- Email: [email protected]
Analog-to-digital converter nonlinearity due to the loss current
NUCLEAR
INSTRUMENTS
AND METHODS
39
(~966) 293-296; :~?, N O R T H - H O L L A N D
PUBLISHING
CO.
A N A L O G - T O - D I G I T A L C O N V E R T E R N O N L I N E A R I T Y DUE T O T H E LOSS C U R R E N T O. SZAVITS Institute " Rudjer Boskovic", Zagreb, Yugoslavia
Received 2 August 1965 A pulse amplitude error is analyzed in this paper, which occurs in pulse stretchers due to the RC-shaped input pulses when the loss current cannot be neglected. An expression is derived showing
the dependence of this error on pulse amplitude, loss current and pulse shape.
1. Introduction
exists and with the linear discharge current I D switched off, the memory capacitor C would be charged to the peak value of the RC-shaped input pulse. The charging diode current would be interrupted exactly in the moment when the input pulse attains its peak value and no reason for an amplitude distortion could occur. Errors due to charge storage and capacity effects of the charging diode are neglected here. The analysis of these errors is given in 2).
Pulse stretching is an often required operation in nuclear instrumentation. A typical application is in the pulse stretching section of the analog-to-digital converter, where a memory capacitor is charged to the peak value of the input pulse preserving so the amplitude for evaluation. Since this operation is an analog one a very good linearity is required. A pulse stretcher causes specific errors giving deviations from linear operation. In this paper an error will be discussed, caused by the loss current and in connection with the pulse shape in the pulse stretcher of an A-D converter, in which tile conversion begins immediately after memory capacitor charging. Other pulse stretcher errors have been discussed elsewherc. The charging error analysis iscarried out in 1) and the analysis of some other errors occurring in different conversion phases of an ADC in 2). The error evaluation of a voltage type pulse stretcher of an A D C will serve as an example though the considerations are of more general validity.
INPUT
O
LINEAR GATE
2. Determination of the error
The pulse stretcher of an analog-to-digital converter is shown in fig. 1 which begins with the memory capacitor discharge immediately after the input pulse has reached its peak value. The pulse is fed through a linear gate (not shown in fig. I) to the stretcher input. It charges the memory capacitor C through the differential amplifier DA, the emitter follower EF1 and the charging diode D. The constant discharge current generator ! o is switched off during the charging process by GL. This is required in order to avoid enormous errors at low amplitudes as will bc shown later. So the only capacitor discharging current remains the loss current 1L. The feedback loop of the charging system is closed through the emitter follower EFz. This ensures the charging error caused by the nonlinear characteristics of the diode D to be held Iow~). In an idealized situation where no loss current 1e 293
C
Fig. l. Pulse stretcher.
CONST. DISCH. CURRENT SWITCHING SIGNAL
Io
In the actual situation the loss current always exists causing a delayed interruption of the diode current. The loss current consists of the leakage current and of the current required for driving subsequent circuits. (in fig. 1 the emitter follower EF2). This delay of the charging diode current interruption depends on the input pulse shape and introduces a pulse amplitude dependent linearity error caused by the charge loss of the memory capacitor. Let us assume the detector pulse shape v(t) = V o [ e x p ( - l / r l ) - - e x p ( - - t / r 2 ) ] ,
(1)
with % the leading edge time constant of the pulse and rl the trailing edge time constant and r~ > r 2. Here Vo = K1Vm.
(2)
/'% is the pulse amplitude and K~ a constant given by K1 = [ e x p ( - T m / z ~ ) - e x p ( - T , , / r z ) ] -1.
(3)
Tm is the time when the voltage pulse reaches its peak value. It is
294
o. SZAVITS (4)
T m = {'t'1l'2/('t" ' - - ' r 2 ) } ]n 2"1/~ 2.
The expressions (2) and (3) follow by inserting (4) into (1) and setting v(t)= Vm which is true for t = T,,. if the total charging diode current is i and thc loss current is I 2, the memory capacitor current becomes i - I c. It follows from 1) that with a loop gain of about ten the leading edge of the pulse on the memory capacitor has already the detector pulse shape. During the short time interval of capacitor charging a constant loss current IL can be assumed. The diode current until the interruption moment is then
i(t)=CK,vm[lexp(-t)-lexp(--t)]+lc 2
2
.
TI
TI,
(5) At the moment t = Tm, when the voltage pulse reaches its peak value Vm, the diode current i(t) is not decreased to zero but reaches the loss current valuc I L as can be verified by insertion of (4) into (5). At this moment only the capacitor current is decreased to zero. The diode current decrcases to zero later at T 1. The time interval T t - Tm, rcpresenting the time elapsed from pulse peak to diode current interruption is a function of the loss current I L. In an idealized situation where no loss current exists, the diode current interruption point T~ would be the peak point Tm and the only current delivered by the diode would be the capacitor charging current. The time interval ) " 1 - Tm can be calculated by Taylor series expansion of the expression (5) at point Tm and by setting i(t) = O. So we have
i(t) = i(Tm) + T1 ~.I.T~ i,(Tm) +
( T 1 -- Tin) 2
i"(Tm) + . . . .
2!
(6) With an approximation to the second order this gives the time interval 7"1 - T m = KI K3
l K3
2K3 K2 _ K I-
V-~
the actual situation the loss current 1 L is of the order o f p A and the pulse time constants zl and r2 are such that the second term under the square root in expression (7) is much smaller than the first one. Then (7) reduces to T1 - T m ,~ [ L / ( K I K 2 C V m ) .
(10)
During the time interval T~ - T m the charging diode remains conductive causing the discharge of the memory capacitor. This affects a voltage loss 1 1 "r~ + AV = - C./ T,,,( i - I t ) dr.
(1 I)
With an approximation to the second order of the Taylor series expansion for the capacitor current at the point Tm we have
AV= - ~K,Vm(Tt-Tm)2[K2-JK,(T,-Tm)]
(12)
which becomes by insertion of (10) --
AV=
]
2
__
2KI~ ( - ~ , ' ) V l m [ I
-
K3
This is the total peak voltage loss. The fractional peak voltage error follows from (13) as d(AV) 1 , l c l / 1 I K3 ( I L l l _1] ,=5 -~tVoV = - K i K 2 \ c ] Vm2 L-3-K~;-K- I \C-IVe, - 2 J '
(14) The expressions become more intuitive if one introduces the pulse shape factor B defined as
B = rl/%.
(15)
By some arithmetical manipulation the constants K1, K2 and K 3 defined in (3), (8) and (9) take then by insertion of (15) the following forms
K , = B - e / ( ' - B ) / ( B - I), K2
=
(7)
(16)
(17)
1
l" 2"
l
K 3 = ~13"
1)].
(18)
Here the constant _l.exp
(
T., j _ ..l. exp
-
(8)
follows from the first derivative of the expression (5) and the constant K3 =
1 exp
.z3 .
.
.
.
Tm
1 exp 1:3l
--
•
The fractional crror in percent reaches by inscrtion of (16), (17) and (18)into (14) d(AV) ~ 100BrlZ(~_£L)2 ! X
a(%) = lOO dVm....
V2m
1
[-
(9)
from the second derivative of the same expression. In
x [B--;t~(B2+I)(B+I)(~--~c) Vm
_
;]"
(19)
The first term in the brackets is usually smaller than
ADC NONLINEARITY
DUE TO THE
0.5 so that the whole expression (19) is negative. This arises from the fact that B > I as ~1 > z2 and z~ < CVm/IL= T. T is the time required to discharge completely the capacitor with IL and is very much longer than "t"t. Thus the expression (19) reduces to
6(%)
=
50B~Z~(I,./C)E(1/V~).
-
(20)
This indicates that real stored amplitudes are always smaller than original amplitudes to be stored. The error increases with decreasing pulse amplitude. As in an analog-to-digital converter the pulse amplitude is proportional to the channel number and the number of channels is a fixed quantity, the lower channels act as being wider due to the amplitude error A V. Hence the expressions (19) and (20) give at the same time the fi'actional channel width error when taken with positive sign. v,i i i
/J"
,, I
I
,V' 'i', it
•
",
;
i i -. ",.
,L ?' I
~0
. . . 1 " m ~ "h-i - ' "~-"4"3"
t
I
--~" ~. h-Tm Fig. 2. Pulse shapes for high and low pulse amplitudes.
In fig. 2 voltage and current pulses are shown for high and low pulse amplitudes, v" is the voltage pulse on the memory capacitor for a high pulse amplitude and i" the corresponding diode current pulse, v' denotes the memory capacitor voltage pulse for a low pulse amplitude and i' the corresponding diode current. IL is the loss current, which is proposed to be constant during the short time interval of capacitor charging. Tm denotes the pulse peak instant on the memory capacitor, which is synchronous with the input pulse peak. TI denoting moments when the diode current decreases to zero. It is seen from the figure that the diode current reaches zero value sooner for higher pulse amplitudes than for lower ones. That is T"I < T'~ as prescribed by expression (10). At moments T~ the charging diode current is interrupted and the voltage pulse value at this moment, which is by A V lower than
LOSS CURRFNT
295
the peak value, will be stretched on the memory capacitor. It can be seen from fig. 2 that the case can arise when the diode current cannot be interrupted at all. This happens either at very low pulse amplitudes of flat pulses or when the loss current I L too is high. I f the loss current were reduced to zero, the current curve in the diagram were translated downward. The current interruption point T t would fall onto the point Tm and so no error in pulse amplitude could occur. This shows the importance of loss current reduction. 3. Discussion Some examples illustrate the consequences of expression (19). Let us choose a pulse stretcher with a relatively high loss current 1L of 3 / t A and a memory capacitor of C = 0.002/~F. If a semiconductor detector with pulse time constants zl = 1.5 and z 2 = 1 ps resp. serves as pulse source, the pulse shape factor B becomes 1.5. If the lowest available amplitude corresponding to the first channel is 10 mV, the expression (20) gives the channel width error of the first channel 6.6°/'0. If on the other hand a scintillation detector with zt = 6 and r2 = 0.6 ps i.e. with the pulse shape factor B = 10 is the pulse source, the lowest permissible amplitude for the same error of about 6% will be 1 V! Only by decreasing the loss current the threshold can be lowered in this case. It is apparent from these considerations that the constant discharge current must be switched off during the charging process as was mentioned at the beginning if the stretcher (fig. 1) is used. The constant discharge current is some orders of magnitude higher than the loss current and it would cause enormous errors in the converter. It is also clear that one can achieve a smaller conversion error with input pulses that have shorter decay times that is with pulse shape factors of lower value as is the case with the semiconductor detector. Additional pulse shaping with line clipping or RC-shaping gives steeper pulses too, though each of these methods has disadvantages. 4. Conclusion A voltage error is caused in the pulse stretcher due to the loss current in connection with the pulse shape. For a given pulse shape and loss current this error depends on pulse amplitude and increases with decreasing amplitude. This error can be reduced by decreasing the loss current. Input pulses with shorter decay time constants give smaller errors as is the case
296
o. SZAVITS
with s e m i c o n d u c t o r detectors c o m p a r e d with scintillation detectors. The a u t h o r is grateful to Dr. M. K o n r a d for his suggestions and many helpful discussions.
References l) K. Kandiah, Nucl. Electronics 2, Proc. Conf. Nucl. Elcctr. Belgrade (1961) 11. 2) p. F. Manfredi and A. Rimini, Energia Nucleare 10, no. 6 (1963).
INSTRUMENTS
AND METHODS
39
(~966) 293-296; :~?, N O R T H - H O L L A N D
PUBLISHING
CO.
A N A L O G - T O - D I G I T A L C O N V E R T E R N O N L I N E A R I T Y DUE T O T H E LOSS C U R R E N T O. SZAVITS Institute " Rudjer Boskovic", Zagreb, Yugoslavia
Received 2 August 1965 A pulse amplitude error is analyzed in this paper, which occurs in pulse stretchers due to the RC-shaped input pulses when the loss current cannot be neglected. An expression is derived showing
the dependence of this error on pulse amplitude, loss current and pulse shape.
1. Introduction
exists and with the linear discharge current I D switched off, the memory capacitor C would be charged to the peak value of the RC-shaped input pulse. The charging diode current would be interrupted exactly in the moment when the input pulse attains its peak value and no reason for an amplitude distortion could occur. Errors due to charge storage and capacity effects of the charging diode are neglected here. The analysis of these errors is given in 2).
Pulse stretching is an often required operation in nuclear instrumentation. A typical application is in the pulse stretching section of the analog-to-digital converter, where a memory capacitor is charged to the peak value of the input pulse preserving so the amplitude for evaluation. Since this operation is an analog one a very good linearity is required. A pulse stretcher causes specific errors giving deviations from linear operation. In this paper an error will be discussed, caused by the loss current and in connection with the pulse shape in the pulse stretcher of an A-D converter, in which tile conversion begins immediately after memory capacitor charging. Other pulse stretcher errors have been discussed elsewherc. The charging error analysis iscarried out in 1) and the analysis of some other errors occurring in different conversion phases of an ADC in 2). The error evaluation of a voltage type pulse stretcher of an A D C will serve as an example though the considerations are of more general validity.
INPUT
O
LINEAR GATE
2. Determination of the error
The pulse stretcher of an analog-to-digital converter is shown in fig. 1 which begins with the memory capacitor discharge immediately after the input pulse has reached its peak value. The pulse is fed through a linear gate (not shown in fig. I) to the stretcher input. It charges the memory capacitor C through the differential amplifier DA, the emitter follower EF1 and the charging diode D. The constant discharge current generator ! o is switched off during the charging process by GL. This is required in order to avoid enormous errors at low amplitudes as will bc shown later. So the only capacitor discharging current remains the loss current 1L. The feedback loop of the charging system is closed through the emitter follower EFz. This ensures the charging error caused by the nonlinear characteristics of the diode D to be held Iow~). In an idealized situation where no loss current 1e 293
C
Fig. l. Pulse stretcher.
CONST. DISCH. CURRENT SWITCHING SIGNAL
Io
In the actual situation the loss current always exists causing a delayed interruption of the diode current. The loss current consists of the leakage current and of the current required for driving subsequent circuits. (in fig. 1 the emitter follower EF2). This delay of the charging diode current interruption depends on the input pulse shape and introduces a pulse amplitude dependent linearity error caused by the charge loss of the memory capacitor. Let us assume the detector pulse shape v(t) = V o [ e x p ( - l / r l ) - - e x p ( - - t / r 2 ) ] ,
(1)
with % the leading edge time constant of the pulse and rl the trailing edge time constant and r~ > r 2. Here Vo = K1Vm.
(2)
/'% is the pulse amplitude and K~ a constant given by K1 = [ e x p ( - T m / z ~ ) - e x p ( - T , , / r z ) ] -1.
(3)
Tm is the time when the voltage pulse reaches its peak value. It is
294
o. SZAVITS (4)
T m = {'t'1l'2/('t" ' - - ' r 2 ) } ]n 2"1/~ 2.
The expressions (2) and (3) follow by inserting (4) into (1) and setting v(t)= Vm which is true for t = T,,. if the total charging diode current is i and thc loss current is I 2, the memory capacitor current becomes i - I c. It follows from 1) that with a loop gain of about ten the leading edge of the pulse on the memory capacitor has already the detector pulse shape. During the short time interval of capacitor charging a constant loss current IL can be assumed. The diode current until the interruption moment is then
i(t)=CK,vm[lexp(-t)-lexp(--t)]+lc 2
2
.
TI
TI,
(5) At the moment t = Tm, when the voltage pulse reaches its peak value Vm, the diode current i(t) is not decreased to zero but reaches the loss current valuc I L as can be verified by insertion of (4) into (5). At this moment only the capacitor current is decreased to zero. The diode current decrcases to zero later at T 1. The time interval T t - Tm, rcpresenting the time elapsed from pulse peak to diode current interruption is a function of the loss current I L. In an idealized situation where no loss current exists, the diode current interruption point T~ would be the peak point Tm and the only current delivered by the diode would be the capacitor charging current. The time interval ) " 1 - Tm can be calculated by Taylor series expansion of the expression (5) at point Tm and by setting i(t) = O. So we have
i(t) = i(Tm) + T1 ~.I.T~ i,(Tm) +
( T 1 -- Tin) 2
i"(Tm) + . . . .
2!
(6) With an approximation to the second order this gives the time interval 7"1 - T m = KI K3
l K3
2K3 K2 _ K I-
V-~
the actual situation the loss current 1 L is of the order o f p A and the pulse time constants zl and r2 are such that the second term under the square root in expression (7) is much smaller than the first one. Then (7) reduces to T1 - T m ,~ [ L / ( K I K 2 C V m ) .
(10)
During the time interval T~ - T m the charging diode remains conductive causing the discharge of the memory capacitor. This affects a voltage loss 1 1 "r~ + AV = - C./ T,,,( i - I t ) dr.
(1 I)
With an approximation to the second order of the Taylor series expansion for the capacitor current at the point Tm we have
AV= - ~K,Vm(Tt-Tm)2[K2-JK,(T,-Tm)]
(12)
which becomes by insertion of (10) --
AV=
]
2
__
2KI~ ( - ~ , ' ) V l m [ I
-
K3
This is the total peak voltage loss. The fractional peak voltage error follows from (13) as d(AV) 1 , l c l / 1 I K3 ( I L l l _1] ,=5 -~tVoV = - K i K 2 \ c ] Vm2 L-3-K~;-K- I \C-IVe, - 2 J '
(14) The expressions become more intuitive if one introduces the pulse shape factor B defined as
B = rl/%.
(15)
By some arithmetical manipulation the constants K1, K2 and K 3 defined in (3), (8) and (9) take then by insertion of (15) the following forms
K , = B - e / ( ' - B ) / ( B - I), K2
=
(7)
(16)
(17)
1
l" 2"
l
K 3 = ~13"
1)].
(18)
Here the constant _l.exp
(
T., j _ ..l. exp
-
(8)
follows from the first derivative of the expression (5) and the constant K3 =
1 exp
.z3 .
.
.
.
Tm
1 exp 1:3l
--
•
The fractional crror in percent reaches by inscrtion of (16), (17) and (18)into (14) d(AV) ~ 100BrlZ(~_£L)2 ! X
a(%) = lOO dVm....
V2m
1
[-
(9)
from the second derivative of the same expression. In
x [B--;t~(B2+I)(B+I)(~--~c) Vm
_
;]"
(19)
The first term in the brackets is usually smaller than
ADC NONLINEARITY
DUE TO THE
0.5 so that the whole expression (19) is negative. This arises from the fact that B > I as ~1 > z2 and z~ < CVm/IL= T. T is the time required to discharge completely the capacitor with IL and is very much longer than "t"t. Thus the expression (19) reduces to
6(%)
=
50B~Z~(I,./C)E(1/V~).
-
(20)
This indicates that real stored amplitudes are always smaller than original amplitudes to be stored. The error increases with decreasing pulse amplitude. As in an analog-to-digital converter the pulse amplitude is proportional to the channel number and the number of channels is a fixed quantity, the lower channels act as being wider due to the amplitude error A V. Hence the expressions (19) and (20) give at the same time the fi'actional channel width error when taken with positive sign. v,i i i
/J"
,, I
I
,V' 'i', it
•
",
;
i i -. ",.
,L ?' I
~0
. . . 1 " m ~ "h-i - ' "~-"4"3"
t
I
--~" ~. h-Tm Fig. 2. Pulse shapes for high and low pulse amplitudes.
In fig. 2 voltage and current pulses are shown for high and low pulse amplitudes, v" is the voltage pulse on the memory capacitor for a high pulse amplitude and i" the corresponding diode current pulse, v' denotes the memory capacitor voltage pulse for a low pulse amplitude and i' the corresponding diode current. IL is the loss current, which is proposed to be constant during the short time interval of capacitor charging. Tm denotes the pulse peak instant on the memory capacitor, which is synchronous with the input pulse peak. TI denoting moments when the diode current decreases to zero. It is seen from the figure that the diode current reaches zero value sooner for higher pulse amplitudes than for lower ones. That is T"I < T'~ as prescribed by expression (10). At moments T~ the charging diode current is interrupted and the voltage pulse value at this moment, which is by A V lower than
LOSS CURRFNT
295
the peak value, will be stretched on the memory capacitor. It can be seen from fig. 2 that the case can arise when the diode current cannot be interrupted at all. This happens either at very low pulse amplitudes of flat pulses or when the loss current I L too is high. I f the loss current were reduced to zero, the current curve in the diagram were translated downward. The current interruption point T t would fall onto the point Tm and so no error in pulse amplitude could occur. This shows the importance of loss current reduction. 3. Discussion Some examples illustrate the consequences of expression (19). Let us choose a pulse stretcher with a relatively high loss current 1L of 3 / t A and a memory capacitor of C = 0.002/~F. If a semiconductor detector with pulse time constants zl = 1.5 and z 2 = 1 ps resp. serves as pulse source, the pulse shape factor B becomes 1.5. If the lowest available amplitude corresponding to the first channel is 10 mV, the expression (20) gives the channel width error of the first channel 6.6°/'0. If on the other hand a scintillation detector with zt = 6 and r2 = 0.6 ps i.e. with the pulse shape factor B = 10 is the pulse source, the lowest permissible amplitude for the same error of about 6% will be 1 V! Only by decreasing the loss current the threshold can be lowered in this case. It is apparent from these considerations that the constant discharge current must be switched off during the charging process as was mentioned at the beginning if the stretcher (fig. 1) is used. The constant discharge current is some orders of magnitude higher than the loss current and it would cause enormous errors in the converter. It is also clear that one can achieve a smaller conversion error with input pulses that have shorter decay times that is with pulse shape factors of lower value as is the case with the semiconductor detector. Additional pulse shaping with line clipping or RC-shaping gives steeper pulses too, though each of these methods has disadvantages. 4. Conclusion A voltage error is caused in the pulse stretcher due to the loss current in connection with the pulse shape. For a given pulse shape and loss current this error depends on pulse amplitude and increases with decreasing amplitude. This error can be reduced by decreasing the loss current. Input pulses with shorter decay time constants give smaller errors as is the case
296
o. SZAVITS
with s e m i c o n d u c t o r detectors c o m p a r e d with scintillation detectors. The a u t h o r is grateful to Dr. M. K o n r a d for his suggestions and many helpful discussions.
References l) K. Kandiah, Nucl. Electronics 2, Proc. Conf. Nucl. Elcctr. Belgrade (1961) 11. 2) p. F. Manfredi and A. Rimini, Energia Nucleare 10, no. 6 (1963).