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Variable frequency tunnel diode relaxation oscillator
NUCLEAR I N S T R U M E N T S AND METHODS 55 (1967) 120--124; (~) NORTH-HOLLAND P U B L I S H I N G CO.
VARIABLE F R E Q U E N C Y T U N N E L D I O D E R E L A X A T I O N O S C I L L A T O R R. J. SMITH-SAVILLE The Physical Laboratories, The University of Manchester, England Received 2 May 1967 A current controlled wide range variable frequency tunnel diode relaxation oscillator circuit is described and analysed. It is shown that the oscillation frequency is linearly proportional to control current over a wide range. Experimental results are
presented for two circuits, one with a frequency range greater than 200 to 1, and both are shown to be in good agreement with theoretical predictions.
1. Introduction Conventional tunnel diode relaxation oscillator circuits consist of a tunnel diode biased by an inductor in series with a voltage source in such a way that the diode is unable to remain permanently in either its low or high voltage positive resistance states but switches between the two at a frequency determined by the rate of change of current in the inductor 1). The oscillation frequency of this circuit can be altered by varying the bias, and so changing the dwell periods in each of the two states, or else by varying the value of the inductance. The former method is unsatisfactory if a wide change in mark to space ratio and stability cannot be tolerated. In the latter case the frequency change which can be achieved will be proportional to the range over which the inductance can be altered. Unless the inductor is tapped the frequency range will not normally be as great as 5 to 1. In certain applications, where the oscillation frequency must be synchronized or equalized with another widely varying frequency, this range may be insufficient. In addition, since the inductance itself must be varied, means must be provided for effecting this which are usually either mechanically inconvenient or else involve the addition of considerable circuit strays due to additional lead lengths to panel controls. These strays can be troublesome, especially at high frequencies. The circuit described below was designed to overcome both of these problems by providing a frequency variation, which can be greater than 100 to I, in response to variation of a steady control current. The dependence of frequency on a current, rather than on the value of a variable inductance, also means that this circuit can easily be incorporated in self adjusting loops. In addition, the oscillation frequency is linearly proportional to control current over a wide range and in this region can be calculated simply from known circuit parameters which
do not involve any of the tunnel diode nonlinearities.
120
2. Description and analysis In the conventional tunnel diode relaxation oscillator circuit the dwell times in the low and high voltage states, TI and 7"2, may be found by solving eqs. (1) and
(2) I o - I , = L -1
Ip-1,
= L -'
f?
[Vb(t ) - Vf(t)]dt,
(1)
Vb(t)]dt,
(2)
[•t)-
Here L is the series inductance, Ip and I, are the tunnel diode peak and valley currents, Vb(t) is the bias voltage and Vf(t) is the tunnel diode voltage in the state under consideration, both as a function of time. From these expressions it will be observed that the frequency, 1/(T 1 + T2), will be inversely proportional to inductance but will be difficult to calculate exactly because of the nonlinear tunnel diode characteristics represented by In the circuit to be described the bias level, Vb(t ), which remains effectively constant in the conventional circuit, is varied in such a way that the difference between this and the tunnel diode voltage, Vf(t), is constant and proportional to control current I. In this case the periods can be calculated directly and the frequency will be proportional to IlL. Fig. 1 shows a circuit in which this condition is satisfied. The transistor, QI, provides a convenient means of biasing the tunnel diode from a low source impedance whilst at the same time enabling the bias potential to be varied by a signal applied to the relatively high impedance base circuit. Transistor bias of tunnel diodes also has several advantages over back diode 2) and high conductance diode 3) nonlinear
VARIABLE
FREQUENCY
RELAXATION
biasing in circuits such as univibrators in which the voltage excursions are large. The "knee" potential is easily adjustable, the supply current drain is lower than for back diode biasing and the stray capacitance is lower than that of most high conductance diodes. Also, it is frequently more economical. :V+
~source, /]] controlcurrent ,
rt mlt*~
J
r:'
D
~ c
~'~O'
2t OV @C
outptX waveforrn
L
~
oo "Vsource 2
2
Fig. 1. Circuit diagram.
The circuit of fig. 1 operates as follows. Assume initially that diode D 2 is short-circuited and that all the circuit time constants are much longer than the period of oscillation. The voltage on the base of QI is adjusted by varying r I and r2 so that the constant bias potential appearing at the emitter of Q~ has the value which causes the tunnel diode to dwell for equal periods in its low and high voltage states (corresponding to phases 1 and 2, respectively, in fig. 1). Under these circumstances, the dwell times /'1 = 7"2, can be found from either expression 1 or 2 and the frequency will be the maximum attainable, given approximately by f , , , , = A V t / { 4 L ( I p - I,)},
(3)
where A Vf is the effective difference between the forward voltages in states 1 and 2. If D2 now functions normally and a small direct current,/, flows from source 1 the mean diode current will dearly be L However, when the tunnel diode switches to state 2 the resultant negative pulse will be transmitted through R and C to the anode of D 2. Because of the exponential dependence of diode current on forward voltage, the current in D 2 will be reduced to virtually zero and all of the current I will then flow through R, C, and the tunnel diode. This
OSCILLATOR
121
will remain true until I becomes so large that the voltage developed by it across R is sufficient to support appreciable the current flow through D2, During the next half cycle, when the tunnel diode has reverted to state 1, the charge previously stored in C by I must be withdrawn, since in equilibrium the mean capacitor voltage will remain constant. The resultant discharge current again flows through R, D2, and the tunnel diode. Together with the c u r r e n t , / , still flowing from source 1, this will ensure that the mean current through D 2 is I, as required. It will be assumed that the dwell times in each state continue to be equal with D 2 in circuit, and also that the discharge current remains constant throughout phase 1, in which case it will be equal to - L These assumptions are experimentally verified. Under these circumstances, the voltage at the anode of D 2, and so at the base of QI, will follow the tunnel diode voltage changes but will be offset by a steady voltage, + I R , when the tunnel diode is in state 2 and, - I R in state 1. Expressions (1) and (2) can be combined and written in differential form, with the addition of a further term to represent the bias source resistance, r, Ldi/dt + ir = Vr(t) + Vb(t).
(4)
Here i is the inductor current, the positive and negative signs refer to phases 1 and 2 respectively, and r = re+ R/~ where re is the incremental emitter resistance of Qt and fl is its common emitter current gain. The bias voltage, Vb(t), is equal to the changing voltage appearing at the anode of D2, VD(t), but with a mean level determined by r I , r2 and the emitter diode characteristics of Qt. This level is equal to the voltage necessary to cause the mean inductor current, ½(lp + Iv), to flow in the resistance r. Therefore eq. (4) can be rewritten: L d i / d t + i r = ½(lp+Iv)r+ Vf(t)+ VD(t).
(5)
VD(t) will now be considered in greater detail. If it is assumed that the base circuit time constant, 1/[R + r t r2/ ( q + r 2 ) ] is infinite then, as described above, VD(t) will differ from Vf(t) by a constant term, IR, the sign of which will be negative in state I and positive in state 2. However, in this case, when I = 0 the voltage appearing at the emitter of Q, will be almost equal to the voltage across the tunnel diode, with the result that insufficient current will flow through L to enable the diode to switch to its other state. This latching will not
122
R.J. SMITH-SAVILLE
occur in practice and in order to calculate the minimum frequency the base circuit time constant must be considered. If it is assumed that the time constants associated with C and c are so large that their effects may be neglected, then a current will build up in / because of the voltage, Vb- Vu(t), appearing across it. (Here Vb is the bias source voltage). This current will develop a voltage across R which will eventually become large enough to enable sufficient current to flow in Qi to allow the tunnel diode to switch to its other state. Since r will be small, this voltage will be very much less than Vb- Vo(t) and so if VD(t) is assumed constant the current in l can be considered to change linearly with time and will be equal to [V b - VD(t)]t/l. In fact, however, Vo(t) will be virtually equal to Vf(t), which is by no means constant, especially in state 2. It is consequently not easy to calculate the time dependence accurately and in the subsequent analysis this expression will be represented by ~tt. a can be estimated as above but for accurate work its effective value must be measured for the particular circuit. It is only in this term, which is of small importance t h r o u g h o u t most o f the frequency range, that the tunnel diode nonlinearities enter the analysis. Immediately after a transition between states the current flowing in l and R will be that appropriate to the end o f the previous phase. A further term, ½RAT', must therefore be added to Vb(t) to represent this initial condition, where T' is the duration o f the previous half cycle or phase. VD(t) now consists o f the four terms below VD(t) = V,(t)-t-tR+_ctR(t-½r').
If this is substituted into eq. (8) and T' put equal to T l , since the dwell periods in each state are assumed equal, the resultant eq. (9) can be solved for T~. ½(I~,-1,) = / { I +(R/r)} +(na/r){½T, - ( L / r ) }
+ [I{1 - (n/,')} + (na/r){½T, + (L/r)} - } ( l p - I,)] exp (-rT,/L).
(9)
The expression for T2 is identical, as required by the assumption of equality between T, and T2. As it stands, eq. (9) is not in a convenient form for the determina. tion o f TI. However, two sets of approximations m a y be made. Firstly, if r(Ip-Iv)/(Ra) >> L/r then the exponential term will be effectively zero for small I and when I = 0 we have
To = {r/(Rct)}(lp-l,)+2L/r,
(10)
so that the minimum frequencyfmln = ½To ~. Secondly, when expression (9) is re-arranged and logarithms taken, the result will simplify to
T
=
L(I~- Iv - 2I)/(IR),
provided that (7)
so that the inductor current, i, as a function o f time is given by
i = Ae-"/L + ½(Ip+ 1,) + [(Rlr){I- (Lair)} +(otR/r)(t-½T')].
A = (~Ra T ' / r ) + (LaR/r 2) + I{1 - ( R / r ) } - ½(lp - I,).
(6)
Substitution of this into eq. (5) gives
Ldi/dt+ir = ½(l,+tv)r+_lR-I-otR(t-½T'),
i has increased to I p - I the tunnel diode current will be Ip and the diode will switch to state 2. Since in this second state the current I subtracts from the diode current the reverse transition to state 1 will occur when the inductor current has decayed to lv+I. If we consider phase I in which the tunnel diode current is increasing towards Ip the positive sign version of eq. (8) is applicable, the initial value o f i at t = 0 is i = Iv+l and the final value at t = T, is i = lp-l. Then
(8)
The constant A must be chosen to satisfy the initial conditions. It will be assumed that the tunnel diode is not significantly loaded by any other circuits so that only the effect of I need be considered. During phase 1 this current adds to the tunnel diode current so that when
[{½(Ip-lv)-l}-(g~tT/r)]/(R/r){(otL/r)-l}
<< 1. (11)
In this case the frequency is: ½./.-1 = Ig/{2L(lp-lv-21)}.
(12)
As can be seen from the condition (I 1), which must be satisfied if the simplified expression for the frequency (12), is to be accurate, the ratio R/r must be as large as possible. This will result in a small control current I < < (Ip-Iv). The term in I in the denominator o f (12) can then be neglected and the frequency will be linearly dependent on the control current. Under the above circumstances the ratio of maximum to minimum fie-
VARIABLE FREQUENCY RELAXATION OSCILLATOR 1.2
quencies can be found approximately from eqs. (3) and (10) L../L,.
~- rA Vy/(2RctL).
1.0
(13)
3. Experimental results and practical considerations
Figs. 2 and 3 show comparison of measured results, for R = 117 f2 and 1 kt2, respectively, with thetheoretical values computed from the full expression (9), and from the approximation (12). At very low control currents the frequency is mainly determined by the base circuit time constant and is independent of 1, giving the relatively horizontal sections of the curves in the low current region. As the control current is increased, the behaviour approaches that predicted by the simple expression, (12), until at sufficiently high currents the curve again becomes markedly non linear. This frequency saturation at high currents occurs because the diode D 2 becomes progressively more forward biased throughout the whole cycle. This invalidates the assumption made in section 2 and means that as the control current increases the base voltage of Qt changes less and less, eventually
/2/
0.8
07
0.6
0.5
/
*
R
117~t 2.21
=
r •
,
t
J
~
ILi ) - I v
= 811.8,~H .8~A
1.19 A sec"
0
0.2
0L
I
0.6 0.8 !'O control current m A
I
i
1.2
Fig. 2. Comparison of measured values (Q), with values calculated from expression 9 (+) and expression 12, (o), for R .= 117.(2.
123
0.8 tJ3
Zofi
:-~oL q-
/
r = 3.23a Ip - Iv: 8.82 m A L = 11.8 p 14 c~ = 30.6 A s e c -~
/*
g'.2 o
0
20
60
100 40 80 control, current~A
20
Fig. 3. Comparison of measured values (O), with values calculated from expression 9 (+) and expression 12 (o), for R
= Ik[2. approaching the situation in the simple case in which the base is completely decoupled. The approximate current at which saturation will occur is given by 21R = lip- Vv, where Vp and Vv are the tunnel diode peak and valley voltages. It will be observed from expression (10) that if L/r << (Ip-l~)r/(~R) the minimum frequency will be proportional to ctR/r. Consequently, if the ratio R/r is maximized to improve linearity the minimum frequency will also be increased. There will, therefore, be a reduction in the frequency range because the maximum frequency remains fixed at the saturation value determined by eqs. (I) and (2). Conformity with expression (12) and a very wide frequency range are therefore somewhat incompatible and a compromise which suits the application must be made. This is apparent from expression (13). It will be observed, however, that the frequency ranges can still be very large, being 230 to 1 infig. 2and 10to 1 infig. 3. The other major determinant of the minimum frequency is the base circuit time constant, represented by ct in eq. (10). Unfortunately, this cannot be increased indefinitely to reduce the minimum frequency because transistor Q1 will eventually become unstable when the inductive impedance of the driving source becomes too large compared with the emitter circuit impedance 4). This instability will manifest as pulse length jitter and oscillation frequency instability.
124
R. J. S M I T H - S A V I L L E
The ratio R/r will approach a limiting value equal to the current gain o f Q t , as R is increased to very large values. The practical upper limit on R is the condition that the bias source impedance r(~- R/fl) must be as small as possible and certainly less than the limiting value above which relaxation oscillations no longer occur ~). Since r = rc+R/fl a suitable compromise value for R is that for which Rift = r c. Because the maximum attainable value of R/r is largely determined by the current gain of the transistor Q~ high gain, high cutoff frequency devices are necessary. The incremental emitter resistance, re, which also contributes to the source resistance, r, is inversely proportional to emitter current and so can be reduced by increasing the steady emitter current from source 2. The minimum value attainable in this way will be limited either by the transistor power dissipation or else by the equivalent emitter series resistance of the transistor. The device used for the measurements displayed in figs. 2 and 3 was a 2N 2926 transistor with a low frequency current gain of approximately 400 and a cutoff frequency of 120 Mc/s, operated at a standing collector current of about 15 mA in fig. 2 and 35 mA in fig. 3. The detailed characteristics of the diode D 2 are unimportant but for high frequency operation its junction capacitance and forward recovery time must be
small, as must also be the stray capacities associated with the inductors L and L
4. Conclusions Operation of this circuit can be extended to considerably higher frequencies than those shown in figs. 2 and 3 but at very high frequencies accurate calculation of the frequency is more complex because of circuit strays and the frequency dependence of transistor parameters, particularly current gain and emitter resistance. At the frequencies used, figs. 2 and 3 demonstrate good agreement between the predicted and measured values and also demonstrate the reasonably linear dependence of oscillation frequency upon control current. Particular advantages of the circuit are the ease with which a relatively accurate prediction of operating frequency may be made and the wide frequency range obtainable, which can be greater than 200 to 1.
References ~) Tunnel diodes, RCA Manual. 2) R. H. Bergman, M. Cooperman and H. Ur, RCA Rev. 23, no. 2 (June 1962) a) A. 1_.. Whetstone, Rev. Sci. Inst. 34 (1963) 412. ') M. V. Joyce and K. K. Clark, Transistor circtdl analysis (Addison-Wesley, 1961) p. 264.
VARIABLE F R E Q U E N C Y T U N N E L D I O D E R E L A X A T I O N O S C I L L A T O R R. J. SMITH-SAVILLE The Physical Laboratories, The University of Manchester, England Received 2 May 1967 A current controlled wide range variable frequency tunnel diode relaxation oscillator circuit is described and analysed. It is shown that the oscillation frequency is linearly proportional to control current over a wide range. Experimental results are
presented for two circuits, one with a frequency range greater than 200 to 1, and both are shown to be in good agreement with theoretical predictions.
1. Introduction Conventional tunnel diode relaxation oscillator circuits consist of a tunnel diode biased by an inductor in series with a voltage source in such a way that the diode is unable to remain permanently in either its low or high voltage positive resistance states but switches between the two at a frequency determined by the rate of change of current in the inductor 1). The oscillation frequency of this circuit can be altered by varying the bias, and so changing the dwell periods in each of the two states, or else by varying the value of the inductance. The former method is unsatisfactory if a wide change in mark to space ratio and stability cannot be tolerated. In the latter case the frequency change which can be achieved will be proportional to the range over which the inductance can be altered. Unless the inductor is tapped the frequency range will not normally be as great as 5 to 1. In certain applications, where the oscillation frequency must be synchronized or equalized with another widely varying frequency, this range may be insufficient. In addition, since the inductance itself must be varied, means must be provided for effecting this which are usually either mechanically inconvenient or else involve the addition of considerable circuit strays due to additional lead lengths to panel controls. These strays can be troublesome, especially at high frequencies. The circuit described below was designed to overcome both of these problems by providing a frequency variation, which can be greater than 100 to I, in response to variation of a steady control current. The dependence of frequency on a current, rather than on the value of a variable inductance, also means that this circuit can easily be incorporated in self adjusting loops. In addition, the oscillation frequency is linearly proportional to control current over a wide range and in this region can be calculated simply from known circuit parameters which
do not involve any of the tunnel diode nonlinearities.
120
2. Description and analysis In the conventional tunnel diode relaxation oscillator circuit the dwell times in the low and high voltage states, TI and 7"2, may be found by solving eqs. (1) and
(2) I o - I , = L -1
Ip-1,
= L -'
f?
[Vb(t ) - Vf(t)]dt,
(1)
Vb(t)]dt,
(2)
[•t)-
Here L is the series inductance, Ip and I, are the tunnel diode peak and valley currents, Vb(t) is the bias voltage and Vf(t) is the tunnel diode voltage in the state under consideration, both as a function of time. From these expressions it will be observed that the frequency, 1/(T 1 + T2), will be inversely proportional to inductance but will be difficult to calculate exactly because of the nonlinear tunnel diode characteristics represented by In the circuit to be described the bias level, Vb(t ), which remains effectively constant in the conventional circuit, is varied in such a way that the difference between this and the tunnel diode voltage, Vf(t), is constant and proportional to control current I. In this case the periods can be calculated directly and the frequency will be proportional to IlL. Fig. 1 shows a circuit in which this condition is satisfied. The transistor, QI, provides a convenient means of biasing the tunnel diode from a low source impedance whilst at the same time enabling the bias potential to be varied by a signal applied to the relatively high impedance base circuit. Transistor bias of tunnel diodes also has several advantages over back diode 2) and high conductance diode 3) nonlinear
VARIABLE
FREQUENCY
RELAXATION
biasing in circuits such as univibrators in which the voltage excursions are large. The "knee" potential is easily adjustable, the supply current drain is lower than for back diode biasing and the stray capacitance is lower than that of most high conductance diodes. Also, it is frequently more economical. :V+
~source, /]] controlcurrent ,
rt mlt*~
J
r:'
D
~ c
~'~O'
2t OV @C
outptX waveforrn
L
~
oo "Vsource 2
2
Fig. 1. Circuit diagram.
The circuit of fig. 1 operates as follows. Assume initially that diode D 2 is short-circuited and that all the circuit time constants are much longer than the period of oscillation. The voltage on the base of QI is adjusted by varying r I and r2 so that the constant bias potential appearing at the emitter of Q~ has the value which causes the tunnel diode to dwell for equal periods in its low and high voltage states (corresponding to phases 1 and 2, respectively, in fig. 1). Under these circumstances, the dwell times /'1 = 7"2, can be found from either expression 1 or 2 and the frequency will be the maximum attainable, given approximately by f , , , , = A V t / { 4 L ( I p - I,)},
(3)
where A Vf is the effective difference between the forward voltages in states 1 and 2. If D2 now functions normally and a small direct current,/, flows from source 1 the mean diode current will dearly be L However, when the tunnel diode switches to state 2 the resultant negative pulse will be transmitted through R and C to the anode of D 2. Because of the exponential dependence of diode current on forward voltage, the current in D 2 will be reduced to virtually zero and all of the current I will then flow through R, C, and the tunnel diode. This
OSCILLATOR
121
will remain true until I becomes so large that the voltage developed by it across R is sufficient to support appreciable the current flow through D2, During the next half cycle, when the tunnel diode has reverted to state 1, the charge previously stored in C by I must be withdrawn, since in equilibrium the mean capacitor voltage will remain constant. The resultant discharge current again flows through R, D2, and the tunnel diode. Together with the c u r r e n t , / , still flowing from source 1, this will ensure that the mean current through D 2 is I, as required. It will be assumed that the dwell times in each state continue to be equal with D 2 in circuit, and also that the discharge current remains constant throughout phase 1, in which case it will be equal to - L These assumptions are experimentally verified. Under these circumstances, the voltage at the anode of D 2, and so at the base of QI, will follow the tunnel diode voltage changes but will be offset by a steady voltage, + I R , when the tunnel diode is in state 2 and, - I R in state 1. Expressions (1) and (2) can be combined and written in differential form, with the addition of a further term to represent the bias source resistance, r, Ldi/dt + ir = Vr(t) + Vb(t).
(4)
Here i is the inductor current, the positive and negative signs refer to phases 1 and 2 respectively, and r = re+ R/~ where re is the incremental emitter resistance of Qt and fl is its common emitter current gain. The bias voltage, Vb(t), is equal to the changing voltage appearing at the anode of D2, VD(t), but with a mean level determined by r I , r2 and the emitter diode characteristics of Qt. This level is equal to the voltage necessary to cause the mean inductor current, ½(lp + Iv), to flow in the resistance r. Therefore eq. (4) can be rewritten: L d i / d t + i r = ½(lp+Iv)r+ Vf(t)+ VD(t).
(5)
VD(t) will now be considered in greater detail. If it is assumed that the base circuit time constant, 1/[R + r t r2/ ( q + r 2 ) ] is infinite then, as described above, VD(t) will differ from Vf(t) by a constant term, IR, the sign of which will be negative in state I and positive in state 2. However, in this case, when I = 0 the voltage appearing at the emitter of Q, will be almost equal to the voltage across the tunnel diode, with the result that insufficient current will flow through L to enable the diode to switch to its other state. This latching will not
122
R.J. SMITH-SAVILLE
occur in practice and in order to calculate the minimum frequency the base circuit time constant must be considered. If it is assumed that the time constants associated with C and c are so large that their effects may be neglected, then a current will build up in / because of the voltage, Vb- Vu(t), appearing across it. (Here Vb is the bias source voltage). This current will develop a voltage across R which will eventually become large enough to enable sufficient current to flow in Qi to allow the tunnel diode to switch to its other state. Since r will be small, this voltage will be very much less than Vb- Vo(t) and so if VD(t) is assumed constant the current in l can be considered to change linearly with time and will be equal to [V b - VD(t)]t/l. In fact, however, Vo(t) will be virtually equal to Vf(t), which is by no means constant, especially in state 2. It is consequently not easy to calculate the time dependence accurately and in the subsequent analysis this expression will be represented by ~tt. a can be estimated as above but for accurate work its effective value must be measured for the particular circuit. It is only in this term, which is of small importance t h r o u g h o u t most o f the frequency range, that the tunnel diode nonlinearities enter the analysis. Immediately after a transition between states the current flowing in l and R will be that appropriate to the end o f the previous phase. A further term, ½RAT', must therefore be added to Vb(t) to represent this initial condition, where T' is the duration o f the previous half cycle or phase. VD(t) now consists o f the four terms below VD(t) = V,(t)-t-tR+_ctR(t-½r').
If this is substituted into eq. (8) and T' put equal to T l , since the dwell periods in each state are assumed equal, the resultant eq. (9) can be solved for T~. ½(I~,-1,) = / { I +(R/r)} +(na/r){½T, - ( L / r ) }
+ [I{1 - (n/,')} + (na/r){½T, + (L/r)} - } ( l p - I,)] exp (-rT,/L).
(9)
The expression for T2 is identical, as required by the assumption of equality between T, and T2. As it stands, eq. (9) is not in a convenient form for the determina. tion o f TI. However, two sets of approximations m a y be made. Firstly, if r(Ip-Iv)/(Ra) >> L/r then the exponential term will be effectively zero for small I and when I = 0 we have
To = {r/(Rct)}(lp-l,)+2L/r,
(10)
so that the minimum frequencyfmln = ½To ~. Secondly, when expression (9) is re-arranged and logarithms taken, the result will simplify to
T
=
L(I~- Iv - 2I)/(IR),
provided that (7)
so that the inductor current, i, as a function o f time is given by
i = Ae-"/L + ½(Ip+ 1,) + [(Rlr){I- (Lair)} +(otR/r)(t-½T')].
A = (~Ra T ' / r ) + (LaR/r 2) + I{1 - ( R / r ) } - ½(lp - I,).
(6)
Substitution of this into eq. (5) gives
Ldi/dt+ir = ½(l,+tv)r+_lR-I-otR(t-½T'),
i has increased to I p - I the tunnel diode current will be Ip and the diode will switch to state 2. Since in this second state the current I subtracts from the diode current the reverse transition to state 1 will occur when the inductor current has decayed to lv+I. If we consider phase I in which the tunnel diode current is increasing towards Ip the positive sign version of eq. (8) is applicable, the initial value o f i at t = 0 is i = Iv+l and the final value at t = T, is i = lp-l. Then
(8)
The constant A must be chosen to satisfy the initial conditions. It will be assumed that the tunnel diode is not significantly loaded by any other circuits so that only the effect of I need be considered. During phase 1 this current adds to the tunnel diode current so that when
[{½(Ip-lv)-l}-(g~tT/r)]/(R/r){(otL/r)-l}
<< 1. (11)
In this case the frequency is: ½./.-1 = Ig/{2L(lp-lv-21)}.
(12)
As can be seen from the condition (I 1), which must be satisfied if the simplified expression for the frequency (12), is to be accurate, the ratio R/r must be as large as possible. This will result in a small control current I < < (Ip-Iv). The term in I in the denominator o f (12) can then be neglected and the frequency will be linearly dependent on the control current. Under the above circumstances the ratio of maximum to minimum fie-
VARIABLE FREQUENCY RELAXATION OSCILLATOR 1.2
quencies can be found approximately from eqs. (3) and (10) L../L,.
~- rA Vy/(2RctL).
1.0
(13)
3. Experimental results and practical considerations
Figs. 2 and 3 show comparison of measured results, for R = 117 f2 and 1 kt2, respectively, with thetheoretical values computed from the full expression (9), and from the approximation (12). At very low control currents the frequency is mainly determined by the base circuit time constant and is independent of 1, giving the relatively horizontal sections of the curves in the low current region. As the control current is increased, the behaviour approaches that predicted by the simple expression, (12), until at sufficiently high currents the curve again becomes markedly non linear. This frequency saturation at high currents occurs because the diode D 2 becomes progressively more forward biased throughout the whole cycle. This invalidates the assumption made in section 2 and means that as the control current increases the base voltage of Qt changes less and less, eventually
/2/
0.8
07
0.6
0.5
/
*
R
117~t 2.21
=
r •
,
t
J
~
ILi ) - I v
= 811.8,~H .8~A
1.19 A sec"
0
0.2
0L
I
0.6 0.8 !'O control current m A
I
i
1.2
Fig. 2. Comparison of measured values (Q), with values calculated from expression 9 (+) and expression 12, (o), for R .= 117.(2.
123
0.8 tJ3
Zofi
:-~oL q-
/
r = 3.23a Ip - Iv: 8.82 m A L = 11.8 p 14 c~ = 30.6 A s e c -~
/*
g'.2 o
0
20
60
100 40 80 control, current~A
20
Fig. 3. Comparison of measured values (O), with values calculated from expression 9 (+) and expression 12 (o), for R
= Ik[2. approaching the situation in the simple case in which the base is completely decoupled. The approximate current at which saturation will occur is given by 21R = lip- Vv, where Vp and Vv are the tunnel diode peak and valley voltages. It will be observed from expression (10) that if L/r << (Ip-l~)r/(~R) the minimum frequency will be proportional to ctR/r. Consequently, if the ratio R/r is maximized to improve linearity the minimum frequency will also be increased. There will, therefore, be a reduction in the frequency range because the maximum frequency remains fixed at the saturation value determined by eqs. (I) and (2). Conformity with expression (12) and a very wide frequency range are therefore somewhat incompatible and a compromise which suits the application must be made. This is apparent from expression (13). It will be observed, however, that the frequency ranges can still be very large, being 230 to 1 infig. 2and 10to 1 infig. 3. The other major determinant of the minimum frequency is the base circuit time constant, represented by ct in eq. (10). Unfortunately, this cannot be increased indefinitely to reduce the minimum frequency because transistor Q1 will eventually become unstable when the inductive impedance of the driving source becomes too large compared with the emitter circuit impedance 4). This instability will manifest as pulse length jitter and oscillation frequency instability.
124
R. J. S M I T H - S A V I L L E
The ratio R/r will approach a limiting value equal to the current gain o f Q t , as R is increased to very large values. The practical upper limit on R is the condition that the bias source impedance r(~- R/fl) must be as small as possible and certainly less than the limiting value above which relaxation oscillations no longer occur ~). Since r = rc+R/fl a suitable compromise value for R is that for which Rift = r c. Because the maximum attainable value of R/r is largely determined by the current gain of the transistor Q~ high gain, high cutoff frequency devices are necessary. The incremental emitter resistance, re, which also contributes to the source resistance, r, is inversely proportional to emitter current and so can be reduced by increasing the steady emitter current from source 2. The minimum value attainable in this way will be limited either by the transistor power dissipation or else by the equivalent emitter series resistance of the transistor. The device used for the measurements displayed in figs. 2 and 3 was a 2N 2926 transistor with a low frequency current gain of approximately 400 and a cutoff frequency of 120 Mc/s, operated at a standing collector current of about 15 mA in fig. 2 and 35 mA in fig. 3. The detailed characteristics of the diode D 2 are unimportant but for high frequency operation its junction capacitance and forward recovery time must be
small, as must also be the stray capacities associated with the inductors L and L
4. Conclusions Operation of this circuit can be extended to considerably higher frequencies than those shown in figs. 2 and 3 but at very high frequencies accurate calculation of the frequency is more complex because of circuit strays and the frequency dependence of transistor parameters, particularly current gain and emitter resistance. At the frequencies used, figs. 2 and 3 demonstrate good agreement between the predicted and measured values and also demonstrate the reasonably linear dependence of oscillation frequency upon control current. Particular advantages of the circuit are the ease with which a relatively accurate prediction of operating frequency may be made and the wide frequency range obtainable, which can be greater than 200 to 1.
References ~) Tunnel diodes, RCA Manual. 2) R. H. Bergman, M. Cooperman and H. Ur, RCA Rev. 23, no. 2 (June 1962) a) A. 1_.. Whetstone, Rev. Sci. Inst. 34 (1963) 412. ') M. V. Joyce and K. K. Clark, Transistor circtdl analysis (Addison-Wesley, 1961) p. 264.