Oscillators III. by Werner Wiesbeck and Manfred Thumm. Forschungszentrum Karlsruhe in der Helmholtz - Gemeinschaft

Transcription

1 Oscillators III by Werner Wiesbeck and Manfred Thumm Forschungszentrum Karlsruhe in der Helmholtz - Gemeinschaft Universität Karlsruhe (TH) Research University founded 1825

2 Electrical Properties (I) The resonant frequency can be derived from the equivalent circuit in Fig C 1 and L 1 describe the mechanical resonance and R 1 describes the damping. Connection to the quartz crystal is achieved by metallizing both sides of the quartz crystal plate, and this causes a not to be disregarded capacitance which is embodied by C 0. If R 1 is neglected, the impedance Z q j LC 1 C C L C C results, by which a series and parallel resonance can be defined. The series resonant frequency results from setting the numerator to 0 (Z q = 0) : f s 1 2 LC (4.29) (4.30) For the AT thickness oscillator one thus obtains: 1.67 MHz f s d/mm (4.31) The parallel resonant frequency is determined by setting the denominator to zero (Z q = ) : 1 C1 C1 fp 1 fs 1 2 LC C C0 (4.32) 1 1 2

3 Electrical Properties (II) The series resonance depends only on the defined product L 1 C 1, but the parallel resonance also depends on the less well defined electrode capacitance C 0. At the same time, the series resonant frequency lies below the parallel resonant frequency. Typical Frequency Dependence of I m (Z) of a Quartz Crystal Oscillator Fundamental Tone: f s 1.67 MHz d / mm 3

4 Adjustment of Resonance Frequency (I) If the resonant frequency is to be changed slightly ( pulling ), then the series resonance can be generated by series connection with a capacitance C s as shown in Fig. 4.17, whereby a new impedance Z q results: 2 1 C1 C0 Cs L1C 1 C0 Cs Z ' (4.33) q 2 jc C C L C C Fig Adjustment of the resonant frequency with a series capacitance. s By reshaping one obtains the resonant frequency f s as f f 1 C C ' 1 s s C0 s (4.34) 4

5 Adjustment of Resonance Frequency (II) Under the condition C 1 << C 0 + C s, as a series expansion in the simplified form f f 1 C wher e f f f ' 1 ' s s s s p 2C0 Cs (4.35) Thus the relative frequency change for series resonance is fs C 1 f 2 C C s 0 s (4.36) and the pulled series resonance approaches the parallel resonance for very small values of C s. C s allows a very accurate setting of the resonant frequency. The parallel resonance is not changed. Frequency Range : f s < f s < f p = const. 5

6 Quartz Crystal Oscillator Circuit (I) NF Quartz Crystal Oscillator For large values of R 1 in the equivalent circuit of the quartz crystal (Fig. 4.14) the amplifier stage must show a high input impedance. This can e.g. be done by an impedance transformation as shown in Fig Fig khz quartz crystal oscillator with impedance transformation. 6

7 Quartz Crystal Oscillator Circuit (II) Fundamental Frequency AT Quartz Crystal Oscillators (Parallel Resonance) A circuit with a Darlington stage is shown in Fig as an example of a Colpitts oscillator. The dividing capacitances C 1 and C 2 can have very large values due to the high input impedance. Thus the effect of the transistor stage on the resonant frequency is very small. The effective load-capacitance of the quartz crystal is represented by the series connection of C 1 and C 2. To obtain sensible standard values (about 30 pf, typical range pf), a capacitance of this order of magnitude is connected in series to the quartz crystal as compensation. 7

8 Quartz Crystal Oscillator Circuit (III) Fig Colpitts oscillator with a Darlington stage for fundamental frequency quartz crystals. 8

9 Quartz Crystal Oscillator Circuit (IV) Overtone AT Quartz Crystal Oscillators (Series Resonance) The series resonant circuit shown in Fig is of advantage. The dimensioning of C 1 and C 2 is done in such a way as to ensure sufficient loop gain. Apart from by the quotient C 1 /C 2, this is also reduced by the voltage division between the quartz crystal resistance and the input impedance at the emitter. For the choice of transistors, the rule of thumb that the transit frequency should reach at least the tenfold of the oscillator frequency is used. It is also recommended that transistors with high DC gain and a small base impedance are used. The resonant frequency of the Pierce oscillator lies above f s, since the quartz crystal has an inductive dummy impedance in the frequency range f s < f r < f p. To prevent undesired overtone oscillations one can use the Pierce-Miller circuit, which is obtained by replacing L 1 of the inductive three-point circuit by a quartz crystal. 9

10 Quartz Crystal Oscillator Circuit (V) Fig Overtone quartz crystal oscillator to 200 MHz (Colpitts oscillator in common-base connection). 10

11 Dielectric Resonator Oscillators (DROs) Forschungszentrum Karlsruhe in der Helmholtz - Gemeinschaft Universität Karlsruhe (TH) Research University founded 1825

12 Dielectric Resonators for Different Frequencies 12

13 Dielectric Resonator Oscillators (DROs) As analogue to the cavity resonator, dielectric resonators have been used increasingly in the last few years. These are cylindrical disks made of a high permittivity like barium-titanium oxide or zircon oxide, with ε r values of between 5 and 150. A material with an ε r 39 is typically used. The unloaded quality factor lies between 7000 and and is thus ten times higher than the Q of a resonant quartz crystal. Fig shows the TE 01δ -field (H 01δ ) of a dielectric resonator. The longitudinal mode index δ (δ < 1) shows that the field also extends to the external space. Fig Dielectric resonator, model of the electrical and magnetic fields of the TE 01δ -resonance of a dielectric resonator. 13

14 Lowest Mode H 01 of Dielectric Cylinder Resonators E Y a) H Z b) H H L c) E z y x D 14

15 Whispering-Gallery Mode TE 50 of a Dielelectric Resonator 15

16 Modes and Fields of Dielectric Resonators H 01 E-Field H-Field 16

17 Single Sided Stripline Coupler (I) The coupling takes place by the magnetic field of the stripline. The equivalent circuit of Fig can be simplified in the vicinity of the resonant frequency so that it only consists of the series connection of two equally long line lengths with a phase shift Φ and resonant equivalent impedance Z T (Fig. 4.22). Fig Single sided connection of a dielectric resonator to a stripline. a) Construction; b) Equivalent circuit. Fig Equivalent circuit in the vicinity of the resonant frequency. 17

18 Single Sided Stripline Coupler (II) In the vicinity of the resonant frequency ω 0, Z T can be calculated to be: 2 M 1 Z Q, x 2 Q T 0 0 Lr 1 jx L with Q, r Rr and M is the mutual inductance 1 L C r r (4.37) At the resonant frequency, with x = 0, it follows that ZT R 0Q0 L r (4.38) The open circuit quality factor Q 0 only takes into account the losses of the dielectric resonator itself. If the coupling factor is defined as ZT 0 / 2Z0at the resonant frequency, then one obtains for the loaded quality factor Q Q : e 0 Q e 2ZL 0 r 2 0M M 2 (4.39) 18

19 Single Sided Stripline Coupler (III) The relation L r /M 2 can be determined by the so-called H/I method. The following is assumed when this method is used: magnetic fields H result solely from the current on the stripline (width w, thickness negligible) and the ground plane; the current on the ground plane is assumed to be constant over a width 3w. One obtains: with M L r H 0 A (4.40) (4.41) Here h S is the substrate height, d is the distance between the stripline and resonator centre, L is the resonator height and A is that surface within which the magnetic and electric fields can be regarded as constant. It is obtained either by calculation of the fields or approximation. With the calculated sizes, the transfer function of the twoport (see Fig. 4.22) can be determined: ZT Z0 2j S11 e 2 Z Z (4.42) 2 2j S21 e 2 ZT Z0 where Z 0 is the characteristic impedance of the stripline. I 2 L L H I ln d w 2 d w 2 T 0 2 h S L h S L 2 2 d 3w 2 ln d 3w 2 (4.43) 19

20 DR Bandpass Filter in Microstrip Two microstrip lines coupled via a dielectric resonator (DR) form a bandpass filter whose centre frequency coincides with the resonant frequency of the resonator. Using multiple DRs results in multipole DR filters. Construction Equivalent Circuit 20

21 Single Mode Filter with Dielectric Resonators 21

22 Resonant Frequency and Detuning (I) The resonant frequency of a dielectric resonator is a function of the relation between its height L and diameter D. Fig shows the resonant frequency of cylinders with D = 16.5 mm and ε r = 83 as a function of the height L for different resonant modes. TE 01 f r GHz 34 a r a L 3.45 a and L in mm 2% precision: 0.5 a/ L 2 30 r 50 Fig Resonant frequencies of a dielectric cylinder with D = 16.5 mm and ε r =

23 Resonant Frequency and Detuning (II) At the same time, the environment is to be considered when determining the resonant frequency. The resonant frequency is dependent on the distance between the resonator and the metallic boundary. This effect can also be exploited for the targeted detuning of the resonant frequency. For this a metallic or dielectric disk is lowered toward the resonator from above, whereby the frequency rises or falls. Fig shows the quality factor and resonant frequency of the TE 01δ mode of a cylinder with D = 8.3 mm, L = 3.41 mm, and ε r = 83 as a function of the distance L 2 from a metallic tuning stub. Fig Changes in the quality factor and resonant frequency of the TE 01δ mode due to a metallic tuning stub at a distance L 2. 23

24 Transistor Oszillator with a Dielectric Resonator 24

25 Circuits with Dielectric Resonator Oscillators (I) The dielectric resonator can fundamentally be used for two methods of stabilisation. Either it is placed at the input, i.e. in the feedback branch, and thus directly determines the frequency (an example is shown in Fig in this case the oscillator does not oscillate without the resonator). Fig Oscillator with dielectric resonator at the input. 25

26 Circuits with Dielectric Resonator Oscillators (II) Or the resonator is placed at the output and causes strong reflections in the oscillator due to its marked series impedance at frequencies other than the resonant frequency (Fig. 4.27). This latter procedure is only successful for strongly load dependent circuits; the oscillator in general also oscillates without the resonator. Fig Oscillator with dielectric resonator at the output. 26

27 Materials and Properties for Dielectric Resonators 27