VCO-Based FM Modulator — Output Expression, Demodulation, Carson Bandwidth: 17-Phys-B4 Signals and Communications, December 2016, Q6

Question 6: VCO-Based FM Modulator — Output Expression, Demodulation, Carson Bandwidth (1/6 of paper)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. VCO characteristic: a straight line through $(f,V)=(f_c,0)$ with slope $dV/df=10^{-4}\text{ V/Hz}$. Modulator output power $=10$ (part a). Message bandwidth $W=10\text{ kHz}$, peak value $m_p=2\text{ V}$ (part c).

VCO characteristic: V is linear in f near f_c, slope dV/df = 10⁻⁴ V/Hz.

Find. (a) FM modulator output $s(t)$ in terms of $m(t)$; (b) a demodulator block diagram; (c) approximate FM bandwidth.

Approach. Invert the VCO's given $V$-vs-$f$ slope to obtain its frequency sensitivity $K_v$ (in Hz/V, the quantity that actually drives instantaneous frequency); write the standard FM instantaneous-phase expression using $K_v\,m(t)$ as the frequency-deviation term, with amplitude set by the given output power; choose a standard FM discriminator for the demodulator; and apply Carson's rule with the peak deviation $K_v\times$(peak of $m$).

Part (a) — FM modulator output. The given slope is $dV/df=10^{-4}\text{ V/Hz}$ (volts needed per hertz of shift); the VCO's frequency SENSITIVITY is its reciprocal, $$K_v=\frac{df}{dV}=\frac{1}{10^{-4}}=\boxed{10^4\text{ Hz/V}}$$ so driving the VCO with $m(t)$ volts produces an instantaneous frequency $f_i(t)=f_c+K_vm(t)$ and phase $\theta(t)=2\pi f_ct+2\pi K_v\int_0^tm(\lambda)\,d\lambda$. The output power is $A_c^2/2=10$, so $A_c=\sqrt{20}=2\sqrt5$: $$s(t)=\boxed{2\sqrt5\,\cos\!\Big[2\pi f_ct+2\pi\!\cdot\!10^4\!\int_0^tm(\lambda)\,d\lambda\Big]}$$
Part (b) — FM demodulator. A standard non-coherent FM discriminator recovers $m(t)$ without needing a phase-locked local oscillator: limit the signal to remove amplitude noise/variation, differentiate (which converts instantaneous-frequency variation into amplitude variation, since $d\theta/dt=2\pi f_c+2\pi K_vm(t)$), envelope-detect, then low-pass filter to strip the DC term left by $f_c$: $$s_{FM}(t)\ \xrightarrow{\text{limiter}}\ \xrightarrow{d/dt}\ \xrightarrow{\text{envelope det.}}\ \xrightarrow{\text{LPF, no dc}}\ \propto m(t)$$ (a PLL-based demodulator, using a second VCO of the same $K_v$ locked to $s_{FM}(t)$, would be an equally standard alternative.)

FM discriminator: limiter → differentiator → envelope detector → LPF, recovering m(t).

Part (c) — approximate FM bandwidth (Carson's rule). The peak frequency deviation is the VCO sensitivity times the message's peak value: $$\Delta f=K_v\times m_p=10^4\text{ Hz/V}\times2\text{ V}=\boxed{20\text{ kHz}}$$ Carson's rule then bounds the FM bandwidth using the message bandwidth $W=10\text{ kHz}$: $$BW_{FM}\approx2(\Delta f+W)=2(20+10)\text{ kHz}=\boxed{60\text{ kHz}}$$

Question 6 — final results
Quantity	Result
VCO sensitivity $K_v$	$10^4\text{ Hz/V}$
Modulator output $s(t)$	$2\sqrt5\cos\big[2\pi f_ct+2\pi\cdot10^4\!\int m\,d\lambda\big]$
Demodulator	limiter → differentiator → envelope det. → LPF
Peak deviation $\Delta f$	$20\text{ kHz}$
FM bandwidth (Carson)	$\approx60\text{ kHz}$

Concept & Theory

Topic. Direct FM generation from a voltage-controlled oscillator, discriminator-based demodulation, and Carson's-rule bandwidth.

Key relations. Instantaneous frequency $f_i(t)=f_c+K_vm(t)$; instantaneous phase $\theta(t)=2\pi\int f_i\,dt$; Carson's rule $BW\approx2(\Delta f+W)$.

Why this works. A VCO's output frequency IS the modulating variable by construction, so once its voltage-to-frequency sensitivity $K_v$ is known, the FM signal's phase is simply the running time-integral of that instantaneous frequency deviation; Carson's rule bounds the (in principle infinite) FM spectrum by the sum of the peak deviation and the message bandwidth because the significant Bessel-function sidebands die off rapidly beyond that point.

Common pitfall. Using the given VCO slope directly as $K_v$ (it is $dV/df=10^{-4}\text{ V/Hz}$, the RECIPROCAL of the frequency sensitivity, not $K_v$ itself), and using the message's average or RMS value instead of its PEAK value when computing the Carson-rule deviation $\Delta f$.

Source. S. Haykin and M. Moher, Communication Systems, 5th ed., §4.2–4.5 (FM generation via VCO, Carson's rule, discriminator demodulation); B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed., Ch. 5 (angle modulation, frequency deviation and bandwidth).

Question 6 of 6: VCO-Based FM Modulator — Output Expression, Demodulation, Carson Bandwidth

Question 6: VCO-Based FM Modulator — Output Expression, Demodulation, Carson Bandwidth (1/6 of paper)