By

V ANNAi and L. P.A.P

InStitute of Communication Electronics. Technical University, Budapest Received Ocwber 5, 1979

Presented by Prof. Dr. S. CSIBI

A sinewave a very sman harmonic content can be generated according to several principies. The so-called "exact sinewave oscillator" [1]

has consIderable distortion when realized and this distortion can be reduced only at the cost of decreasing stability of the amplitude and increasing transient times. This is equally true for the V AN der POL oscillator [2J, moreover distortion can't be even theoretically zero in this case. Osciilators with quasi linear amplitude stabilization [3,4] have two main sources of distortion: the quasi linear components are not perfectly linear in practice; and the control signal (proportional to the amplitude) necessarily contains a certain amount of ac especially at low frequencies.

Small distortion, great stability of the amplitUde and fast transients are contradictory requirements, demanding new concepts for signal generation.

Such concepts are described in the thesis by D. MEYER-EBRECHT [5J, in the

paper by E. VANNERSON and K. C. SMITH [6J and in the technical description of some products.

The block-diagram of the system introduced in the following sections is shown in Fig. 1. The frequency of the oscillation is determined by the linear system consisting of two integrators and an inverting amplifier. This system is characterized by the following differential equation:

(1) Damping is determined by the -l/e: resistor (the resistor can be of either positive or negative sign).

*Fig. *1. State variable oscillator with fast amplitude control

-1~~--~-=pL_+--4

-x1T--r~~-t--+--t

Time of control circuit activation '::=0

*f *4

### r

, I

01

*Y2 *
0

*Y3 *
, I
0

*y+ *

### j

1 0

I

I I

### I

I I

### !I

### HI

t=o

*Fig. *2. Generation of the fast control signal
I

I

I

1

I I

I I *t *

I ^{I }

~

! !

I

I

1 *t *

I

### -

I

### 1'

_{I }

^{t }

t

*RC OSCILLATOR * 61

Operation of the so-called fast amplitude control circuit can be analyzed studying the waveforms in Fig. 2. (These are the waveforms in the case of 8=0.) The control circuit interferes with the oscillator circuit for a short time when

*- x *

reaches its positive maximum: closing the analog switch *AS*it restores the initial condition of -

*x *

^{when x }

^{= }O. the oscillator starts every period with the same initial conditions and the distortion is determined by the properties of the free running linear The time interval of the control is set by the

the gate G: a narrow is generated

*- x *

is positive width of the pulse is constant at all
* -X= * 2 cos exp cos

*1 1 = - - - = = * [ = *cv=1 *

2

HalrrrlOfllCs can be calculated case in the IImA/mQ" way:

**2;: **

*X**n**= *

### ~ *f *

^{exp }

^{(-at) }^{cos }

^{cos }

^{dt= }

o

1

### r

^{11 }

^{a }^{l }

=-[1-exp(-21117:)]? _{217: }

_{La-} _{+ }

_{(n }_{-1 f }

### +

_{a}*2*

### +

*(n*

### +

1)-### 'J'

*Yn= *

### ~ *f *

^{exp }

*(t)sin (nt)dt=*

^{(-at)cos }o

1 *[ n - 1 * *n+l * ]

= -- [1-exp(-2a17:)J *217: * *a-*?

### +

^{(n }*- ,*

^{-1)-}### +

*a2+(n+l)2 .*These yield for the amplitude of the harmonics:

(3)

(4)

1

*Ja*

^{2}

^{+n}

^{2 }*Zn=JX;+ Y/ = -* [l-exp (-2a17:)J (5)

17: J[a^{2}+(n- 1fJ [a*2 +(n+l)2J *

Distortion can be *small only if a *~ 1 and in this case the following
approximations are valid:

'7 ' "

### ~ ~ #+1 '"

^{1 }

*L * 1

### =

*2an*

*u-:-;:*

### = '"

- 11: *a;j a*^{2 }*+4 *

*n * *2a *

Zn~2a ~-,

(n-l)(n+l) *n * *n>l *
As

### <12a~l,

_{n - l }

. the distortion factor is

<

### =Ial

From results it is clear realized if the figure of merit

### IQI

~ 1.28 .1,

-exp

if the capacitor's is

At the moment of fast control activation a step

*=-b *1
*b *

211:

(6)

(7)

(8)

**can be **

(U)

*RC OSCILLATOR * 63

appears at the output of the integrating amplifier (Fig. 3) and changes the value
of [; through the function *G(z). *

*x *

Offset control

*Fig. *3. State variable osdHator with fast and slow amplitude control

*001 *+----,----,---r----.----,---,---~----._----,_--~

o

*- 0.01 *

*Fig. *4. Transients of the Q value

### Q

is changed according to the information gained at the end of the cycle;in the next cycie there'll be a new Q. * ^{Z }*doesn't change between fast controls.

Operation of the system can be described by

(

_ Gn . *2n * )
2 _{\ }

## R2 1-~)

_{4 }

^{P }

(12)

and

(13 )

*c>O; * *n=O, *1, ...

where x~ is the amplitude at the end of the cycle and C is the gain of the slow
control circuit. *8(Z) *was assumed to be linear in the interval where (13) is valid.

80 is given.

Fig. 4 shows the typical waveforms. It is obvious from the figure that C;;;:; 0.3 is the optimum if we want Q to be settled as fast as possible.

system in Fig. 3 has been actually realized. Fig. 5 shows one possible

solution the fast control problem.

Fig. Sa the fast control circuit is shown. Loop gain is set by the
differential amplifier *T*_{1 , }*T** _{2 • }*This differential amplifier also serves the.purpose
of comparing the reference to the attenuated and value of -~. It gives a
signal at its symmetrical output (on resistor proportional to the difference.

This signal charges or discharges the integrator. Diodes and *D2 help to *
sh,orl:en trctnS:lellt times. At the moment operation switching
transistor is swltc:neC! on.

*-x *

### @

*Fig. *5. Simplified schematic of the amplitude control circuitry

Fig. 5b shows the slow control circuit. means of *R *^{1,8 }is set to a negative
v§l.lue to help the oscillation build up. In the end e is adjusted to zero with the aid
of the inverting amplifier, the attenuator formed by *R*_{2 , }*R3 and Tt, functioning *
as a controlled resistor.

*RC OSCILLATOR * 65

adjustment of the distortion can be done with the offset poten- tiometer shown in 3. The following list of measured values illustrate the range of realizable distortions.

**Frequency of **
oscillation

29.7 Hz
100 Hz
270 Hz
937 Hz
2.8kHz
**9.7kHz **
17.8kHz

**2. harm. **

<iO

10
12
16
**24 **

17 24

3. harm.

Il V

<iO 7

7 7

10 16

the References

4. harm.

pV 6 7

6

lO 10

Dist. factor

<1.17·lO-^{5 }
1.07 ·lO-5

L4 . lO -5

1.68,10-^{5 }
2.33' lO-5

range and can

A novel concept is presented for the generation of a sinewave with ultra low distortion in the 1 Hz -100 kHz band. A short summary of the presently known methods is followed by some calculations using a simple model of the system that yield the theoretical limitations inherent in the concept. The realized circuit and the results of measurements are presented as welL

1. PORTER, S. N.: Signal generator with rapid automatic amplitude stabilization, US-Patent, 3.419.815. 1968.

XII. 31.

2. KOR:-:, G. A.-KoRl'i, T. M.: Matematikai kezik6nyv muszakiaknak. Muszaki K6nyvkiad6. Budapest.

1975.

3. KOMARIK, J.: Nemlinea.ris aramk6r6k. Tankonyvkiad6, Egyetemi jegyzet, J5-1046, 1973.

4. PAP, L.-NEMES. M.: Oszcillatorok stabilitasa, Hiradastechnika, XXVI. evf. 12. sz., pp. 364-369.

5. MEYER-EBRECHT, D.: Schnelle Amplitudenregelung harmonischer Oszillatoren, Thesis, Technische Universitiit Braunschweig, May 1974.

6. V ANNERSOl', E. and SMITH, K. c.: A low-distortion oscillator with fast amplitude stabilization, Proc. of IEEE Symp. on Circuit and Systems, pp. 142 -146. 1974.

7. PAP, L.-VAl':-:AI, N.-FIXfl.:. L.: Kapcsolasi elrendezes ultra kis torzitasu ketfazisu RC oszcillator megval6sitasara. Magyar Szabadalom, H 03 BS/20, EE-2469. 1976. XII. 31.

Nandor V ^{ANNAI }

Laszi6 ^{PAP } ^{} H-1521 }Budapest

5 Periodica Po\yteChnica El. *24iI-2 *