lc and quartz oscillators -...

Universität Karlsruhe (TH) Research University•founded 1825in der Helmholtz - Gemeinschaft

Forschungszentrum Karlsruhe

by Manfred Thumm and Werner Wiesbeck

LC and Quartz Oscillators

Institut für Hochfrequenztechnik

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For the frequency range to about 500 MHz, it is possible to construct

oscillators from monolithic circuit elements. For LC-oscillators, the

frequency-selective feedback two-port K(ω) consists of concentrated

spools and capacitors. If one considers the equivalent circuits of the

amplifier- and feedback-two-ports together, the equivalent circuit

diagram in Fig. 4.10.

Fig. 4.10. Oscillator two-port (LF equivalent circuit).

LC Oscillators


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If positive inductances and capacitances are assumed (XL = ωL, XC = -1/ωC),

then the following applies for the reactive elements X1, X2, and X3:

(G+gm)X1 + GX2 = 0, thus |X2| > |X1| since G, gm are positive and real.

Further analysis reveals a further equation: X1 + X2 + X3 = 0. In summary, the

following results apply:

X1 and X2 must be of opposite reactance type, and the reactance type of X2

must be that of X1 + X2.

X3 and the series combination of X1 and X2 are of opposite reactance type,

i.e. the reactance type of X3 is the same as that of X1.

In the course of time empirical and theoretical developments led to the circuit

variations shown in Fig. 4.11. Fig. 4.12 summarises the basic circuits for the

oscillator types described in Fig. 4.11.

Classification of LC Oscillators (I)


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Fig. 4.11.

Systematics of LC-oscillators

(different combinations of the

dummy impedances X1, X2,

and X3, ).

Classific. of LC Oscillators (II)


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Fig. 4.12.

LC-oscillators

from Fig. 4.11.

In three basic

circuits.

Classification of LC Oscillators (III)


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Colpitts-Oscillator:

C1 a. C3 Capacitances,

Cr controller negative feedback

C1 = C3= variable capacitor

if L2 is small, then in common source

configuration

Clapp-Oszillator

L2 large and

Pierce-Oscillator

L2 = Quartz

Frequency Stability

Meißner-Oscillator:

X1 : Inductance

Common Base and Collector

X2 : Parallel Resonance circuit (capacitive : )

X3 : Inductance

Common Emitter

X2 : neg. Inductance

counter-wound windings

X3 : Parallel Resonance circuit (induktive : )

Hartley-Oscillator:

L1 a. L3 Autotransformer

.r oszf f

.r oszf f

1

3

m

LG g

L

2 2( )X L

3

1

m

CG g

C

2 21/L C

2 2 2 21 1/ 2eff rL L L C L

55 10

1

3

m

LG g

L

LC Oscillators


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The functioning of a quartz crystal as a resonator depends the

piezoelectric effect. If mechanical pressure or traction is exerted on a

slab cut from a quartz crystal at a suitable angle, an electric charge is

created. The effect is reversible. Thus it can be deduced that the quartz

crystal is in principle a mechanical oscillator. In the following the varoius

quartz crystal forms and their equivalent resonant circuits will be

described along with their temperature dependence.

Fig. 4.14.

Circuit symbol and dynamic equivalent circuit of an oscillating quartz crystal.

Circuit symbol

Q

Equivalent circuit:

mech. resonance Attenuation

Electrode capacitance

Quartz Crystal Oscillators


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z

yx

F

F

Si

O

+

+

-

-

a) no force b) longitutinal force (d33) c) transversal force (d31)

Quartz Crystal (three Si4+ -, six O2- -Ions):

Piezoelectric Effect


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LF Quartz Crystals:In the frequency range to 1 MHz, bending, stretching and area shear

oscillators with the properties shown in Fig. 4.13 are used.

Fig. 4.13.

Overview of

the most

popular

non-AT

quartz

crystals in

the range

to 1 MHz.

Quartz Crystal Types


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The temperature dependence is described by the parabola

(4.25)

The slope of a varies between 2∙10-8K-2 and 5∙10-8 K-2, depending on type.

The reversal temperature TUKP can be specified for most oscillator types.

An exception to the temperature dependence described by Eq. 4.25 is

the H-bending oscillator which has a linear frequency dependence.

2

UKP

fa T T

f

Temperature Dependence of Frequency of LF Quartz

Crystals


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At high frequencies quartz crystals with an AT cut are utilised. AT quartz

crystals are thickness shear quartz resonators with a fundamental

frequency range of between about 750 kHz and 25 MHz. The dependence

of the relative frequency change on the temperature can be well

approximated by a cubic parabola

(4.26)

where a1 = 84∙10-9 Δθ K-1, where Δθ is in radian minutes, and a3 = 10-10 K-3.

Δθ is the deviation from the so-called zero angle. It is the cutting angle Δθ

for which the turning tangent in the temperature dependence (curve (1) in

Fig. 4.15) is horizontal. TINV is the temperature at the turning point. Fig.

4.15 qualtitatively shows the relative frequency change for two different

cutting angles as a function of temperature.

Modern micromechnical etching techniques (inverted mesa) allow the

manufacturing of AT fundamental tone quartz oscillators with resonant

frequencies to 250 MHz.

3

1 3INV INV

fa T T a T T

f

AT Fundamental Tone Oscillator (I)


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Fig. 4.15.

Relative frequency

change of AT quartz

crystals (qualitative) with

different cutting angles

relative to the

crystallographic axes.

Table 4.1. Substitute data of AT fundamental tone quartz crystals.

AT Fundamental Tone Oscillator (II)


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If a thickness shear oscillator is excited by a higher harmonic the quartz

crystal plate’s subplates oscillate out of phase to each other. Only odd

overtones can be excited since the electrodes would have the same

polarity for even overtones. Fig. 4.16 schematically shows the excitation

of the fundamental resonance and third overtone.

Fig. 4.16.

Thickness shear

oscillator operating at

the fundamental

frequency and the third

overtone (schematic).

The fundamental frequency of an AT quartz crystal is inversely

proportional to the plate thickness. So, e.g., a 30 MHz fundamental

frequency quartz crystal is about 55 μm thick. If it is excited in the third

overtone, i.e. at 90 MHz, then the electrically active „subplate“ thickness

is a third of that, i.e. about 18 μm.

AT Overtone Quartz Crystals (I)


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The overtone frequency is not an exact multiple n of the fundamental

frequency, but this so-called anharmony decreases with growing overtone.

Thus it is relatively easy to operate quartz crystal oscillators to a

frequency of about 300 MHz, even though the „conventional“ upper

frequency limit is at about 200 MHz (9th overtone): one operates the quartz

crystal at the 11th or 13th overtones, which are fairly exactly multiples of

11/9 and 13/9 of the 9th overtone. However, a quartz crystal with as high a

fundamental frequency as possible should be chosen (20-30 MHz) so that

the overtones are far apart. Typical equivalent circuit values are given in

Table 4.2.

Table 4.2. Equivalent circuit values of AT overtone quartz crystals.

AT Overtone Quartz Crystals (II)


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The dynamic capacitance C1 decreases by the square of the overtone n:

(4.27)

The values of R1 increase in the same way. They typically lie between 20

and 200 Ω. Thus the achievable quality factor decreases with increasing

frequency, and the static capacitance C0 provides an increasingly large

bypass for the quartz crystal. Thus the capacitance should be

compensated by a parallel inductance

(4.28)

from a certain frequency (for fs see Eq. 4.30). As a rule of thumb, C0 should

be compensated for when |XC0| < 5R1, or generally above about

70 MHz.

1, 2

1typC

n

2 2

0

1

4p

s

Lf C

AT Overtone Quartz Crystals (III)

lc and quartz oscillators -...

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