# frequency modulation direct method

Xavier University – Ateneo de Cagayan

Submitted by:Submitted to:

Dapal, Jamie DaisyletteEngr. Mary Jean Apor

Fermano, Jerick James

Oraiz, Gerard James

Sabunod, Shane John

Siman, Lester C.

Objectives:

1. To obtain a frequency modulated signal via direct method

2. To understand the process of frequency modulation by direct method Background Theory:

Frequency modulation uses the information signal, Vm(t) to vary the carrier frequency within some small range about its original value. Here are the three signals in mathematical form: Information: Vm(t)

Carrier: Vc(t) = Vco sin ( 2 fc t +

FM: VFM (t) = Vco sin (2 fc + (f/Vmo) Vm (t)t + We have replaced the carrier frequency term, with a time-varying frequency. We have also introduced a new term: f, the peak frequency deviation. In this form, you should be able to see that the carrier frequency term: fc + (f/Vmo) Vm (t) now varies between the extremes of fc - f and fc + f. The interpretation of f becomes clear: it is the farthest away from the original frequency that the FM signal can be. Sometimes it is referred to as the "swing" in the frequency. We can also define a modulation index for FM, analogous to AM: = f/fm , where fm is the maximum modulating frequency used. The simplest interpretation of the modulation index, is as a measure of the peak frequency deviation, f. In other words, represents a way to express the peak deviation frequency as a multiple of the maximum modulating frequency, fm, i.e. f = fm. Here is a simple FM signal:

Mathematical process of generating FM: (direct method)

Consider a CW signal with constant envelope but time-varying phase, so

Upon defining the total instantaneous angle

we can express xc(t) as

Conceptually, direct FM is straightforward and requires nothing more than a voltage controlled oscillator (VCO) whose oscillation frequency has a linear dependence on applied voltage. It’s possible to modulate a conventional tuned-circuit oscillator by introducing a variable-reactance element as part of the LC parallel resonant circuit. If the equivalent capacitance has a time dependence of the form C(t) = C0 – Cx(t)

And if Cx(t) is “small enough” and “slow enough,” then the oscillator produces xc(t) = Ac cos θc(t) where

However, instead of a LC parallel resonant circuit, here we used a RC parallel resonant circuit. Therefore

Also since we are using a 2 resistor resonant circuit R = R1 + R2 Letting = and assuming |(C/C0)x(t)|

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