26
E x p e r i m e n t s o f RC P h a s e ‑ S h i f t O s c i l l a t o r
Takej i SATO CReceived on 31 October. 1975)
1. Introduction
Positive feedback results when the feedback factor μs is positive and less than 1. where the term μis called the open ‑loop voltage gain. and the symbol s is used here to denote the fractional gain of the feedback network. Ifμs is in‑ creased to unity. the gain with feedback becomes theoretical1y infinite and the amplifier functions as an oscillator. The condition μs
=
1 is true at a single and precise frequency. and at this frequency the feedback signal appering at the input is exact1y in phase with the input signal. Ifμs = 1. oscillations sti11 occur and the amplitude of oscillation increases theoretically without limit. In actual practice. nonlinearity limits the the theoretically infinitive gain to some finite value for both μs?;.l.The important characteristics of an oscillator are its frequency stability. amplitude stability. outout power. and harmonic content. And osci11ators can also be classified as either negative ‑impedance. or feedack oscillator. But all oscillators can be regarded as negative ‑impedance oscil1ators. if we aim at only bui1d ‑up of the osci11ation.
The purpose of this paper is to propose new experimental results about con‑
ditions for build up osci11ation using RC phase ‑shift oscillator.
2. Condition for Oscillation and Its Build‑UP
It was pointed out. as is well known. that the osci11ators must be satisfied as a condition for sustained os.clillations. orμs
=
1. However. oscillators are theoretically designed so thatμs> 1 in order to ensure bu ild ‑up of oscilla‑tion. Under this condition. an oscillator must build IIp oscillations. which maintain constant stable waveform when reaching the steady state.
Although it is possible that under consideration of minor changes in circuit or device parameters in actual practice.μs might become larger than 1. on the ass.umption that the parameters have no changes. papers have never been found about what percentageμs might become more than 1 to hold bui1d up possility. As the result that we made. then. experiments on a RC phase ‑shaift oscillatory circuit of Fig. 1 in order to solve the above question， practical oscillators would
E玄perimentsof RC Phase‑Shift Oscillator 27
‑+ lj C C C ‑+ '.0
R
Fig. 1. Transistor phase shift or RC oscillator
C.
Fig. 2.
be designed so thatμs> 1 by about 5 percent.
The experiments is as follows.
Equivallent circuit of transistor RC oscillator
First. this circuit would be driven by rectangular pulse waveform whose re‑ petitive frequency and voltage are about 40Hz. and 6V. respectively. In this case. transistor being kept extremely over saturation state on account of litt1e transistor base current f10ws by variable resistor connected from the base to the earth. loop gain of this circuit would almost become zero. Under these condi‑ tion. sinusoidal signal whose frequency 950 Hz equals that of RC phase ‑shift osci11ator is applied to the transistor base by external oscil1ator in order to seek the critical osci11atory condition. Actually the Toshiba 2 SB 54 being selected as the transistor of oscillatory circuit. oscillatory frequency is 950 Hz as above‑
mentioned.
Consequently the feedback voltage at transistor base. when this RC oscil1atory circuit is self‑sustaining. is 38mV. col1ector output voltage is 2.23V. and vol‑ tage amplification factor 1μ1:::
印
However. in accordance with the experimen‑tal results. Tase input voltage by e玄ternalosci11ator at critical oscillatory con‑ ditions have the value of 0.4 to 1.7 mV.
Taking the percentage as the ratio of the base input voltage at the critical osci11atory conditions to the feedback voltage to the base when the RC osci11ator maintains a constant oscillatory amplitude. it is found that. about from 1 to 5
直 /
/0・And so if an osci11ator is designed so that μ戸=1 at a particular frequency. μs might become less than 1 for some reason. for example. with minor changes in device parameters. and oscillations would cease. To overcome this possibi‑ lity. practical osci11ators should be designed so thatμs> 1 by about 5 percent.
on the assumption that oscillatory frequency and the fractional gain of the feed‑ back network are constantly kept.
28 Takeji SATO 3. QuaIity Factor Q and Logarithmic
Decrement o in the Circuit
The equivalent circuit of a transistor RC phase ‑shift oscil1ator which is called a lead network is given in Fig. 2， if R;: R1 1 1hie' ， and R2)> R ..
where hie = input resistance with output short ‑circuited for common emitter
and so the Q in the feedback circuit can be calculated as follows; Q = ~o 2 I I
旦 ￨
dω￨ω→ωo
where 0)0 = the resonant frequency
。 =
the phase shift between input and outputWhen considering output current io and input current ii> we can calculate s = io / ii and phase difference e between output current and input current， they become
where
n‑l
X(6‑xE)e = tan‑1
~一一一一一一一
( 1 ‑5 xりQ= ‑ 2 ~o I I ~ι| dω￨ω ー豆K三""̲!
→ω0‑ 29 = 2
As the logarithmic decrement o when critical build ‑up of oscil1ation is found nearly 6 using oscilloscope experimentally. The quality factor Q of this circuit may be found by using next equation
Q=+=0.52
Since it is seen that the Q of the RC phase ‑lead network is theoretically about 0.5， experimental value 0.52 is considerably better in checking respect. To find what percentage of increase of μs is necessary for ensure oscillation， it may be useful to measure the frequency of the RC oscillator which must build up osci‑ llation before reaching the steady state. The frequency at the beginning of build‑up of os.cil1ation， which is now found on the oscilloscope， is within from 870 to 900 Hz .
Consequent1y， taking the ratio of the frequency for the build ‑up to that for steady state of oscillation， it is seen to be the range from 5 to 9 %. However， since above calculated res:ult is only my suspicion， practical oscillator should be designed so thatμs> 1 by about 5 percent in order to they bui1d up osci‑ llations.
