An improved pseudo-exponential, pseudo-logarithmic circuit Un circuit pseudo-exponentiel et ´ e´ pseudo-logarithmique amelior Brent Maundy, David Westwick, and Stephan Gift∗ In this paper a useful pseudo-exponential and pseudo-logarithmic circuit is proposed that offers improved performance compared to a current-conveyorbased design. The circuit employs two operational amplifiers and several resistors and mimics the exponential and logarithmic functions over predefined ranges. Control of the pseudo-functions is achieved by a single resistance, which may be digitally switched or implemented by tunable transconductors. Measured results using digitally switched resistors and commercially available components demonstrate the versatility of the circuit for audio/video applications. Dans cet article, on propose un circuit pseudo-exponentiel et pseudo-logarithmique utile qui offre une performance d’ex´ecution am´elior´ee par rapport a` une conception bas´ee sur un convoyeur de courant. Le circuit utilise deux amplificateurs op´erationnels et quelques r´esistances. Il imite des gammes pr´ed´efinies de fonctions exponentielles et logarithmiques. Le contrˆole des pseudo-fonctions est r´ealis´e par une r´esistance simple qui peut eˆ tre num´eriquement commut´ee ou impl´ement´ee par des transconducteurs r´eglables. Les r´esultats mesur´es en utilisant les r´esistances num´eriquement commut´ees et les composants commerciaux disponibles d´emontrent la polyvalence du circuit pour les applications audio/vid´eo.
I.
Introduction
order terms in (3), we obtain
In the field of communications and signal processing, there is a need for exponential and logarithmic amplifiers. Exponential amplifiers typically allow wide gain control through a parameter, and logarithmic amplifiers are useful in signal compression and conditioning. Both types of amplifiers can be easily implemented by exploiting the natural current-voltage (I-V ) characteristics of bipolar transistors or by using MOS transistors in weak inversion. Recently several circuits have been proposed that provide approximations to the exponential and logarithmic functions [1]–[4]. Invariably these circuits make use of the functions 1+x (1) e2x ≈ 1−x and log10 (ˆ x) ≈
x ˆ−1 1+x ˆ
– (ˆ x − 1)3 (ˆ x − 1)5 x ˆ−1 + + + · · · . 1+x ˆ 3(1 + x ˆ)3 5(1 + x ˆ)5
«
„ = 0.8686
x ˆ−1 1+x ˆ
«
(4)
and note that e2x = ex /e−x , we arrive at the bilinear approximation 1+x e2x ∼ . = 1−x
(6)
x = −ˆ x
(7)
and substitute (7) into (6), we get
(3)
Noting that log 10(ˆ x) = ln(ˆ x)/ ln(10) and neglecting the higher∗ Brent Maundy and David Westwick are with the Department of Electrical and Computer Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4. E-mail:
[email protected]. Stephan Gift is with the Faculty of Engineering, University of the West Indies, St. Augustine, Trinidad.
Can. J. Elect. Comput. Eng., Vol. 32, No. 3, Summer 2007
x ˆ−1 1+x ˆ
If we choose
To understand the relationship between x and x ˆ and hence (1) and (2), we consider the Taylor series expansion of ln(ˆ x), x ˆ > 0, which is given by »
„
if the scaling factor of 0.8686 is rounded up to one. Likewise, if we consider the series expansion of e±x (x2 < ∞) to a first order, as given by e±x ∼ (5) = 1 ± x,
(2)
or their variants to approximate the exponential and logarithmic functions. Here x and x ˆ are control parameters that may be voltage, current, or resistance ratios.
ln(ˆ x) = 2
2 ln(10) x ˆ−1 ∼ = 1+x ˆ
log10 (ˆ x) ∼ =
2x
e
−2ˆ x
=e
∼ =−
„
x ˆ−1 1+x ˆ
« .
(8)
Equation (8) reveals the odd symmetry of the two approximations, that is, in the form f2 (x) = −f1 (−x). Note, however, that even though the functions have odd symmetry, the approximations to log10 (ˆ x) and e2x themselves have different errors since each is a truncated approximation of the respective terms [5]. With regard to the respective errors in each approximation, it is well known that the pseudo-exponential approximation in (1) yields an output error of 0.5 dB for x in [−0.42, 0.42] or 1 dB in the range [−0.52, 0.52]. Conversely, the pseudo-logarithmic function of (2) has an output linear error of 3.3% for x ˆ in [0.2, 5]. This result corresponds to about 28 dB of dynamic range in the input. If the error is referred back to the input as is sometimes done, (2) yields a maximum log error θ of 1.05 dB or 0.6529 dB over the ranges 0.2 ≤ x ˆ ≤ 5 and 0.22 ≤ x ˆ ≤ 4.65, respectively [2], [6]. By definition the log error can
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CAN. J. ELECT. COMPUT. ENG., VOL. 32, NO. 3, SUMMER 2007
be thought of as the error in the input x ˆ that yields the ideal answer to log10 (ˆ x) and can be expressed mathematically as
15
θ = 20 log10
1 + log10 (ˆ x) 1 − log10 (ˆ x)
«
–
1 . x ˆ
(9)
Gain in dB
10
»„
Because of the symmetry of the function in (2), the magnitude of the log error for x ˆ < 1 is identical to the magnitude of the log error for x ˆ > 1. Fig. 1 shows a comparison between the approximations in (1) and (2) and their ideal curves.
5 0 −5 −10 −15 −0. 5
−0. 4
−0. 3
−0. 2
−0. 1
0
0.1
0.2
0.3
0.4
0.5
3
4
x 1 θ
0.5 Gai n
In [1] a novel pseudo-exponential circuit that employs a positive second-generation current conveyor (CCII+) and an operational amplifier (opamp) was presented. In that paper the circuit was used only for the realization of the pseudo-exponential function, but it can be shown that the circuit is capable of realizing the pseudo-logarithmic function with minor modifications. A new circuit capable of realizing (1) and (2) is therefore presented in this paper. The proposed circuit eliminates the CCII+ of [1] and hence any errors associated with input buffer resistance rx and current and voltage transfer errors. As such, it is a significant improvement over the work in [1]. This work and the work of [1] are the only two such circuits with dual functionality known at this time. Tuning in the circuit can be achieved digitally by switched resistors, active resistors, or tunable transconductors. In this paper, Maxim digitally switched resistors are used to demonstrate and prove the validity of the concept. Tunable transconductors or active resistors as such will not be dealt with here. The remainder of this paper is devoted to a description of the proposed circuit, sources of error, bandwidth effects, and measured and simulation results.
0
−0. 5
−1 0.2
0.3
0.5 0.6 0.7 0.8
0.4
1
2
Figure 1: Upper panel: Comparison of e2x and its pseudo-exponential approximation in (1). Lower panel: Comparison of log10 (ˆ x) and its pseudo-logarithmic approximation in (2). The log error θ is also illustrated. (Dashed lines are the ideal functions, and solid lines are the approximations.)
R7 = R
+
Vy Vin
II.
Circuit description
5
xˆ
+ R6 = R
Vo
A2
R3
-
A1
-
The proposed pseudo-exponential, pseudo-logarithmic circuit is shown in Fig. 2. It consists of two amplifiers A1 and A2 , each having localized feedback around them. Resistor R1 is a grounded variable resistor that may be digitally controlled or voltage- or current-controlled. Along with fixed resistors R2 and R3 , it serves to generate and control the parameters that determine the function of this circuit, either exponential or logarithmic. Positive feedback around the entire circuit is introduced by equal-valued resistors R6 and R7 . Straightforward analysis at the inverting terminal of A2 and subsequent solution for the output of this circuit yields „
«
1 1 − R1 R3 „ « Vin , Vo = 1 1 1 − R2 − R1 R3 1 + R2
(10)
1+x Vin , 1−x
„
1 1 − R1 R3
R5 = Rˆ
Figure 2: The proposed pseudo-exponential, pseudo-logarithmic circuit.
restriction that Vin < 0, (11) can be rewritten as « „ x ˆ−1 Vo = − Vin 1+x ˆ ∼ x)Vin = − log10 (ˆ ∼ = −Kl Vin ,
(14)
(11)
(12)
to the power Vin .
where x = R2
R2
which mimics the pseudo-logarithmic function with Kl = log10 (ˆ x). Here the role of Vin is to raise x ˆ, defined as „ « 1 1 x ˆ = R2 − , (15) R3 R1
which simplifies to Vo =
R1 R4 = Rˆ
« .
If Vin > 0, then the output approximates the pseudo-exponential function ∼ Ke Vin , Vo = (13) where Ke = e2x . Equation (11) is of the same form first proposed in [1] except that the circuit of Fig. 2 eliminates the need for a current conveyor. Also, resistor R3 is floating and not grounded as in [1]. Furthermore, if R1 > R3 , then x < 0, whereas R1 < R3 corresponds to x > 0, so that the generation of the pseudo-exponential function between the limits of ±0.42 for a 0.5 dB linear error requires that Vin > 0 and resistors R1 , R2 , and R3 be appropriately chosen. To generate a pseudo-logarithmic function requires use of (7). With the
A. Control of x and x ˆ When deciding on the generation of the variables x and x ˆ in their respective ranges, [−0.42, 0.42] and [0.2, 5], we first need to understand the function proposed in (12). Fig. 3 illustrates the relationships among the variables R1 , R2 , and R3 . Here ratios of R1 /R3 are plotted against different values of the ratio R2 /R3 to yield the variables x and x ˆ. From Fig. 3 it can be observed that the larger the R2 /R3 ratio chosen, the smaller the R1 /R3 ratio required. In fact, to satisfy a lower limit on x of −0.42, a minimum R2 /R3 ratio of 0.42 is required for a 0.5 dB linear error. Choosing a ratio of, say, R2 /R3 = 1 requires that the maximum spread of element values (MSEV) in R1 be 2.448; that is, R1max /R1min = 2.448 to satisfy x in [−0.42, 0.42]. Likewise, for
MAUNDY / WESTWICK / GIFT: AN IMPROVED PSEUDO-EXPONENTIAL, PSEUDO-LOGARITHMIC CIRCUIT where 1
R2 = 4 R3
x
0.5
R2 = 1.5 R3
R2 R2 = 1 = 0.5 R3 R3
R2 = 0.1 R3
−0. 5
−1 0
1
2
3
4
R1 R3
5
6
7
8
9
10
SδG1 =
10
R2 = 10 R3
8
R1 R3
. (20) R1 −1 R3 To gain an appreciation of the effect of the mismatch errors on the transfer function, let us define the gain, G, of the circuit as G = Vo /Vin . It can then be easily shown that the sensitivities of G with respect to mismatch errors δ1 and δ2 are given by µ=
0
147
R2 = 7 R3
xδ1 (µδ2 − 1) (δ1 − 1) [1 − x(1 − µδ2 ) − δ1 ]
(21)
and
xˆ
6 4
R2 = 0.1 R3
2 0 0
1
2
3
4
R1 R3
5
6
7
8
9
10
Figure 3: Upper panel: Plot of x with resistance ratio R1 /R3 for varying R2 /R3 values. Lower panel: Plot of x ˆ with resistance ratio R1 /R3 for varying R2 /R3 values.
x ˆ, which is always greater than zero, a theoretical minimum ratio of R2 /R3 = 5 is needed to ensure that x ˆ is in [0.2, 5] for a 1.05 dB log error. In addition, R1 is always greater than R3 for x ˆ in [0.2, 5]. Note that for the pseudo-logarithmic function, choosing a large R2 /R3 ratio is sometimes advantageous, as the MSEV in R1 will then be small. Finally, if a more direct relationship with R1 is desired as opposed to, say, x, we may choose R2 = R3 , reducing (13) to (2R /R ) Vo ∼ = 0.1353e 2 1 Vin ,
(16)
which increases exponentially as R1 decreases. Likewise, if R2 = 10R3 , then (14) reduces to » „ «– R3 , (17) Vo ∼ = −Vin · log10 10 1 − R1 which increases logarithmically as R1 increases for R1 > R3 . III.
In the previous section we ignored offset voltages and the fact that, in practice, resistances R4 and R5 , and R6 and R7 , will be mismatched. In reality these nonidealities must be considered, as they lead to tradeoffs in the performance of the circuit. Examining first the opamp’s offset voltage, which we shall define in the usual fashion, and assuming that Vos1 ∼ = Vos2 = Vos , we find through routine circuit analysis that (11) becomes 1+x (Vin + 2Vos ) . (18) Vo = 1−x The effect of Vos is therefore to decrease or increase the value of Vin by a factor of 2Vos . Since in general |Vin | |Vos |, the error in the scaling factor can be easily compensated for. This capability is in direct contrast to the shifting effect (to the right) of rx on the pseudo-exponential curve of [1]. This circuit therefore has an advantage in that regard. In Section II we assumed that resistors R6 and R7 , and R4 and R5 , were identical, so that (11) was satisfied. In reality, R6 is never exactly equal to R7 , and the same can be said about R4 and R5 . It is therefore instructive to examine the effect of mismatch. To that end, let R6 = (1−δ1 )R7 , where δ1 is a small error that in the limit approaches zero. Likewise, let R4 = (1 − δ2 )R5 . Then it can be easily shown that (11) is modified to (1 + x)(1 − δ1 ) + xµδ2 (δ1 − 1) Vin , 1 − x(1 − µδ2 ) − δ1
If the gain sensitivities of the circuit with respect to x and x ˆ are examined, it can be shown that SxG = −SxˆG =
2x . 1 − x2
(23)
For x in [−0.42, 0.42], |SxG | ≤ 1.02, which is low. However, for x ˆ in [0.2, 5], the gain sensitivities can become large near x ˆ = 1 or when R1 = R2 R3 /(R2 − R3 ). This effect is unavoidable, as it arises because of the approximation.
Sources of error and noise considerations
Vo =
(δ1 − 2)µxδ2 . (22) [1 − x(1 − µδ2 ) − δ1 ] [1 + x(1 − µδ2 )] From (21) and (22) two observations can be made. First, the denominators of (21) and (22) go to zero at x1 = (1 − δ1 )/(1 − µδ2 ) and x2 = −1/(1 − µδ2 ). Since µ is a function of R1 /R3 and |µ| > 1, therefore for practical tolerances in resistors, |x1 | is always greater than the upper bound on x for the pseudo-exponential function. As an example, at x = 0.42, |xmin | = 0.967 and |xmin 1 1 | = 0.727 for 1% and 10% resistance tolerances, respectively. Furthermore, for x in [−0.42, 0.42], and using 1% resistors, we have |SδG1 | ≤ 0.0079 and |SδG2 | ≤ 0.025. With 10% resistors, these numbers change to |SδG1 | ≤ 0.155 and |SδG2 | ≤ 0.835 respectively, both of which are still small and less than one. Second, for the pseudo-logarithmic function which requires that x ˆ > 0 (x < 0), only SδG2 increases around the point x ˆ2 = −x2 . Since x ˆ2 falls in the range [0.2, 5] for the pseudologarithmic function, care must be exercised in the choice of tolerance for R5 and R6 . From simulation, x ˆ2 ≈ 1 and x ˆ2 ≈ 2 for 1% and 10% resistance tolerances, respectively. Since the log error θ is zero when x ˆ = 1, use of resistance tolerances of 1% or less is preferable to the use of larger tolerances. SδG2 =
R2 = 5 R3
R2 = 1 R3
(19)
Finally, in order to gauge the effects of noise on this circuit it is instructive to examine thermal noise behaviour to a first order. To this end we model each resistor with an equivalent series mean-square noise voltage e2l = 4kT Rl (l = 1, . . . , 7) in series with a noiseless resistor, with k and T having their usual meanings. For each opamp, we model the resistor with an input referred mean-square noise voltage of e2noi (i = 1, 2) at the non-inverting input of each opamp. Then it can be shown that the output mean-square noise voltage is given by „ «2 1+x 4R22 ˆ + 16kT R2 · 8kT R e2o = · 8kT R + 2 1−x R3 (x − 1)2 (x − 1)2 » „ «– 2 1 1 16R2 × 1 + R2 + + 2 · e2no1 R1 R3 R3 (x − 1)2 » „ «–2 1 1 + 4 1 + R2 R1 R3 + · e2no2 . (24) (x − 1)2 Noting that x is never equal to one, we observe that the dominant terms in (24) are the last two for all x and x ˆ by several orders of magnitude. In the case of the pseudo-logarithmic function, as x ˆ increases, all the right-hand-side (RHS) terms decrease in value except for the first term, which is parabolic and insignificant compared to the remaining terms. Conversely, for the pseudo-exponential function, all the RHS terms increase as x increases.
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CAN. J. ELECT. COMPUT. ENG., VOL. 32, NO. 3, SUMMER 2007 IV.
Bandwidth effects 2
H(s) =
κ
2
GBP21
[2(α − βx)s + αGBP1 (1 + x)] , a3 s3 + a2 s2 + a1 s + a0
8
x = 0.42 x = −0.42
γ= 0
1.5
1 γ= ∞
0.5 imaginary part
Of interest is the bandwidth and transient response of this amplifier, as we wish to quantitatively define the behaviour of this circuit to transient changes in both Vin and x (or x ˆ). To accomplish this, we model each opamp in the circuit as single-pole with open loop gain Ai (s) ∼ = (GBPi /ωti )/(1 + s/ωti ), where GBPi is the gainbandwidth product of the opamp and ωti is the opamp’s low-frequency pole. Let us also assume that at the non-inverting terminal of the amplifiers there exists a grounded lumped capacitor Cp . It may be a parasitic capacitor or, as will be shown later, a capacitor introduced deliberately for stability purposes. This capacitor serves to introduce an extra pole at that node. Furthermore, if we assume that the GBP values of the opamps are related to each other by GBP2 = κ · GBP1 and that GBPi ωti (i = 1, 2), then the transfer function Vo /Vin of the circuit can be derived as
x 10
γ= ∞
0
−0.5
−1
−1.5
−2 −5
(25)
−4.5
−4
−3.5
−3
−2.5 real part
−2
−1.5
−1
−0.5
0 x 10
8
with α = R1 − R3 , β = R1 + R3 ,
(26)
γ = κ(RCp )GBP1
(27)
Figure 4: Root locus plot of the circuit of Fig. 2 for varying γ, κ = 1, R2 /R3 = 1, with x in [−0.42, 0.42] in the pseudo-exponential mode.
and denominator coefficients a3 = 2γ(α − βx), a2 = [4κ(α − βx) + γ [α(1 + 2κ) − βx]] GBP1 , a1 = 2κGBP21 [α − βx + κ(α + βx)] + αγκGBP21 ,
2
x 10
8
xˆ = 0.2 xˆ = 5
1.5
γ= 0
(28) 1
ao = ακ2 GBP31 (1 − x). imaginary part
0.5
From (25) it can be seen that the transfer function is third-order with a zero which may be in the left-hand plane (LHP) or right-hand plane (RHP), depending on the value of x. For the exponential function the zero is always in the LHP, whereas for the logarithmic function the zero is in the LHP for x ˆ < 1 and in the RHP for x ˆ > 1. Satisfying the first Routh-Hurwitz criterion for stability requires that coefficients of the denominator of (25) be positive. Initial observation of (28) reveals that only ao is satisfied if x < 1. This condition is easily realized since the limits of the pseudo-exponential, pseudo-logarithmic functions fall within x < 1. The remaining coefficients become tedious to quantify since α and x can be positive or negative, and hence a root locus plot of the poles of (25) as a function of γ, such as the one shown in Figs. 4 and 5, reveals a better picture of the relationships between the variables. Fig. 4 shows the MATLAB root locus plot of the amplifier poles as a function of γ to resistor ratio R2 /R3 = 1, κ = 1, with x in [−0.42, 0.42] for a 0.5 dB linear error. Fig. 5 illustrates the root locus plot for γ to resistor ratio R2 /R3 = 10, κ = 1, with x ˆ in [0.2, 5]. The gain-bandwidth product of amplifier A1 is assumed to be 100 MHz in both cases. Several observations can be made: First, for κ = 1 in both modes, as γ is increased by increasing R or by introducing added capacitance to Cp , more damping in the circuit occurs. Also, in general more damping occurs for the pseudo-exponential function than for the pseudo-logarithmic function for the same κ and R · Cp product. This occurs because the poles for the pseudo-logarithmic function are closer to the jω-axis for identical γ values. Note that for the pseudo-exponential function, less damping occurs for x = 0.42 than for x = −0.42. Conversely, for the pseudo-logarithmic function, more damping occurs for x ˆ = 0.2 than for x ˆ = 5. Second, for both functions (but not shown in Figs. 4 or 5) damping can be increased by choosing κ < 1 without increasing R or Cp . This is akin to saying that by making the bandwidth of A1 larger than A2 , the amplifier poles will be more widely separated, and hence damping is automatically increased. Finally, for the pseudo-exponential mode the bandwidth decreases as x increases from a negative to a positive value, whereas in the pseudo-logarithmic mode the converse is true; that is, the bandwidth increases as x increases. Finally, the analysis presented earlier looked at the transfer function when Vin was the input and x was held constant. However, in a typical amplifier application, Vin may be held
γ= ∞
γ= ∞
0 γ= ∞
−0.5
−1
−1.5
−2 −4
−3.5
−3
−2.5
−2 real part
−1.5
−1
−0.5
0 x 10
8
Figure 5: Root locus plot of the circuit of Fig. 2 for varying γ, κ = 1, R2 /R3 = 10, with x in [0.2, 5] in the pseudo-logarithmic mode.
constant while x is modulated. The response to step changes in x (or x ˆ) can be predicted from the unit step response due to Vin . For any fixed value of x (or x ˆ), Fig. 2 is a linear time-invariant system. Hence, the response to a step change in x (or x ˆ) can be viewed as the response of the system corresponding to the value of x (or x ˆ) after the step change, to a non-zero set of initial conditions. Those initial conditions are determined by the state of the circuit immediately before the step change in x (or x ˆ).
V.
Measured and simulation results
The circuit of Fig. 2 was implemented using 1% resistors and National Semiconductor LM6172IN high-speed opamps with a 100 MHz bandwidth. The opamps were operated from a ±15 V power supply. Resistor R1 was implemented using the Maxim Max5468 10 kΩ, 32-tap linear digital potentiometer in series with an 820 Ω resistor. The purpose of the 820 Ω resistor was to limit the current in the Max5468 potentiometer to within specification. For all the tests, resistors R4 and
MAUNDY / WESTWICK / GIFT: AN IMPROVED PSEUDO-EXPONENTIAL, PSEUDO-LOGARITHMIC CIRCUIT
149
2.5 x = - 0.42 x = 0.42 x ˆ = 0.2 x ˆ=5
2
o
V (Volts)
1.5
1
0.5
0
-0.5
-1 0
100
200
300
400
500
Time(ns)
Figure 6: Measured dc transfer characteristics of the circuit of Fig. 2 in the pseudoexponential mode, alongside the ideal e2x and (1 + x)/(1 − x) responses.
improve from current levels. By using discrete resistors in place of the Max5468, we were able to verify the full range of x and x ˆ, but these results are not shown.
0.8 log10 (ˆ x) (ˆ x − 1) /(ˆ x + 1) Measured Gain
0.6
0.4
Gain
0.2
0
-0.2
-0.4
-0.6
-0.8
0.1
1
Figure 8: Simulated transient response of the circuit of Fig. 2 in the exponential mode with x = 0.42 and x = −0.42 (small squares). Simulated transient response of the circuit of Fig. 2 in the logarithmic mode with x ˆ = 0.2 and x ˆ = 5 (small triangles).
10
x ˆ Figure 7: Measured dc transfer characteristics of the circuit of Fig. 2 in the pseudologarithmic mode alongside log10 (ˆ x) and (ˆ x − 1)/(ˆ x + 1) responses.
R5 around amplifier A1 were set at 1 kΩ. For the pseudo-exponential function, resistors were set as follows: R6 = R7 = R = 4.7 kΩ, R2 = 1 kΩ, R3 = 1.8 kΩ, and R1 varied in 32 steps over its full range. For the pseudo-logarithmic function, resistors were set as follows: R6 = R7 = 4.7 kΩ, R2 = 5.6 kΩ, R3 = 1kΩ, and R1 varied, also in 32 steps. In the first of the tests, Vin (exp) ≡ −Vin (log) = 300 mV, and the dc results are shown in Figs. 6 and 7 along with the ideal response (dashed line), approximation (solid line), and measured results (small circles). Clearly the measured results closely approximate the respective functions in (1) and (2). The same outcome was observed for PSPICE simulation results, which are not shown. Note that because of the coarse resolution of only 32 steps and the presence of the 820 Ω resistor, the full range could not be attained for either x or x ˆ. It is expected, however, that as more digitally switched resistors become commercially available, the resolution and number of steps will
A full transient response of the circuit to the extremity of its resistance value could not be observed because the resistance of the Max5468 changes only in sequential steps. However, the expected results can be observed in PSPICE simulations using the supplied models for the LM6172IN. For the simulated transient tests, the output responses to a step change in Vin of ±1 V are shown in Fig. 8 for the full range of values of x and x ˆ, respectively. Note that the final values in each case are consistent with the expected results. In the case of the pseudo-logarithmic response of Fig. 8, some overshoot can be observed for x = 5. If R is decreased to 1 kΩ, the overshoot in the output response will increase for both x = 0.2 and x = 5, confirming what was noted in Section IV; namely, that as γ is decreased, the amount of damping is reduced. Note that a 12 pF capacitor was added to Cp to stabilize the circuit for the values of R6 and R7 chosen. In another test, the resistance R1 was rapidly stepped down though its full 32 values in both the exponential and logarithmic modes. The exponential mode is shown in Fig. 9, and the logarithmic mode in Fig. 10. In the exponential mode of Fig. 9 the output steps in Vo (t) are initially small, but they become progressively larger in an exponential manner as R1 decreases or as x increases. Conversely, in the logarithmic mode of Fig. 10 the output steps in Vo (t) are initially large as R1 increases or as x ˆ increases. Finally, at the time of this writing it was not possible to measure noise or total harmonic distortion on the circuit. VI.
Conclusions
A novel circuit capable of realizing a pseudo-exponential or pseudologarithmic function has been proposed. The circuit is versatile in that only one grounded resistor must be altered to achieve either of its functions. The theory, sources of error, and measured and simulation results have been presented. The use of the circuit with digitally controlled switched resistors has been demonstrated, and the circuit has been shown to be accurate and stable. It also represents an improvement over the work of [1] in that no current conveyor is needed, so that errors associated with its input resistance, rx , are eliminated. The new circuit should find uses in many audio, video and high-speed applications requiring digital or analogue control.
150
CAN. J. ELECT. COMPUT. ENG., VOL. 32, NO. 3, SUMMER 2007 Brent Maundy received the B.Sc. degree in electrical engineering and the M.Sc. degree in electronics and instrumentation in 1983 and 1986, respectively, from the University of the West Indies, Trinidad. In 1992 he received the Ph.D. in electrical engineering from Dalhousie University, Halifax, Nova Scotia, Canada. He completed a one-year postdoctoral fellowship at Dalhousie University, where he was actively involved in its Analogue Microelectronics Group. Subsequently he taught at the University of the West Indies and was a visiting professor at the University of Louisville, Louisville, Kentucky, U.S.A., for seven months. He later worked in the defense industry for two years on mixed-signal projects. In 1997 Dr. Maundy joined the Department of Electrical and Computer Engineering at the University of Calgary, Calgary, Alberta, Canada, where he is currently an associate professor. Dr. Maundy’s current research is in the design of linear circuit elements, high-speed amplifier design, CMOS circuits for signal processing and communication applications, active analogue filters and RF circuits. Dr. Maundy is a member of the IEEE and is a past associate editor of the IEEE Transactions on Circuits and Systems I.
2R /(R1 (t)) Figure 9: Oscilloscope plot showing Vo (t) ∼ Vin when R1 is = 0.33e 2 rapidly decreased from its largest value through 32 steps to its smallest value with Vin = 300 mV.
Figure 10: Oscilloscope plot showing Vo (t) ∼ = − log10 [5.6(1 − R3 /(R1 (t)))]Vin when R1 is rapidly increased through 32 steps with Vin = −300 mV.
References
[1] B. Maundy and S. Gift, “Novel pseudo-exponential circuits,” IEEE Trans. Circuits Syst. II, vol. 52, Oct. 2005, pp. 675–679. [2] J. Greer, “Error analysis for pseudo-logarithmic amplification,” Measurement Sci. Technol., vol. 3, Oct. 1992, pp. 939–942. [3] R. Harjani, “A low-power CMOS VGA for 50 Mb/s disk drive read channels,” IEEE Trans. Circuits Syst. II, vol. 42, no. 6, 1995, pp. 370–376. [4] A. Motamed, C. Hwang, and M. Ismail, “A low-voltage low-power wide-range CMOS variable gain amplifier,” IEEE Trans. Circuits Syst. II, vol. 45, no. 7, 1998, pp. 800–811. [5] H.B. Dwight, Tables of Integrals and Other Mathematical Data, New York: The Macmillan Company, 3rd ed., 1960. [6] G. Acciari, F. Giannini, and E. Limiti, “Theory and performance of parabolic true logarithmic amplifier,” IEE Proc. Circ. Devices Syst., vol. 144, no. 4, 1997, pp. 223– 228.
David Westwick received the B.A.Sc. in engineering physics from the University of British Columbia in Vancouver, British Columbia, Canada, the M.Sc.E. in electrical engineering from the University of New Brunswick in Fredericton, New Brunswick, Canada, and the Ph.D. in electrical engineering from McGill University in Montreal, Quebec, Canada. After earning his Ph.D., he spent two years as a research associate in the Department of Biomedical Engineering at Boston University in Boston, Massachusetts, U.S.A., and a year at the Technical University of Delft, Delft, the Netherlands, as a research fellow in the Systems and Control Engineering Group. Since 1999, he has been a faculty member in the Department of Electrical and Computer Engineering at the University of Calgary, Calgary, Alberta, Canada, where he is currently an associate professor. He has written over 60 publications, including a textbook, The Identification of Nonlinear Physiological Systems, published in 2003 as part of the IEEE Engineering in Medicine and Biology book series. His research interests include using system identification techniques to construct mathematical models of various physiological systems, and the development of identification techniques that are suitable for these applications. Stephan J.G. Gift received the B.Sc. (first class honours) degree in electrical engineering and the Ph.D. in electrical engineering from the University of the West Indies (UWI), Trinidad and Tobago, West Indies, in 1976 and 1980, respectively. He was head of a Telecommunications Research and Development Centre for 12 years, where he directed the design and development of advanced microelectronic systems. He is currently professor of electrical engineering and deputy dean, undergraduate student affairs, in the Faculty of Engineering, University of the West Indies, where he teaches electronic circuit analysis and design. His research interests include microelectronics, linear integrated circuit application, control systems and field theory, and he has published many technical papers. He holds one international patent for an electronic test system and received a Young Innovator award in 1986 for this system. Professor Gift was president of the Association of Professional Engineers of Trinidad and Tobago and is now a fellow of the Association. He is a senior member of the Institute of Electrical and Electronics Engineers, a member of the Caribbean Academy of Sciences, and a past president of the Rotary Club of St. Augustine West. He was the recipient of the Prime Minister’s Award of Merit for Innovation in Electronics, 2002, the UWI Alumni Association’s Pelican Award for Excellence in Science and Technology, 1993, and a BPTT Fellowship, 2002, for scholarly work.
