Donald T. Hess and Kenneth K. Clarke
Abstract – This paper presents experimental evaluation of the performance of more than twenty-five production units of a 100 A, mind dc to 100 kHz transconductance amplifier. In addition to the stability, health uncertainty, distortion and frequency response, experimental results seem to indicate that the output impedance may be closely modeled by a parallel combination of a resistor and a capacitor in series with a resistor. This model is then used to estimate, and to correct for, the effect of load inductance on high-frequency, high-current measurements.
A 100 Ampere, dc to 100 kHz transconductance amplifier has been produced and experimental data have been taken on the performance of more than 25 units. The amplifier comprises five decade current ranges from 2 mA to 20 A plus a 100 A range with each range having a compliance voltage of at least 7 V. The five low ranges operate to twice their full scale value. To circumvent the requirement for a very good high current shunt as the output current sampling element on the highest ranges, the design follows the one outlined by Owen Laug  by having these ranges generate their output current by combining the output currents of lower current (typically 5 A) current modules. Each of these modules incorporate feedback from a temperature stable (typically less than 1 mW/W/°C), high frequency (1 GHz) shunt which results in gain stability, low distortion and an excellent high frequency response. Nineteen modules are placed in parallel on the 100 A range, ten are placed in parallel on the 20 A range and a single module is employed for the 2 A range. The lower current ranges employ current amplifiers which are similar to the current modules with the exception that their output stages deliver smaller currents.
The transconductance amplifier measures and displays both the compliance voltage and the frequency of the input signal. It also contains an IEEE-488 interface which permits automated reading of these displays as well as automated range setting.
II. AMPLIFIER CHARACTERISTICS
Some of the important properties of such an amplifier are the stability and uncertainty of the transconductances, the distortion, the linearity with output current levels, the output impedance, and the variation of these properties with frequency, compliance voltage and temperature. These characteristics are not unrelated. For example, to maintain the gain within limits as the compliance voltage varies implies that a minimum value for the output impedance of the range in question exists. Unless the distortion is low, the definitions of the other quantities becomes blurred. If variations occur with time, or temperature, then measurements become very difficult.
Based on the measurements of more than 25 amplifiers, the short term stability (10 minutes) of the transconductance, for all current ranges, is better than ±40 mS/S for dc through 1000 Hz and better than ±75 mS/S up to 100 kHz. The stability has been designed to be, and is, independent of compliance voltage.
The transconductance uncertainty is better than ±400 mS/S for dc and better than ±600 mS/S for frequencies up to 20 kHz which degrades gradually to ±1500 mS/S by 50 kHz and to 5000 mS/S by 100kHz for all current ranges. These uncertainties remain the same for compliance voltages up to 7 V and for temperatures within the range of 18°C to 28°C. The preservation of the uncertainties with increasing compliance voltage at low frequencies implies that the parallel resistance component of the output impedance is quite high. Values for each range are shown in Table 1. To determine the dc transconductance, an average is taken of the value obtained with a positive voltage input and the value obtained with a negative voltage input. This eliminates the effect of the small dc offset currents in the amplifier as well as thermocouple effects in the output shunts.
The total harmonic distortion, THD, measured for all current ranges is less than -70 dB at low frequencies, less than -60 dB at 15 kHz and less than -40 dB at 100 kHz.
For all ranges the linearity is within ±100 ppm of the full scale range value.
III. AMPLIFIER MODEL AND ERROR CORRECTIONS
Based on direct measurements, a very accurate model for the transconductance amplifier, which is valid for each current range with different parameters, has been obtained. In this model, which is shown in Fig.1, V1 is the input voltage to the amplifier, G is the transconductance, R0 is the parallel output resistance, C0 is the parallel output capacitance and RS is its series loss resistance. For each current range, the value of R0, the value of G, the value of C0 and the value of RS are shown in Table 1. The values shown are an average of a number of measurements. The spread of the values among different units is within ±4%. As the frequency is increased the effect of R0 becomes negligible compared with the effect of RS and C0. For any given frequency, the series combination of C0 and RS may be modeled as an equivalent parallel resistor-capacitor combination where the resistor is a decreasing function of frequency.
Figure 1. Transconductance Amplifier Model
Table 1. Transconductance Amplifier Parameters
On the lower current ranges (2 mA, 20 mA and 200 mA), this capacitance and resistance have the effect of shunting some of the current away from the load connected to the amplifier with the result that the load current is less than the value which would be calculated by multiplying the input voltage by the transconductance. For load resistors which keep the compliance voltage less than 1 V, the effect is less than 200 ppm at 100kHz. For higher load resistors the current reduction may be calculated by employing the model from Fig. 1 along with the parameter values from Table 1. Agreement within 200 mA/A at 100kHz between the measured and the calculated output current, with load resistors causing compliance voltages greater than 6 V, are consistently obtained.
On the higher current ranges, the larger capacitance resonates with the inherent inductance (typically between 40 nH and 200 nH) in series with the load at a low enough frequency to affect the output current. In particular, the load current becomes the circulating current of a tuned circuit and is thus larger than the generated output current of the amplifier. Figure 2 illustrates this situation. For the circuit of Fig. 2, the ratio of the current flowing through the load, IL, to the input voltage, V1, given by
where w is the radian operating frequency, RL is the load resistance and L is the inductance of the load and its interconnections. It is apparent that at low frequencies the ratio reduces to the transconductance while at higher frequencies the ratio becomes the effective transconductance which increases with frequency. For small deviations from the low frequency transconductance, the increase is almost directly proportional to w2, to L and to C0.
Figure 2. Output Circuit Including Load Inductance.
The calculation of the effective transconductance requires the value of the inductance. An expression for L in terms of the measured compliance voltage, VC (Fig. 2), is given by
where RL is the low frequency ratio of the measured compliance voltage to the load current which is equal to GV1. (The transconductance amplifier has a meter which indicates compliance voltage directly.) To obtain the greatest accuracy for L, (2) should be evaluated for the highest frequency of operation, typically 100kHz. Since the value of IL, which can be determined from (1), is also a function of L, a more complicated quadratic expression for L, which depends only on circuit constants, can be obtained by combining the two equations. An easier approach is to assume that IL is equal to its low frequency value of GV1 and to evaluate L from (2). Using this value, IL can be evaluated from (1) and used with (2) to refine the calculation of L. This procedure usually converges in one or two cycles. Once L is obtained, (1) yields the effective transconductance.
The validity of the model shown in Fig. 2 and the results derived from it have been verified by calibrating the transconductance amplifier with one artifact using the corrections generated from (1) and then measuring it with a completely different device. The amplifier was calibrated at the 20 A level on the 100 A range using a broadband resistor and a precision voltmeter to determine the output current and the internal voltmeter to determine the compliance voltage. The effective inductance was calculated and internal current gain vs. frequency corrections were made in software such that the measured current was equal to the current determined from (1) The amplifier was then set up to drive a two stage, zero flux, broadband transformer (with a trans-resistance of 10 mV/A) at the 100 A level. The transformer output was monitored with a precision voltmeter and the compliance voltage was again monitored by the internal meter. From the compliance voltage, the output capacitance of 580 nF and the series resistance of 0.16W, the inductance was calculated to be 87.5 nH from which the transformer output was calculated using (1).
Table 2 presents the compliance voltage, the measured transformer output, the calculated transformer output and the difference between the two vs. frequency.
Tracking of the measured and the calculated output at the 100A level to better than 700 uV/V from 30 kHz to 100 kHz has been realized.
Table 2. Comparison of Measured and Calculated Results at 100 A
The validity of the model was also checked by placing different inductances in series with the same broadband resistor, applying a constant amplitude input voltage over the frequency range of 100 Hz to 100 kHz, measuring the compliance voltage and the output current through the broadband resistor and finally calculating this output current using (1) and (2). A fixture was constructed to allow the series connection of two devices. This procedure was conducted at the 20 A level on the 100 A range with inductances of 47 nH, 133 nH and 330 nH. The 47 nH inductance was obtained with the broadband coaxial resistor connected directly to the amplifier, the 133 nH inductance was obtained with the coaxial resistor connected to the fixture with a short circuit in series with the resistor and the 330 nH inductance was obtained with the same fixture but with a 62 cm narrow loop of “Monster” audio cable in series with the resistor. The effective transconductances, for each inductance, increased significantly from its low frequency value. The values measured in each case were within ±400 mS/S at all frequencies of the values calculated from (1). Table 3 shows the measured effective transconductance at 50 kHz and 100 kHz for the three values of inductance and the errors between these values and the calculated values. With this ability to calculate the effective transconductance in terms of the measured variation in the compliance and the parameters of the output circuit of the amplifier, the user can set any desired output current quite accurately by adjusting the input voltage at higher frequencies.
In the cases where two series-connected current measuring devices are being compared with each other, the accuracy requirement disappears. In this case, the effective transconductance merely lets the user have a good idea of the actual current level through the devices as a rough check on the device being used as the reference.
In testing the transconductance amplifiers, extensive use is made of computer controlled instrumentation. This allows easy correction for variations of input voltage with frequency and allows the mechanization of (1) and (2) so that results are obtained without the need of further calculation.
Table 3. Comparison of Measured and Calculated Results with Variations in Load Inductance
The experimental evaluation of the characteristics of a number of wide-band, high current transconductance amplifiers has been presented. In addition, a method for accurately calculating the output current at high frequencies when a significant amount of series inductance is present in series with the load has been outlined. Such an amplifier, with its low measured uncertainties, should provide a valuable tool for creating and evaluating current sensing artifacts which extend to 100 A and 100 kHz. (National laboratory certification to the 20 A, 100 kHz level only became available in 1997 .)
 O.B. Laug, “A 100 A, 100 kHz Transconductance Amplifier”, IEEE Trans. Instrum. Meas., vol 45, pp 440-444, June 1996
 J.R. Kinard, etal., “Extension of the NIST AC-DC Difference Calibration Service for Current to 100 kHz, Jour. of Res. of NIST, vol 102, pp 75-83, Jan-Feb 1997