Piezoelectric crystalsexhibit the property that, when an electric potential is applied across the faces of the crystal, it physically bends or deforms. Conversely, when the same crystal is mechanically deformed by pres- sure, an electric potential is developed between the crystal faces. The crystal also exhibits the phenomenon of Figure 1.13.3 (a) Low-pass filter circuit. (b) Equivalent circuit consisting of two end sections and one intermediate T- section. (c) Low-pass filter circuit with an intermediate -section. (d) High-pass filter circuit with an intermediate T-section.
End section
Intermediate T-section
End section L1
2L2 2L2
2L2
2L2
2L2 L2 2L2
2C1 C1 2C1
C1 C1
C2
C2 C2 C2
C2
C2
(a)
(b)
(c)
(d)
C2
C2
C2
C2 1 2
L1
L1
L1 L1 L1
L2
L1 L1 L1
L2
C2
C2
1
2 1
2 1
2 1
2
1 2
1 2 1
2
1 2
1 2 2 1 2
1
mechanical resonance when it is excited with an alternating potential of the correct frequency. The frequency of mechanical resonance is determined by the size and shape of the crystal sample in question and can be con- trolled over several orders of magnitude, from about 20 kHz to about 50 MHz, with considerable precision. In form, the packaged crystal is a slice of crystal cut in such a way as to give the desired mechanical resonant fre- quency, with electrodes deposited on opposite sides so that a capacitive device is made.
Electrically, the mechanical resonance of this device makes the crystal look like a very high Q series res- onant circuit, with a capacitor in parallel with it. This capacitor causes a second parallel resonance, which occurs at a frequency that is very close to the mechanical resonant point. The reactance of a quartz crystal is plotted in Fig. 1.13.4 and shows that, for low frequencies up to the series mechanical resonance, the crystal is capacitive.
For frequencies between the series resonant and parallel resonant points, the reactance is inductive, and for frequencies above the parallel resonance, the reactance is again capacitive. At series resonance XLsXCsand the reactance is zero, and at parallel resonance XLs(XCsser. XCp) and the reactance is infinite. The resonant frequencies of the crystal are very well defined and very stable, provided that the operating temperature is kept constant, making it very well suited as the high-Oresonant circuit that controls the operating frequency of oscillator circuits.
The reactance characteristic of the quartz crystal is changed radically by placing an inductance in par- allel with it. The series resonant frequency remains unchanged, but the parallel resonant frequency is moved higher, so the separation between the two is greater than is the case for the crystal by itself.
Placing an inductor in series with the crystal has similar drastic effects on the reactance characteristic. In this case, however, the parallel resonant frequency remains unchanged, while the series resonant frequency is caused to move lower and a second series resonant frequency is created.
The frequency separation between the series and parallel resonant frequencies of the crystal itself is small, on the order of a few hundred hertz at most for a 1-MHz crystal. Frequency spreading by means of series or parallel inductors can increase this separation to a few thousand hertz, making it possible to use the crystals as band-pass filter elements for IF amplifiers and for sideband separation.
The crystal gate shown in Fig. 1.13.5(a) is a narrow-band sharp-cut-off filter circuit that makes use of the reactance characteristic of the crystal itself. It has been used for separating the sidebands in single sideband (SSB) circuits. When the capacitance of C2is relatively large, a high-pass sharp-cut-off filter with the charac- teristics of Fig. 1.13.5(b) is formed. At the frequency f∞, the reactance of the crystal is capacitive and equal in magnitude to that of the capacitor C2, so the signal fed to the output through the crystal is equal in magnitude and opposite in phase to that fed through C2, causing a complete cancellation at the output. At frequency f0the reactances are again equal in magnitude, but this time the crystal is inductive. The signal through the crystal is
Rs
Ls Cs
Cp XL
XC
fs fp f
×
(a) (b)
Figure 1.13.4 Quartz piezoelectric crystal: (a) the graphic symbol and the equivalent circuit of a quartz crystal; (b) varia- tion of crystal terminal reactance with frequency.
shifted by 90°, while that through the capacitor is shifted by 90°, so both arrive at the output in phase with each other. For frequencies above f0up to f1, just below the parallel resonant frequency of the crystal, the atten- uation remains low as the signal is propagated through the capacitor C2. Severe phase-shift distortion occurs near fp, so the usable passband is only between f0and f1. The cut-off beyond f1is quite gentle, but f0and f∞are only separated by a few hundred hertz, providing a sharp lower cut-off. Frequency shifting by a series inductor can be used to increase the passband to usable widths.
When the reactance of C2 is made considerably higher, a low-pass filter with the characteristic of Fig. 1.13.5(c) results. Again, at f0the crystal reactance is equal to that of the capacitor and inductive, so the signals arrive at the output in phase with each other. At f∞the crystal reactance is equal to that of the capaci- tor and capacitive, resulting in complete signal cancellation, this time at a frequency higher than f0(and fp).
For frequencies below f0down to f1near fs,the attenuation is low, determined by XC2. Phase-shift distortion near fs prevents use of frequencies below this, and again the usable band-pass can be increased by using a series inductor.
The crystal gate is inexpensive to build and uncritical in its adjustment, making it attractive, but it suffers from the disadvantage of providing a very narrow usable passband width. The crystal lattice filter is a more complicated circuit, but it provides bandwidths of a few hundred hertz to several tens of kilohertz and essen- tially flat response characteristics within the passband. Furthermore, sharp cut-off can be provided on both the upper and lower edges of the passband.
Figure 1.13.6(a) shows the circuit of a full-lattice filter, and Fig. 1.13.6(b) shows a half-lattice filter. The attenuation characteristics of both are the same and are shown in Fig. 1.13.6(c). The crystals used in the lattice are matched pairs, so crystals CR1and CR2are identical, and CR3and CR4are identical, but different from CR1 and CR2. The crystals are chosen so that the series resonant frequency of CR3and CR4coincides with the par- allel resonant frequency of CR1and CR2. The inductances of the coils in the input and output circuits are effec- tively in parallel with the crystals and act to space out the separation between series and parallel resonance and provide the second parallel resonance of each crystal.
C2
Vout
Attenuation Attenuation
(a)
(b) (c)
f∞ f0 f1 f f1 f0 f∞ f
Figure 1.13.5 Crystal gate: (a) the circuit; (b) plot of attenuation versus frequency for the case where XC2is small, giving a high-pass characteristic; (c) plot of attenuation versus frequency for the case where XC2is large, giving a low- pass characteristic.
At f∞1the reactances of X1and X3are both inductive and equal, so the in-phase and antiphase signals fed to the output cancel, providing infinite attenuation. This is very near the parallel resonant frequency of X1, and just past this frequency f01occurs, where again the crystal reactances are equal, but X1is capacitive and X3is inductive, so the signals arrive at the output in phase with each other. f∞1and f01delineate the lower transition band of the filter.
At f∞2the reactances X1and X3are equal but capacitive, so cancellation again takes place. This occurs just above and very near the upper parallel resonant frequency of X3. Just below f∞2, the reactances are again equal, but opposite, so the signals are again in phase, at f05. Frequencies f∞1and f05delineate the upper tran- sition band of the filter.
At f03the reactances X1and X3are again equal and opposite, so the signals arrive in phase. At frequency f02, crystal 1 has zero reactance, shorting the input to the output. At frequency f04, crystal 3 has zero reactance and again the output is connected directly to the input. Between these frequencies, the signal is fed to the output with very low attenuation values, providing the band-pass of the filter. Outside the band, attenuation is
L CR4 CR1
CR3
CR2
(a)
(b)
(c) CR1
C
CR3
L
Attenuation
f∞1 f01 f02 f03 f04 f05 f∞2 f
Figure 1.13.6 (a)Full-lattice crystal filter circuit. (b) Half-lattice crystal circuit. (c) Attenuation versus frequency for lattice filter.
relatively high and can be improved by providing additional gradual cut-off filtering in tandem, such as with a normal IF amplifier filter.