Frequency linearization and sensitivity equalization of a frequency modulated crystal oscillator

A direct FM modulator includes an oscillator comprising a current conducting device providing amplification to sustain oscillations and a frequency determining circuit connected thereto for controlling the frequency of oscillation. The frequency determining circuit includes a piezoelectric element in series with a voltage variable capacitance and an adjustable capacitance coupled across this series combination. The capacitance of the voltage variable capacitance is changed by a modulating voltage to provide frequency deviation about the center frequency of oscillation. A frequency linearization of the crystal oscillator is provided by a selected value of capacitance coupled across the varactor and the capacitance value of the adjustable capacitance such that the frequency deviation vs. voltage is maximum at the initial reference bias voltage and is symmetrical about this reference bias.

BACKGROUND OF THE INVENTION 
This invention relates to frequency modulated oscillators. Present art 
direct frequency modulated crystal oscillators using variable capacitance 
diodes can be divided in two main categories or types. The term, "direct 
frequency modulated crystal oscillator" refers to that type of modulation 
frequency with or without temperature compensation where the direct 
frequency modulation is obtained by applying a modulating signal, such as 
an audio signal, to a voltage variable capacitance diode (varicap) biased 
by a fixed D.C. voltage network. 
In a first type of modulated crystal oscillator the required system 
frequency deviation is obtained by further multiplication of oscillator 
frequency deviation, usually by a factor of nine to 36 times. In this type 
of system, the total frequency excursion due to FM modulation is a 
fraction of the spacing between the zero and the pole frequencies of the 
crystal. 
In the second type of system, the oscillator is also modulated in frequency 
by applying the audio signal to a D.C. biased voltage variable capacitance 
diode, but the total system frequency deviation is obtained directly 
without further multiplication from the crystal oscillator. In land mobile 
radio applications, the system peak frequency deviation is limited to 5 
KHz above and below the transmitter carrier frequency that span, at 
present in the U.S., the frequency range of 25 to 512 MHz. Because a total 
of 10 kHz excursion of the crystal frequency is too large a portion of the 
crystal pole-zero frequencies spacing, the oscillator frequency deviation 
vs. modulating signal amplitude characteristic becomes non-linear with 
consequent increase of distortion of the recovered audio signal. The 
linearity is improved and the frequency deviation for a given modulating 
signal amplitude is increased if an inductor is inserted in series with 
the crystal and the feedback capacitances of the circuit. 
Analysis of such circuit shows that the inductor improves the linearity of 
the oscillator characteristic of frequency deviation versus modulation 
voltage amplitude, permits small adjustment of the crystal output 
frequency to offset manufacturing crystal calibration and aging tolerance, 
and increases the frequency deviation sensitivity (the oscillator 
frequency output excursion for a given modulating voltage amplitude). 
These three functions are interdependent; therefore, the adjustment of the 
inductance value that improves the linearity and the distortion most of 
the time does not always coincide with the required value necessary to set 
the frequency or to obtain the desired constant sensitivity necessary. 
This is especially true in multifrequency applications when driven by the 
same audio signal source. The usual solution to these problems is 
invariably a compromise; i.e., the inductor is adjusted for the value that 
permits the setting of the frequency by compensating for the overall 
feedback capacitance and manufacturing crystal tolerances, and in the 
process offers a value that somewhat improves the linearity. The 
characteristic that the compromise sacrifices is the ability to set the 
frequency deviation sensitivity precisely in multifrequencies 
applications. In the latter case, the equalization of the frequency 
deviation in a given frequency range is obtained by applying the audio 
signal through an adjustable potentiometer that can set a voltage 
different in amplitude for each oscillator in order to compensate or 
equalize different oscillator frequency deviation sensitivities. 
The disadvantages of the prior art type of circuit are the following: 
1. It requires a potentiometer for each frequency channel element to 
equalize the different frequency deviation sensitivities of each 
individual channel oscillator. 
2. It requires an inductor of low Q in series with the frequency 
determining elements of the crystal network and therefore the noise and 
frequency stability are now dependent also on the stability of the 
inductor characteristics. 
Usually both noise and frequency stability are degraded with respect to 
those of the crystal and feedback stable capacitors alone. The degradation 
is proportional to the amount the crystal series resonance frequency is 
shifted down by the inductor value. Since this value is unique for a given 
frequency and linearity improvement, no minimization of this effect is 
possible without affecting the demodulated signal distortion. 
3. A certain degree of instability and/or frequency jumping exists, created 
by the inductor-generated new reactance zero above the crystal pole 
frequency where usually large crystal spurious frequencies can produce 
unwanted oscillation, either crystal or non-crystal controlled. 
4. The slope of the frequency deviation versus the modulating voltage 
V.sub.D is not constant, as it ideally should be, but has a constant 
negative slope which gives different frequency deviations for equal 
negative or positive going modulating signal, a phenomenon called 
frequency deviation asymmetry. This lack of symmetry between positive and 
negative going frequency excursions when driven by symmetrical modulating 
input is the most serious drawback of the prior art approaches. Another 
limitation is that when a frequency-temperature compensation voltage is to 
be applied directly at the modulating element, it affects the symmetry and 
the sensitivity of the frequency deviation. The latter is a very important 
characteristic when the oscillator should be compensated to frequency 
stability of better than .+-. 2 ppm in the temperature range of 
-40.degree. to +80.degree. C. 
SUMMARY OF THE INVENTION 
A frequency modulated crystal controlled oscillator is provided which 
includes a current conducting amplifying device and a frequency 
determining circuit connected thereto for controlling the frequency of 
oscillations. The frequency determining circuit includes a crystal element 
resonant at a frequency dependent upon a reactance in series therewith. 
The modulation of the oscillator is provided by a voltage variable 
capacitance connected in series with the crystal element. An initial DC 
reference bias is applied across said voltage variable capacitance for 
setting the center frequency of oscillation. The improvement to achieve 
linearity is achieved by a second capacitance connected across the series 
combination of the voltage variable capacitance and the crystal element 
and a third capacitance across the voltage variable capacitance where the 
value of the third capacitance and the second capacitance is selected to 
provide a frequency deviation versus voltage that is generally symmetrical 
about the initial D.C. reference bias.

DESCRIPTION OF THE INVENTION 
Referring to FIG. 1, curve A illustrates the present art peak frequency 
deviation for a fixed modulating signal amplitude versus the varicap DC 
bias. If the D.C. voltage in the present art deviation schemes should 
change .+-.0.5 volts to compensate for the crystal temperature drift, this 
D.C. voltage change would cause an increase or decrease in the frequency 
deviation of about .+-.400 Hz producing an 800 Hz asymmetrical error. If 
the characteristic above had a constant slope as curve B of FIG. 1, the 
peak frequency deviation would be independent of the varicap DC bias 
variation necessary for oscillator temperature compensation. Curve C 
illustrates the peak frequency deviation versus voltage characteristic 
resulting from the circuit described herein. 
Referring to FIG. 2, there is illustrated a schematic diagram of the 
frequency modulated crystal oscillator of the present invention. The 
circuit includes an NPN transistor 10 having a base electrode 11, emitter 
electrode 13 and collector electrode 15. The D.C. bias for the base 11 is 
provided via voltage dividing network of resistors 17, 18 and 19 with 
resistor 19 connected to a regulated power supply at terminal 21 and 
resistor 18 connected to ground. The junction point 12 of resistors 17 and 
18 are connected to the base 11 of transistor 10. The bypass of the power 
supply is provided by capacitor 16. When the oscillator is to be switched 
on, resistor 32 is connected to ground via switch 35. RF bypass of 
resistor 32 is provided by capacitor 33. Capacitor 37 is coupled across 
resistor 31 with one terminal connected to the emitter 13 of transistor 10 
and the opposite terminal connected to the junction of resistors 31 and 
32. Fixed capacitor 39 and variable capacitor 41 are coupled in parallel 
across the base-emitter junction of transistor 10 and these parallel 
capacitors are connected in series with capacitor 37 to form a voltage 
dividing capacitor network with the junction 45 of the parallel connected 
capacitors 39 and 41 and capacitor 37 being coupled to emitter 13 to 
provide what is sometimes referred to as a modified Colpitts circuit. The 
value of resistor 31 and the value of capacitors 39, 41 and 37 determine 
the signal which is fed back to the base electrode 11 to cause 
oscillations. 
Bias is applied to the collector electrode from the source terminal 21 
through the inductive portion of a three-pole band pass filter 51 at the 
collector. The inductor is part of a tuned output bandpass filter circuit 
51 which is tuned at the third harmonic of the fundamental oscillator 
frequency. The transistor is biased such that with the crystal connected 
between the base and ground and oscillating, it operates in a class C 
condition. Since the transistor 10 is operated in class C, its conduction 
angle is adjusted, by proper dimensioning of the feedback capacitors 39, 
41 and 37 and D.C. bias, for maximum output at the third harmonic of the 
frequency at the output of band pass filter 51. 
The frequency determining circuit includes a crystal 50 in series with the 
parallel capacitances provided by fixed capacitors 54, variable capacitor 
55 and varactor diode 56. The crystal 50 is connected in series with these 
parallel capacitances between the base 11 and ground or reference 
potential. The frequency determining circuit of the crystal 50 and the 
capacitors 54 and 55 and diode 56 are coupled across the voltage dividing 
capacitors 39, 41 and 37 when the oscillator is switched into operation. 
The crystal 50 is operated in an anti-resonant or parallel resonant mode so 
that it presents an effective inductance at the frequency of oscillation. 
The crystal 50 is selected to have a series resonant frequency relatively 
widely spaced from this anti-resonant frequency so that a relatively large 
change in frequency is possible to obtain relatively large deviation in 
the oscillator frequency. Capacitors 54 and 55 are coupled in parallel 
across the variable capacitance diode 56. The variable capacitance diode 
bias is provided via voltage dividing resistors 65 and 66, or 
alternatively by the two arms of a temperature compensation network, and 
resistor 64. Point 68 at the junction of resistors 65 and 66 is coupled to 
a reference DC bias source, resistor 66 is coupled to the regulated source 
at terminal 21, and resistor 65 is coupled to the junction of resistors 66 
and 64 at point 68 at one terminal end and at the opposite terminal end to 
ground or reference potential. Resistor 64 is an isolation resistor to 
isolate the RF at point 70 from the temperature compensation network at 
point 68. Modulating signals at teminal 59 are applied through capacitor 
60 and either a resistor 61 or an inductor 67 across the voltage-variable 
capacitance diode 56 and the parallel capacitances 54 and 55. Capacitor 60 
with resistor 61 (or inductor 67) is adapted to pass frequencies to point 
70 between 10 Hz to 30 kHz. 
The voltage variable capacitance diode 56 in combination with the 
capacitors 54 and 55 form an equivalent capacitance (C.sub.s) or 
EQU C.sub.S = C.sub.54 + C.sub.55 + C.sub.d 
where 
C.sub.54 = the value of capacitor 54. 
C.sub.55 = the value of capacitor 55. 
C.sub.d = the value of the variable capacitance diode 56. 
This equivalent capacitance C.sub.s cooperates with the value of the 
voltage divider capacitors 39, 41 and 37 to form a resonant circuit with 
crystal 50 at the parallel resonant frequency of the crystal. The total 
capacitance between transistor base and ground is referred to as 
oscillator input capacitance and is approximated by the following 
equation: 
##EQU1## 
where C.sub.IN = imaginary part of the base to ground impedance in pF 
(picofarads). 
C.sub.39 = the value of capacitor 39 in pF. 
c.sub.41 = the value of capacitor 41 in pF. 
c.sub.37 = the value of capacitor 37 in pF. 
c.sub.be = the effective base-emitter capacitance in pF. 
C.sub.in may be for example on the order of 45 pF. 
The change in the capacitance of the voltage variable capacitor 56 produces 
a change in the equivalent load capacitance which is applied to the 
crystal to change its frequency. The crystal may be designed to work with 
a given load capacitance. A change in the equivalent load capacitance will 
change the frequency of the oscillation of the crystal. 
The equivalent A.C. circuit of the oscillator portion of FIG. 2 diagram is 
illustrated in FIG. 3. It consists of: 
1. The crystal motional parameters L.sub.1.sbsb.m, C.sub.1.sbsb.m, R and 
C.sub.0. 
2. the equivalent hybrid parameters of the transistor 10: R.sub.b'b, 
R.sub.b'e, C.sub.b'e, C.sub.b'c, and the current source gm V.sub.b'e. 
3. The capacitors C.sub.37, C.sub.41, C.sub.39, C.sub.55, C.sub.54, 
C.sub.56. 
4. the equivalent resistance R.sub.B of the base D.C. bias network, where 
##EQU2## 
where R.sub.17, R.sub.18 and R.sub.19 are the respective values of 
resistors 17, 18 and 19. 
5. The modulation coupling network R.sub.61 C.sub.60 or L.sub.67 C.sub.60, 
where R.sub.61 and L.sub.67 is the value of resistor 61 and inductor 67, 
and C.sub.60 is the value of capacitor 60. 
6. The equivalent resistance of the varicap D.C. bias network R.sub.T, 
where 
##EQU3## 
and R.sub.64 is the value of resistor 64, R.sub.65 is the value of 
resistor 65 and R.sub.66 is the value of resistor 66. 
If it is assumed that the impedance of R.sub.61 C.sub.60 in parallel with 
R.sub.T is much larger than the reactance of the parallel capacitance 
C.sub.54 + C.sub.55 + C.sub.56 and if the losses of the feedback loop are 
lumped in the reactance "Req," the complex circuit of FIG. 3 can be 
approximated by the circuit of FIG. 4. When the input impedance between 
the transistor base and ground presents a negative resistance -R.sub.IN of 
the magnitude necessary to start the oscillation and furnish the desired 
amount of power to the load the oscillating circuit is reduced to the 
crystal and the two capacitors C.sub.IN and C.sub.S in series with each 
other. Referring to FIG. 4 and defining as crystal load capacitance 
C.sub.L the capacitance seen by the crystal across its terminal then one 
can write: 
##EQU4## 
where: C.sub.L = crystal load capacitance, farad 
C.sub.IN = the equivalent reactance of the imaginary part of the transistor 
input impedance Z.sub.IN. 
l.sub.m = crystal motional inductance (henry) 
C.sub.m = crystal motional capacitance (farad) 
C.sub.0 = crystal shunt capacitance (farad) 
C.sub.S = the sum of capacitances C.sub.54, C.sub.55 and C.sub.56 (farad) 
C.sub.T = the total capacitance in parallel with the crystal L.sub.m, 
C.sub.m branch, (farad) 
Defining the crystal series resonance F.sub.S as the frequency at which the 
crystal series branch L.sub.m, C.sub.m would oscillate the crystal 
parallel or anti-resonant frequency F.sub.P as the frequency of 
oscillation of the motional inductance L.sub.m with the series combination 
of C.sub.m and the total capacitance C.sub.T seen by the crystal the 
following equations can be written: 
EQU F.sub.S = (2.pi. .sqroot.L.sub.m C.sub.m).sup.-1 in Hz (1) 
##EQU5## 
EQU .DELTA.F = F.sub.p - F.sub.S = F.sub.S C.sub.m /2 C.sub.T in Hz (4) 
The frequency deviation .DELTA.F of the oscillator, the equivalent circuit 
of which is illustrated in FIG. 4, can be shown to be: 
##EQU6## 
or, normalizing with respect the crystal series resonance frequency 
F.sub.S, to: 
##EQU7## 
Mobile radio with standard maximum frequency deviation of 5 kHz are at 
present subdivided in the three main frequency ranges of low VHF band (25 
to 50 MHz), high VHF band (132 to 174 MHz) and UHF band (400 to 512 MHz). 
In the worst case (at the lowest frequency of each UHF/VHF band), the 
frequency excursion in PPM (parts per million) required at the output 
frequency is .+-. 200 PPM for the low VHF band, .+-. 37.87 PPM for the 
high VHF band, and .+-. 12.5 PPM for the UHF band. 
The frequency deviation sensitivity S is defined herein as: 
##EQU8## 
where: S = frequency deviation sensitivity in PPM/Volt 
.DELTA.F = oscillator frequency deviation in Hz 
F.sub.C = output carrier frequency in MHz 
V.sub.D.sbsb.i = instantaneous varicap voltage in volts 
If the maximum available peak amplitude of the modulating signal is known, 
the required minimum sensitivity can be calculated. For mobile radio 
application, where the available regulated power supply is around 9.5 
volts D.C. typical magnitude of the available undistorted modulating audio 
signal is around 2.2 V.sub.RMS or .+-. 3.1V peak; therefore, the minimum 
frequency deviation sensitivity that can be used is 65 PPM/Volt for the 
low VHF band, 12.5 PPM/Volt for the high VHF band and 4 PPM/Volt for the 
UHF band. However, when simultaneously, the varicap is also used as 
variable capacitance to temperature compensate the crystal drift, a 
compromise sensitivity should be chosen depending on the realizability of 
the compensating voltage generator network for a given sensitivity. 
It was found that at high VHF and UHF frequencies the optimum sensitivity 
is around 20 PPM/Volt and at low VHF around 70 PPM/Volt. This is 
realizable for the present available commercial diodes used. Future new 
diodes or different diodes from the ones used might require different 
optimum sensitivities. From equation (6), solving for C.sub.S and with 
.DELTA.F/F.sub.S = .phi. 
##EQU9## 
Ideally if C.sub.S is matched by the varicap capacitance/voltage 
characteristic at any instantaneous frequency deviation point, the only 
capacitance required to make C.sub.S is the varicap capacitance C.sub.56. 
In practice, varicaps that match exactly the required value of C.sub.S for 
a constant rate of increase of .DELTA.F/F.sub.S versus the instantaneous 
magnitude of the modulating signal are at presently not available 
commercially. Even if they were available reasonable production tolerances 
of the crystal motional capacitance C.sub.m and of the capacitors and 
active device that make up the capacitance C.sub.IN would dictate a 
special matched varicap with unique variable capacity diode (C.sub.56) 
versus voltage characteristic for each combination of crystals and 
component tolerances. This is clearly impractical and it is the purpose 
and embodiment of this invention to describe a practical method that 
synthesizes the required C.sub.S or a close approximation of it for a 
relatively broad range of frequency deviation sensitivities and reasonably 
practical crystal and other component tolerances. 
The synthesis procedure is carried out for the high VHF and UHF 
requirements of 20 PPM/Volt sensitivity, the varicap D.C. reference 
voltage V.sub.D.sbsb.REF of 6V and typical motional and load capacitance 
of 0.03 and 25 pF respectively. It can be shown that similar results are 
obtained for different sensitivities and crystal motional and/or load 
capacitances, the only difference being the degree of linearization 
achieved from a given excursion range of the D.C. varicap bias voltage. 
In FIG. 5, curve D illustrates the required capacitance versus voltage 
varicap characteristic if it alone is to constitute the capacitance 
C.sub.S of FIG. 4. Curve E depicts the capacitance versus voltage 
characteristic of a typical abrupt junction varicap and curve F depicts 
the capacitance versus voltage characteristic of a hyperabrupt junction 
diode. It was found that two conditions should be met apriori for the 
desired approximation of curve D of FIG. 5. First, at any value of the 
varicap voltage the capacitance C.sub.56 should never exceed the lowest 
possible value of C.sub.S within the design range of variability of 
C.sub.IN, V.sub.D.sbsb.REF, C.sub.m, C.sub.0 and S. Second, at any value 
of the varicap voltage the slope of the varicap capacitance versus voltage 
characteristic should be steeper or equal to the steepest possible slope 
of curve D of FIG. 5 within the design range of variability of C.sub.IN, 
C.sub.m, C.sub.0, V.sub.D.sbsb.REF and S. As can be seen in FIG. 5, only 
the hyperabrupt junction diode meets this condition when trying to match 
curve D. When these two conditions are met by either choosing the 
commercial diode that approximates the requirement above for a given 
design set of C.sub.IN, C.sub.m, C.sub.0, V.sub.D.sbsb.REF and S, or by 
matching closely the design set of C.sub.IN, C.sub.m, C.sub.0, S and 
V.sub.D.sbsb.REF to an available commercial diode, the following equations 
give the values of the capacitance C.sub.p, that is the sum of C.sub.54 
and C.sub.55 in parallel to the varicap diode, and C.sub.IN, the 
imaginary parts of the input impedance required: 
##EQU10## 
where A, B, C, D, E, and F are intermediate calculation parameters and 
have the value given by the following equations: 
EQU A=2.phi..sub.1 C.sub.0 -C.sub.m ; B=C.sub.D.sbsb.1 A; C=C.sub.m -2 C.sub.0 
.phi.2-2.phi..sub.2 C.sub.D.sbsb.2 ; D=C.sub.m -2.phi..sub.1 (C.sub.0 
+C.sub.D.sbsb.1); E=C.sub.D.sbsb.2 (2.phi..sub.2 C.sub.0 -C.sub.m); 
F=2.phi..sub.2 C.sub.0 -C.sub.m ; 
and where C.sub.D.sbsb.1 and C.sub.D.sbsb.2 are the varicap capacitances at 
the two extreme voltages where the curves are to be matched perfectly and 
.phi..sub.1 and .phi..sub.2 the correspondent .DELTA.F/F.sub.S, with 
.DELTA.F and F.sub.S in hertz and all the capacitances in pF. To match 
curve D in FIG. 5 as close as possible using a hyperabrupt junction diode 
having curve F, the calculations using the equations (9) and (10) give a 
value for C.sub.p = 25.548 and C.sub.IN = 45 pF, where C.sub.m = 0.03 pF, 
C.sub.0 = 7 pF, S = 20 PPM/V, C.sub.L = 25 pF and V.sub.REF = 6 volts. 
This hyperabrupt junction diode is a KSW Electronics, Inc. of Burlington, 
Mass., type No. KV2202. With the above values of C.sub.p and C.sub.IN, 
there is provided the dashed curve G in FIG. 5. FIG. 6 illustrates the 
measured slope of the normalized frequency deviation produced by a 
modulating signal of .+-. 2 Vpp (1.414 V.sub.RMS) with respect to the 
variation of the varicap D.C. bias. Though the characteristic shown is 
only a good approximation of the ideal straight line required, it is clear 
that it possesses the important propriety required in modulators for 
mobile radio applications of never exceeding the maximum slope set at 
V.sub.D.sbsb.REF, should the varicap D.C. bias be varied for the purpose 
of temperature compensation. In this particular example the frequency 
deviation after being multiplied to the VHF frequency of 125 MHz is 
approximately 5000 Hz at 6V, 4950 Hz at 6 .+-. 0.5V, and 4800 Hz at 6 .+-. 
1V. 
This method of linearization makes possible and practical the use of a 
varicap in the simultaneous function of direct frequency modulator and as 
control element of the temperature compensation network. Also, because the 
slope of the frequency deviation is symmetrical with respect to the fixed 
reference voltage and close to a straight line, very low distortion of 
audio signals is achieved. Typical values are 0.5% at room temperature and 
under 1% within the temperature range of -40.degree. to +80.degree. C. The 
values are for an audio signal of 1 kHz and 3 kHz deviation at VHF and UHF 
frequencies of 132 to 512 MHz. 
Since equations (9) and (10) show that C.sub.p and C.sub.IN can be 
synthesized for various values of S via .phi..sub.1 and .phi..sub.2 
dimensioning, the adjustment of the variable portion of C.sub.p is also an 
exact method of setting the desired varicap modulation sensitivity S. This 
process is called modulation sensitivity equalization and unlike prior art 
multifrequency TCXO's equalization schemes where each modulation voltage 
is separately adjusted to account for variance in individual oscillator 
frequency deviation sensitivity, the new equalization circuit internal to 
the TCXO, permits the elimination of all but one of the adjusting 
potentiometers as illustrated in FIG. 7. The microphone output at terminal 
101 is coupled through the audio processor 103 to potentiometer 105. The 
tapped output from potentiometer 105 is coupled via capacitor 107 to each 
of the temperature compensated crystal oscillators 110 through 114 which 
are each like that described above in connection with FIG. 2. The position 
of control arm 115 determines which of the oscillators 110 through 114 
channel switch is closed to thereby determine which of the oscillator 
outputs is provided to the transmitter output 117. Capacitor 107 is a D.C. 
blocking capacitor and resistor 105 is adjusted to provide a selected 
maximum amplitude of the modulating voltage for determining the maximum 
frequency deviation in all of the oscillators 110 through 114. The 
equalization of frequency deviation sensitivity is achieved by adjusting 
finely the capacitance across the varicap. 
The oscillator described herein is further described in connection with 
application Ser. No. 790,865 filed Apr. 26, 1977 of Bortolo Mario Pradal. 
A temperature compensation network for use herewith is described in 
application Ser. No. 741,405 filed Nov. 12, 1976. The inventor is Bortolo 
Mario Pradal.