# FM Demodulation with RLC Circuits.

Cuthbert Nyack
Commercial FM demodulation is done at the IF frequency of 10.7MHz. With a frequency deviation of ±75kHz, the deviation of the IF carrier is ±0.7%. With this deviation, it is possible to convert the FM to AM or PM and AM using passive RLC circuits. The resulting AM is then demodulated with an envelope detector.
Some basic FM demodulators are examined in the applet on this page. The possibility of converting a FM to an AM signal using first order RC and 2nd order Butterworth filters is first briefly examined.
In the applet the Diode RC combination is assumed to be an ideal envelope detector. The practical envelope will be lower in amplitude and phase delayed compared to that shown in the applet. The demodulators used here are shown below. Although some are no longer used, understanding them makes understanding the more complex ones easier.

The image below shows 4 demodulators, slope, balanced slope, phase discrimminator and quadratic detector. Fn = 1 shows the FM passed through a simple 1st order system. Because of the variation of the response with frequency, then the signal an AM component which can be demodulated. In this case the output is small and distorted.
Fn = 2 to 7 show special cases of Fn = 1. Image below show the case when the FM is centered at the frequency of maximum slope. Green shows an amplified version of the envelope.  Fn = 8 shows the case of the single slope detector shown above. Fn = 9 to 20 show special cases of Fn = 8.
Fn = 9 to 15 shows the output as the center frequency of the FM is moved across the resonant peak when the Q is 15.
Fn = 16 to 20 does similarly with Q = 8.
Fn = 11 is shown below. The signal at node 1 is shown in Yellow and its envelope in Green in the lower part of the image. The envelope is visibly distorted.  Fn = 21 shows the functioning of the balanced slope detector shown above.
Fn = 22 to 32 show special cases of Fn = 21.
Assuming the center frequency and BW of the FM are shown by the brown lines, then the resonant frequencies must be adjusted as shown below. The difference between the two responses is shown by the blue magenta curve. Both the resonant frequencies and Q's should be adjusted so the blue magenta line is linear over the BW of the FM.
The LS1-C1 response is shown by the green to blue resonant peak and the signal at 3 is shown by the blue curve at the top. Assuming ideal envelope detectors, then the signal at 4 is shown by the light magenta curve at the bottom.
The freq response of the lower LS2-C2 resonant circuit is shown by the yellow to green curve at the bottom, the time response by the yellow curve at the top and the envelope at 5 by the orange curve at the bottom. When the orange curve is inverted and added to the light magenta one, the result is shown in green. Compared to the single resonant circuit the output is much less distorted. A measure of the distortion is shown by the magenta curve which plots the output vs the mod signal. The image below shows what happens when the resonant frequencies are not separated enough. Severe distortion results as indicated by the 'S' curve in magenta.  Fn = 33 shows the functioning of the Foster-Seely phase discrimminator.
Fn = 34 to 39 show special cases of Fn = 33.
Image below shows simplified versions of the signals which occur in this circuit.
Red is the FM signal which is applied to the primary. The primary and secondary form a pair of inductively coupled circuits both tuned to the center frequency of the FM. The magnitude and phase of the coupled circuit response are shown by the red to green and cyan lines. Kc is the coupling coefficient expressed in terms of the critical coupling.
Between 6 and 8 and 8 and gnd the signal is Vp ± Vs/2 and is shown as the yellow and blue signals.
The envelopes at 7 and 9 are shown by the orange and blue magenta plots and the signal between 7 and 9 is shown as the green signal which follows the dark green modulating signal at the top.
Magenta line is a measure of the linearity of the output. freq devn/center freq is 0.16/7 = 2.2% > 0.7% required for commercial FM.
Because of the coupled circuits, this demodulator can demodulate wider bandwidth FM than other RLC circuit demodulators. It is not uncommon to see descriptions of the functioning of the phase discrimmator using only the secondary circuit. This approach can illustrate the basic functioning of the circuit but it does not show the property of being able to demodulate wider BW FM than can be done by other RLC circuit demodulators.
Fn = 40 shows this case and Fn = 41 to 45 show special cases of Fn = 40. This is a simpler version of the RLC circuits above. The resonant circuit is tuned to the center frequency of the FM and the circuit is effectively a phase detector. A current produced by the FM is applied at 10.
Fn = 46 shows this case and Fn = 47 to 51 show special cases of Fn = 47.
The image below show the signals in the circuit. red and orange are the current through L1 and the voltage across L1 and blue is the voltage across R2L2C2. The product of the 2 voltages is in yellow and the envelope (or average) of the yellow signal is shown in green which follows the modulating signal. freq devn/carrier freq is ~ 1% just about sufficient for commercial FM. Narrow band FM (only 1 pair of sidebands) can be produced by phase shifting the carrier of an AM signal by 90°.
This signal can be demodulated at the receiver by multiplying it by the FM carrier shifted by 90° ie the original AM carrier.
Fn = 52 shows this case and Fn = 53 to 55 show special cases of Fn = 52.
Image below illustrates this approach. FM is in red, modulating signal is in blue, phase shifted carrier is in cyan and the product of phase shifted carrier and the FM is in yellow. The envelope or average of the yellow is in green which follows the blue modulating signal. As Fn = 55 shows, this only works if d is small. As d becomes greater than one, the FM develops more sidebands which distort the product.