Frequency modulation

Frequency modulation (FM) is a modulation system developed in 1936 as an alternative to AM. Amplitude modulation is not an efficient way to transmit voice and music, and it is vulnerable to noise. FM overcomes both weaknesses: it has strong resistance to noise (better noise immunity) and gives high-fidelity reproduction, which is why it is used at VHF for music and speech broadcasts. There is also a key difference in bandwidth behaviour — in AM the transmission bandwidth grows with the signal bandwidth, whereas in FM the transmission bandwidth is large but is essentially unaffected by the signal bandwidth.

Frequency modulation is the process by which the frequency of the carrier is varied in accordance with the instantaneous amplitude of the modulating signal, while the carrier amplitude stays constant. When the modulating signal is zero, the carrier stays at its rest (centre) frequency $f_c$; a rising modulating amplitude raises the carrier frequency and a falling amplitude lowers it, with the maximum deviation occurring at the positive and negative peaks of the modulating signal (Figures 8.5.1 and 8.5.2).

Mathematical representation of FM

Let the modulating voltage be

and the carrier be

Writing the total instantaneous phase angle as $\phi = \omega_c t + \theta$ (8.29) gives $v_c = V_c \sin\phi$ (8.30), with $\omega_c = d\phi/dt$. During frequency modulation the carrier's angular frequency varies with time in step with the modulating voltage:

where $K$ is the modulation deviation constant, or frequency sensitivity of the modulator.

Frequency deviation and carrier swing

Dividing (8.31) by $2\pi$, the instantaneous carrier frequency is $f = f_c + K\dfrac{V_m}{2\pi}\cos(\omega_m t)$. It reaches its extremes when $\cos(\omega_m t) = \pm 1$:

The frequency deviation $\Delta f$ (also written $\delta$) is the change of carrier frequency from its centre value produced by the modulating signal:

Importantly, the frequency deviation is independent of the modulating frequency and is proportional to the amplitude of the modulating signal. The carrier swing (CS) is the total variation from the minimum to the maximum frequency:

Integrating (8.31) and neglecting the constant of integration gives the phase

so the FM carrier voltage becomes

with $m_f = \dfrac{K V_m}{\omega_m} = \dfrac{K V_m}{2\pi f_m} = \dfrac{\delta}{f_m}$.

Modulation index

The FM modulation index $m_f$ is the ratio of the maximum frequency deviation to the modulating-signal frequency:

It has no unit and is expressed as a decimal. For a fixed frequency deviation, $m_f$ decreases as the modulating frequency rises. The modulation index is what is used to calculate the bandwidth and the number of significant sidebands of an FM wave. (Unlike AM, $m_f$ can be much greater than 1.)

Deviation ratio

The deviation ratio describes the worst case of the modulation index — the ratio of the maximum allowed frequency deviation to the maximum frequency of the modulating signal:

It is widely quoted in FM broadcasting and TV: for FM broadcast it is $75\text{ kHz}/15\text{ kHz} = 5$, and for TV sound it is $25\text{ kHz}/15\text{ kHz} = 1.67$. It is used to find the worst-case bandwidth of a transmitter.

Percent modulation

In FM, percent modulation has a different meaning from AM. Because the modulating signal varies only the carrier frequency, percent modulation compares the actual carrier deviation with the maximum permitted deviation:

Broadcast FM

By FCC allocation the FM broadcast band runs from 88 to 108 MHz. Each station is allowed a deviation of $\pm 75$ kHz, so the FM channel bandwidth is $2\times 75 = 150$ kHz. A 25 kHz guard band is added on either side of the channel to prevent interference between adjacent stations, giving a total of 200 kHz per station (Figure 8.5.3). Commercial FM stations must follow these values: maximum frequency deviation $\pm 75$ kHz; carrier frequency stability $\pm 2$ kHz; maximum allowed audio frequency 15 kHz; guard band 50 kHz (25 kHz on each side); and maximum bandwidth 200 kHz per channel.

Frequency spectrum of FM

Unlike AM, an FM wave modulated by a single tone contains an infinite number of sidebands, whose amplitudes are given by Bessel functions of the first kind of the modulation index. The FM wave expands as

where $J_0, J_1, J_2, \dots$ are the zero-, first-, second-order Bessel coefficients, $\omega_c = 2\pi f_c$, $\omega_m = 2\pi f_m$ and $V_c$ is the peak unmodulated carrier value.

From this expansion we observe: (1) the first term is the carrier; (2) the FM wave contains infinitely many sidebands, each separated from the next by $f_m$; (3) the sidebands are distributed symmetrically about $f_c$, so sidebands equidistant from the carrier have equal amplitudes; and (4) the amplitudes of the carrier and sidebands depend on the $J_n$ coefficients, which depend on the modulation index and are read from a standard Bessel table. Figure 8.5.4 shows the FM spectrum for particular values of $m_f$ and $f_m$.

Significant sidebands

Although an FM wave has infinitely many sidebands, most are too small to matter. Significant sidebands are those whose amplitude is at least 1% of the carrier amplitude. Their number is fixed by the modulation index.

Bandwidth

The theoretical bandwidth of an FM wave is infinite, but in practice it is set by the number of significant sidebands. Bandwidth is defined as the width of the frequency spectrum containing all side frequencies with amplitude $\ge 1\%$ of the carrier. For $n$ significant sidebands,

A convenient estimate is Carson's rule — the bandwidth is twice the sum of the maximum frequency deviation and the highest modulating frequency:

FM modulators

An FM signal is generated with a device whose reactance can be varied by an applied voltage — a FET, a BJT or a varactor diode. If such a device is placed across the tank circuit of an oscillator, then varying its reactance with the modulating voltage varies the oscillator frequency and produces FM. Two common circuits are the varactor-diode modulator and the transistor reactance modulator.

Varactor diode modulator

A varactor diode is a diode whose junction capacitance changes with the reverse voltage applied to it. In the varactor-diode modulator (Figure 8.5.5) the diode is connected across the oscillator tank circuit through a coupling capacitor C of relatively large value, which provides DC isolation between the oscillator and the diode; an RF choke L passes the low-frequency modulating signal but blocks the RF. Working: the modulating signal is applied in series with the DC bias; passing through the RF choke, it appears across the varactor diode and changes its capacitance in step with the modulating signal. This alters the total tank capacitance and hence the oscillator frequency, producing an FM wave. Being a simple two-terminal device, the varactor diode is also used for automatic frequency control and remote tuning.

Reactance modulator

The transistor reactance modulator (Figure 8.5.6) is widely used for FM generation and gives better stability than the varactor circuit; it is used with a Hartley or Colpitts oscillator. A capacitor $C_f$ and resistor $R_B$ introduce a 90° phase shift between the collector voltage and current (the current leads the voltage by 90°), so the stage behaves like a voltage-controlled capacitor placed in parallel with the oscillator's tuned circuit. Working: the information signal applied to the base has the same effect as varying the transistor bias, which increases or decreases this voltage-controlled capacitance. That changes the effective capacitance of the tuned circuit $L_1 C_1$ and hence its resonance frequency, so the oscillator frequency varies with the modulating-signal amplitude and the output is an FM signal.

FM transmitter …