM modulated waves](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Amodulation-am-fm-wave) — Basic Electronics Engineering, chapter 5