Frequency Modulation Equations

FM (Frequency Modulation) has been around for a long time. This system existed beside AM (Amplitude Modulation) and had a few flaws. Even though we couldn’t differentiate the FM transmitter potential, FM itself had no problems. But, when wireless communication was a new concept, people’s requirements were not fulfilled by only using FM. This was because narrower bandwidth which was useful in reducing noise was required. So, to overcome all these obstacles, they started using amplitude signal modulation. This article elaborates on frequency modulation equations, properties of frequency modulation, their advantages and disadvantages, and questions pertaining to frequency modulation. 

What is frequency modulation? 

According to the frequency modulation equation UPSC notes, the frequency of the carrier signal is adjusted by (in line width) the amplitude of the input modulating signal in frequency modulation. A single tone sine wave is also used as the input to gain required signals. If the modulating (input) signal increases, only the carrier frequency (fc) will increase. Also, the input signal will reach its peak when the carrier frequency (fc )max is at its highest. 

The carrier frequency will only deviate from its initial value. As the amplitude of the modulating (input) signal decreases, so will the carrier frequency signals. When the input signal is at its lowest, the carrier frequency is minimum (fc)min. The carrier deviates from its typical value only slightly. When the input signal value is 0V, the carrier frequency will be at its normal value (free-running) fc.

The message signal will hold the specified data, but the carrier signal will hold nothing. FM modulated signals will come from the modulation of these signals. This signal is necessary as the frequency of this signal fluctuates based on the signal’s amplitude. This frequency shift can also be expressed in kHz ( kilohertz). For example, if the frequency variation is 6 kHz up and down, it is written as 6 kHz. 

Frequency modulation voltage derivation

Frequency modulation will happen when the carrier signal’s frequency is modified in response to the modulating signal’s instantaneous amplitude. Meanwhile, the carrier amplitude in FM will remain constant. Frequency deviation ((δ)) is the difference in carrier frequency. That is the difference between the modulated and unmodulated carriers.

FM is a non-linear process containing frequency deviation.

Mathematical derivation of frequency modulation

Let the instantaneous frequency of modulated wave be: 

         Wi = Wc+δ

The frequency variation above or below Wc will be determined by the modulating signal’s instantaneous amplitude, i.e.

         δ=kVm cos Wmt

Here k represents the frequency deviation sensitivity of frequency modulation.

Then, the instantaneous frequency will be: 

Wi = Wc+δ

Wi = Wc+kVm cos Wmt        (i)

The modulated signal will be represented mathematically as: 

Vfm=Vc sin θ                    (ii)

Here, θ represents the instantaneous phase, and you can find this using 

Wi = dθ /dt

Wi dt = dθ

∫dθ= ∫Wi dt

Now substitute the values of equation (1) in the expression given above: 

θ= ∫ (Wc+kVm cos Wmt ) dt

θ= Wct+ (KVm/Wm) sin Wmt

θ= Wct+Mf sin Wmt         (iii)

Here, Mf will be our frequency modulation index which is equal to (KVm/Wm)

Now, we can substitute the values of equation (3) in equation (2) 

Vfm=Vc sin (Wct+Mf sin Wmt)

This is our required equation of frequency modulation

The mathematical equation of frequency modulation

The frequency modulation equation includes: 

v = A sin [ wct + (Δf / fm) sin wmt ]

Here, 

A is the amplitude of frequency modulation signals. 

∆f is the frequency deviation

Mf is the modulation index of frequency modulation.

mf = ∆f/fm

wm = 2π fm WC = 2π fC 

This is our required mathematical equation to represent frequency modulation equations’ importance. 

FM is also known as a constant bandwidth system. But why? 

FM constant is known as a constant bandwidth system. Let’s understand why it is so with this simple example: 

• If Δf = 75 KHz, FM = 500 Hz then the bandwidth of frequency modulation will be  = 2 [75 + (500/1000)] KHz = 151.0 KHz
• If Δf = 75 KHz FM = 5000 Hz then the bandwidth of FM will be = 2 [75 + (5000/1000)] KHz = 160.0 KHz
• If Δf = 75 KHz fFM= 10000 Hz, then bandwidth constant of frequency modulation will be = 2 [75 + (10000/1000)] KHz = 170.0 KHz

You can see from the above examples that the frequency has increased 20 times from 50 Hz to 5000 Hz. But deviation has only increased marginally from 151 Hz to 170 Hz. Therefore this frequency modulation system is known as the constant bandwidth system. 

The advantages and disadvantages of frequency modulation

Advantages of frequency modulation are: 

• The main advantage of frequency modulation is its resistance to signal variations of any magnitude. It is accomplished in such a way that any amount of signal deviation does not influence the audio. And the signal isn’t at a level that the receiver can’t handle. Due to this FM modulated transmissions are excellent options for mobile and radio communications, as well as a variety of other portable uses. 
• Also, frequency modulation signals are not prone to any aberrations, and they are perfect for improving broadcast transmission quality.
• Another advantage is that it is possible to use modulation even at the earliest stages of transmitter development. But it is not necessary to use linear amplification for boosting signal power. 
• It is even more conceivable to use non-linear RF amplifiers to combine FM signals in a transmitter. But these are more effective than the linear ones required for signals with amplitude variations. 

Disadvantages of frequency modulation are: 

• FM demodulators are a little more difficult to build than radio integrated circuits. It is also not the biggest drawback because nowadays radios have built-in demodulators.  
• As frequency modulation signals can be prolonged indefinitely, filters are sometimes required to reduce the sideband’s extensibility. Filters can cause distortion. This can be one of the disadvantages of using frequency modulation.

Conclusion 

Frequency modulation is a technique of encoding information on a signal (analogue or digital) by shifting the carrier wave frequency in line with the modulating signal’s frequency. Here we learned how to derive the equation along with mathematical expression.