FREQUENCY MODULATION PRACTICAL REPORT
𝑡
1. Aim 𝑥𝑐 (𝑡) = 𝐴𝑐 cos[2𝜋𝑓𝑐 + 2𝜋𝑘𝑓 ∫−∞ 𝑚(𝑡)𝑑𝑡] (6)
The aim of the practical is to demonstrate the
concepts of frequency modulation in a
2.3. Theory on Modulation Index & Max
communication system by studying frequency Frequency Deviation
deviation, bandwidth of the signal and sideband
amplitudes. Furthermore, the modulation index 𝛽𝑓 of the signal
is defined as:
2. Prerequisites 𝑘𝑓 𝐴𝑚 𝑘𝑓 ([max(𝑚(𝑡)) ])
𝛽𝑓 = = (7)
2.1. Administrative 𝜔𝑚 𝑊
Students undertaking this practical are required to where W=bandwidth of the message signal in Hz.
have submitted pre-practical reports as evidence of
preparation. These reports are appended to this This can also be written in terms of the maximum
document. frequency deviation which is given by:
2.2. Theory on Angle Modulation Δ𝑓𝑀𝐴𝑋 = 𝑘𝑓 (max |𝑚(𝑡)|) (8)
Angle modulation is the exponential modulation Hence by equation (7) and (8):
type under which frequency modulation and phase Δ𝑓𝑀𝐴𝑋 = 𝛽𝑓 ∗ 𝑊 (9)
modulation fall. Angle modulation offers better
noise and interference immunity as this type of Circuit diagrams are found in the Appendix A.
modulation does not use varying amplitude to define
the message data. Amplitude variations caused by 2.4. Modulated Signal Bandwidth
noise have no bearing on the data transmitted. [1]
The modulated carrier signal is defined as: Using the above theory [2], if we modulate a
message signal 𝑚(𝑡) = 𝐴𝑐 cos[𝜔𝑚 ] onto the carrier
𝑥𝑐 (𝑡) = 𝐴𝑐 cos[𝜔𝑐 + 𝜙(𝑡)] (1) 𝑥𝑐 (𝑡) = 𝐴𝑐 cos[𝜔𝑐 + 𝜙(𝑡)] (equation 1) the
resultant signal y(t) is given by (replacing frequency
𝑥𝑐 (𝑡) = modulated carrier; 𝐴𝑐 =amplitude of the terms as in (6):
carrier; 𝜔𝑐 = carrier frequency also expressed as 𝑘𝑓 𝑓𝑎
2𝜋𝑓𝑐 ; 𝜙(𝑡) = phase of the modulated signal. 𝑦(𝑡) = 𝐴𝑐 cos[2𝜋𝑓𝑐 + sin[2𝜋𝑓𝑚 (𝑡) ]] (10)
𝑓𝑚
The values The theoretical bandwidth of the modulated signal is
𝑡 then about twice the bandwidth hence
𝜙(𝑡) = ∫−∞ 2𝜋𝑘𝑓 𝑚(𝑡)𝑑𝑡 (2)
𝒇𝒎 = 𝟐𝑾
where 𝑘𝑓 is the FM frequency deviation constant.
Where 𝒇𝒎 = the message signal frequency and W
is the modulated signal bandwidth.
(3)
Note that the amplitude of the frequency sidebands
𝑑𝜌
The instantaneous frequency 𝜔𝑖 = and so 𝒇𝒎 Hz apart may be plotted using the Bessel
𝑑𝑡
Function and Bessel tables:
(4)
1 𝜋 𝑖(𝑠𝑖𝑛𝑥−𝑛𝑥)
The instantaneous frequency deviation is 𝐽𝑛 (𝛽) = ∫ 𝑒 𝑑𝑥 (12)
2𝜋 −𝜋
proportional to the message signal and so
Where 𝛽 is the argument and n is the Bessel order.
𝑑𝜙(𝑡)
= 𝑘𝑓 𝑚(𝑡). (5)
𝑑𝑡
Using (1) as the starting point and substituting the
relations for (3), (4), (5) the frequency modulated
carrier is expressed as:
, 3. Materials & Methods The capacitor value, which achieved the results was
120nF which was arguably different from the
3.1. Part 1
calculated 149nF.
The oscillator described in the pre-practical was
built to operate at the nominal 10kHz with the tuning 4.2. Part 2
capacitor C for which values were calculated. The tuning capacitor was removed and the output of
the oscillator was connected to the reactance
3.2. Part 2
modulator, which was constructed as shown in the
The tuning capacitor was removed from the circuit
pre-practical report. Due to unavailability of exact-
and replaced with the reactance modulator circuit,
valued resistors, R1 = 5,7Ω was used instead of the
which was also built per specification.
calculated 4,7 Ω.
3.3. Part 3
At 10KHz oscillator output it was found the value of
The value of –Vm was varied and checked on the
Vm was -2,13V as shown.
DMM. This variation determined the linearity and
hence the sensitivity of the reactance modulator,
both by measurement and extrapolated calculation.
3.4. Part 4
The HP dynamic signal analyser was then used to
plot the spectra of the FM signal for the various Figure 2 Vm as found @ 10KHz
values of the modulation index β at a modulating
frequency of ~30Hz. The BJT of the reactance modulator is switched on
by applying a negative voltage (-Vm) to the emitter
3.5. Part 5 of the transistor. This results in a voltage drop of (-
The output of the modulator was coupled to the Vm+Vbe) across the 10K resistor (where Vbe=0.7V,
frequency demodulator that was provided. is constant). Thus, corresponding current can be
Sensitivity was calculated and thereafter linearity calculated. Since α and VT are constant, Ie may be
was experimentally confirmed. This was done by varied by varying gm. To vary gm, only the voltage -
plotting dc output for varying unmodulated input Vm needs to be varied.
frequencies by varying –Vm of the oscillator.
To calculate C1 we need gm. gm requires finding IE.
3.6. Part 6
Finally the RF spectrum analyser was used to study 𝑉𝑇 = 25𝑚𝑉 & 𝑉𝐸 = −0,7𝑉 𝑎𝑡 𝑉𝑚 = −2,13𝑉
the frequency spectrum of the 60MHz oscillator; the 𝑉𝐸 − 𝑉𝑚 (−0,7 − (−2,13))
harmonics were observed and these were used to 𝐼𝐸 = = = 0,143𝑚𝐴
𝑅 104
investigate carrier frequency and modulation index
multiplication. Only 1/n of the nodulation amplitude To find α from the BC237 NPN datasheet’s value of
is needed to achieve zero carrier; and this hypothesis β:
was confirmed.
4. Analysis & Discussion
𝐼𝐶 𝛼𝐼𝐵 0,9917 ∗ 0,143𝑚𝐴
4.1. Part 1 𝑔𝑚 = = =
The required 10KHz oscillator was built without the 𝑉𝑇 𝑉𝑇 25𝑚𝑉
reactance modulator, to find the correct tuned value 𝑚𝐴
= 5,673
of C and compare it to predictions. The output was 𝑉
verified as being 4Vp-p at 10,12KHz as shown in the Finally finding C1:
image below.
𝐶𝑖𝑛 120𝑛𝐹
𝐶1 = =
1 + 𝑔𝑚 𝑅1 1 + 5,673 𝑚𝐴 ∗ 5,7
𝑉
= 116,241𝑛𝐹
The calculated value is within the margin of error
from the predicted 127nF at which we may consider
component tolerances, capacitive coupling as being
non-negligible. Also the different value of 𝑅1 and
Figure 1 Showing the achieved 10KHz
𝑡
1. Aim 𝑥𝑐 (𝑡) = 𝐴𝑐 cos[2𝜋𝑓𝑐 + 2𝜋𝑘𝑓 ∫−∞ 𝑚(𝑡)𝑑𝑡] (6)
The aim of the practical is to demonstrate the
concepts of frequency modulation in a
2.3. Theory on Modulation Index & Max
communication system by studying frequency Frequency Deviation
deviation, bandwidth of the signal and sideband
amplitudes. Furthermore, the modulation index 𝛽𝑓 of the signal
is defined as:
2. Prerequisites 𝑘𝑓 𝐴𝑚 𝑘𝑓 ([max(𝑚(𝑡)) ])
𝛽𝑓 = = (7)
2.1. Administrative 𝜔𝑚 𝑊
Students undertaking this practical are required to where W=bandwidth of the message signal in Hz.
have submitted pre-practical reports as evidence of
preparation. These reports are appended to this This can also be written in terms of the maximum
document. frequency deviation which is given by:
2.2. Theory on Angle Modulation Δ𝑓𝑀𝐴𝑋 = 𝑘𝑓 (max |𝑚(𝑡)|) (8)
Angle modulation is the exponential modulation Hence by equation (7) and (8):
type under which frequency modulation and phase Δ𝑓𝑀𝐴𝑋 = 𝛽𝑓 ∗ 𝑊 (9)
modulation fall. Angle modulation offers better
noise and interference immunity as this type of Circuit diagrams are found in the Appendix A.
modulation does not use varying amplitude to define
the message data. Amplitude variations caused by 2.4. Modulated Signal Bandwidth
noise have no bearing on the data transmitted. [1]
The modulated carrier signal is defined as: Using the above theory [2], if we modulate a
message signal 𝑚(𝑡) = 𝐴𝑐 cos[𝜔𝑚 ] onto the carrier
𝑥𝑐 (𝑡) = 𝐴𝑐 cos[𝜔𝑐 + 𝜙(𝑡)] (1) 𝑥𝑐 (𝑡) = 𝐴𝑐 cos[𝜔𝑐 + 𝜙(𝑡)] (equation 1) the
resultant signal y(t) is given by (replacing frequency
𝑥𝑐 (𝑡) = modulated carrier; 𝐴𝑐 =amplitude of the terms as in (6):
carrier; 𝜔𝑐 = carrier frequency also expressed as 𝑘𝑓 𝑓𝑎
2𝜋𝑓𝑐 ; 𝜙(𝑡) = phase of the modulated signal. 𝑦(𝑡) = 𝐴𝑐 cos[2𝜋𝑓𝑐 + sin[2𝜋𝑓𝑚 (𝑡) ]] (10)
𝑓𝑚
The values The theoretical bandwidth of the modulated signal is
𝑡 then about twice the bandwidth hence
𝜙(𝑡) = ∫−∞ 2𝜋𝑘𝑓 𝑚(𝑡)𝑑𝑡 (2)
𝒇𝒎 = 𝟐𝑾
where 𝑘𝑓 is the FM frequency deviation constant.
Where 𝒇𝒎 = the message signal frequency and W
is the modulated signal bandwidth.
(3)
Note that the amplitude of the frequency sidebands
𝑑𝜌
The instantaneous frequency 𝜔𝑖 = and so 𝒇𝒎 Hz apart may be plotted using the Bessel
𝑑𝑡
Function and Bessel tables:
(4)
1 𝜋 𝑖(𝑠𝑖𝑛𝑥−𝑛𝑥)
The instantaneous frequency deviation is 𝐽𝑛 (𝛽) = ∫ 𝑒 𝑑𝑥 (12)
2𝜋 −𝜋
proportional to the message signal and so
Where 𝛽 is the argument and n is the Bessel order.
𝑑𝜙(𝑡)
= 𝑘𝑓 𝑚(𝑡). (5)
𝑑𝑡
Using (1) as the starting point and substituting the
relations for (3), (4), (5) the frequency modulated
carrier is expressed as:
, 3. Materials & Methods The capacitor value, which achieved the results was
120nF which was arguably different from the
3.1. Part 1
calculated 149nF.
The oscillator described in the pre-practical was
built to operate at the nominal 10kHz with the tuning 4.2. Part 2
capacitor C for which values were calculated. The tuning capacitor was removed and the output of
the oscillator was connected to the reactance
3.2. Part 2
modulator, which was constructed as shown in the
The tuning capacitor was removed from the circuit
pre-practical report. Due to unavailability of exact-
and replaced with the reactance modulator circuit,
valued resistors, R1 = 5,7Ω was used instead of the
which was also built per specification.
calculated 4,7 Ω.
3.3. Part 3
At 10KHz oscillator output it was found the value of
The value of –Vm was varied and checked on the
Vm was -2,13V as shown.
DMM. This variation determined the linearity and
hence the sensitivity of the reactance modulator,
both by measurement and extrapolated calculation.
3.4. Part 4
The HP dynamic signal analyser was then used to
plot the spectra of the FM signal for the various Figure 2 Vm as found @ 10KHz
values of the modulation index β at a modulating
frequency of ~30Hz. The BJT of the reactance modulator is switched on
by applying a negative voltage (-Vm) to the emitter
3.5. Part 5 of the transistor. This results in a voltage drop of (-
The output of the modulator was coupled to the Vm+Vbe) across the 10K resistor (where Vbe=0.7V,
frequency demodulator that was provided. is constant). Thus, corresponding current can be
Sensitivity was calculated and thereafter linearity calculated. Since α and VT are constant, Ie may be
was experimentally confirmed. This was done by varied by varying gm. To vary gm, only the voltage -
plotting dc output for varying unmodulated input Vm needs to be varied.
frequencies by varying –Vm of the oscillator.
To calculate C1 we need gm. gm requires finding IE.
3.6. Part 6
Finally the RF spectrum analyser was used to study 𝑉𝑇 = 25𝑚𝑉 & 𝑉𝐸 = −0,7𝑉 𝑎𝑡 𝑉𝑚 = −2,13𝑉
the frequency spectrum of the 60MHz oscillator; the 𝑉𝐸 − 𝑉𝑚 (−0,7 − (−2,13))
harmonics were observed and these were used to 𝐼𝐸 = = = 0,143𝑚𝐴
𝑅 104
investigate carrier frequency and modulation index
multiplication. Only 1/n of the nodulation amplitude To find α from the BC237 NPN datasheet’s value of
is needed to achieve zero carrier; and this hypothesis β:
was confirmed.
4. Analysis & Discussion
𝐼𝐶 𝛼𝐼𝐵 0,9917 ∗ 0,143𝑚𝐴
4.1. Part 1 𝑔𝑚 = = =
The required 10KHz oscillator was built without the 𝑉𝑇 𝑉𝑇 25𝑚𝑉
reactance modulator, to find the correct tuned value 𝑚𝐴
= 5,673
of C and compare it to predictions. The output was 𝑉
verified as being 4Vp-p at 10,12KHz as shown in the Finally finding C1:
image below.
𝐶𝑖𝑛 120𝑛𝐹
𝐶1 = =
1 + 𝑔𝑚 𝑅1 1 + 5,673 𝑚𝐴 ∗ 5,7
𝑉
= 116,241𝑛𝐹
The calculated value is within the margin of error
from the predicted 127nF at which we may consider
component tolerances, capacitive coupling as being
non-negligible. Also the different value of 𝑅1 and
Figure 1 Showing the achieved 10KHz
