1. Introduction
Microcalorimeters have been recognized to be an effective solution for radio frequency (RF), microwave, and millimeter-wave power measurements [
1,
2,
3], and have been successfully developed within the National Metrology Institutes worldwide over past few decades [
4,
5,
6,
7,
8]. The main function and application of a microcalorimeter is to determine the effective efficiency
of a transfer standard (e.g., a thermistor mount, and referred as a device under test (DUT) in this paper), and its correction factor g is found to be critical and has been studied in many different ways [
9,
10,
11,
12].
In the millimeter-wave range, waveguide microcalorimeter has been adopted due to its good reliability and accuracy up to 110 GHz [
8,
10,
11,
12] or further. To accurately determine its correction factor g, a method based on the measurement of offset shorts of different length followed by one single calibration measurement of a DUT has been proposed in [
10]. Recently, another method based on attaching a thermistor sensor into the waveguide thermal isolation section (TIS) to accurately measure its temperature change has been proposed in [
12]. Both the methods are found to have good performance during the evaluations of a WR-22 (33–50 GHz) waveguide microcalorimeter.
However, as the frequency of interest further increases, the size of waveguide becomes smaller which motivates us to find other solutions without using extra fixtures/accessories. As a continued work of [
13], theoretical analysis and modeling of the correction factor g of a waveguide microcalorimeter will be performed in this paper, in terms of the physical dimensions of its TIS, based on the electromagnetic field theory analysis. The proposed solution tends to eliminate the usage of external fixtures, and reduce the measurement uncertainty when calibrating a RF/microwave/millimeter-wave power sensor with waveguide connection. For simplicity in the rest of this paper, RF will be synonymous for RF, microwave, and millimeter-wave.
In the remainder of this paper, the theoretical background and operation principle of a waveguide microcalorimeter is discussed in
Section 2. This is followed by the proposal of a new method for determining its correction factor g in
Section 3. In
Section 4, detailed description of electromagnetic field theory analysis for waveguide TIS in foil short measurement mode will be carried out. Performance comparison of the new method will be given in
Section 5. Finally, conclusion of this paper will be drawn in
Section 6.
2. Theoretical Background and Operation Principle
The effective efficiency
of a thermistor mount (a type of power sensor which is widely used for precision RF power measurements) is important for accurate determination of total power
dissipated within the thermistor mount, and is defined as
where
is the direct current (DC) substituted power of the thermistor mount and calculated in term of its bias voltages without and with RF signal applied (i.e.,
V1 and
V2) at a steady status as follows,
Here,
R is the operating resistance of the thermistor mount.
Figure 1 below shows a detailed measurement setup for determining the effective efficiency
of a DUT thermistor mount. Its core part consists of a thermally insulated microcalorimeter (twin-line structure) including a thermopile and a thermal reference (dummy), and a Type-IV power meter.
V1 and
V2 are measured by the Type-IV meter directly.
According to the law of conservation of energy, the unsubstituted portion (labeled as
, and
) of total dissipated power
can cause relative temperature rise of the DUT mount referring to the Dummy mount, which is monitored by the thermopile as shown in
Figure 1, and supposed to be indicated by thermopile output voltage change
, where
e1 and
e2 are the output voltages of the thermopile corresponding to
V1 and
V2 and measured by a nanovoltmeter. It is noted that this unsubstituted power
is the portion of RF power dissipated but does not affect the reading
V2 of the nanovoltmeter.
2.1. Definition of the Correction Factor
However, the power
dissipated at its waveguide TIS can also contribute to
, and therefore has to be differentiated. Without differentiation, the output voltage change
of thermopile includes the contribution from
and
, with the following relationship,
where
is a proportionality constant that depends on the fraction of power that is detectable by the thermopile and the thermopile sensitivity [
11], and
is an equivalence factor that considers the thermal paths which are different comparing to those from the mount to the thermopile.
The uncorrected effective efficiency
[
7] comparing to the effective efficiency
in (1), including the contribution from
to the thermopile output voltage change
, and is defined as
The correction factor g of a microcalorimeter is then defined as,
The correction factor g is used to remove the contribution of
from the directly calculated uncorrected effective efficiency
[
7,
12] with the measured
V1,
V2,
e1, and
e2 using the hardware setup in
Figure 1.
2.2. System Constant
For a thermistor mount with input reflection coefficient of
and incident power of
, the dissipated power
at the TIS due to both the forward and reverse transmissions is,
where
is the power dissipation coefficient of TIS. The net absorbed power by the thermistor mount is
Therefore, it can be obtained from Equations (5)–(7) that
As and are determined by the physical structure and the material property of a waveguide TIS, their product is actually a system constant of the microcalorimeter and denoted as in this study. From Equation (8), note that once the system constant is obtained, the correction factor g of microcalorimeter for calibrating a thermistor mount with known input reflection coefficient can be determined, and thereby the effective efficiency of the thermistor mount.
3. Determination of the Correction Factor
“Foil Short” measurement has been well-accepted for experimental determination of the correction factor g [
7,
11]. A schematic illustration of “Foil Short” measurement is shown in
Figure 2 as a reference, where a foil short is inserted between the DUT (thermistor mount) to be calibrated and the interface plate. During the “Foil Short” measurements, the DUT is dc-biased through the Type-IV power meter in a steady status.
With the RF input on, the power
dissipated at the foil short and the power
dissipated at the TIS cause the output voltage change
of the thermopile. Similar to Equation (3), the following relationship can be arrived at
For the foil short with a reflection coefficient of
and incident power of
to the TIS, similar to Equation (6), the dissipated power
at the TIS can be determined as
Combining Equations (9) and (10), it can be obtained that
Since
, conventionally combing (8) and (11), the correction factor g of a microcalorimeter can be determined as
This relationship has been reported in [
7,
11,
12]. However, a recent bilateral comparison [
14] of scattering parameter magnitude measurements of WR-15 (50–75 GHz) and WR-10 (75–110 GHz) waveguide type between the Istituto Nazionale di Ricerca Metrologica (INRIM), Italy and the National Metrology Center, A*STAR (NMC), Singapore showed that the uncertainty of reflection coefficient for a “Short” traveling standard can vary from 0.005 to 0.02 (at a 95% confidence level). Higher uncertainty of the reflection coefficient
for foil short can then be propagated to the estimated correction factor g, and thereby the determined effective efficiency
. Therefore, it motivates us to find an alternative solution for determining the correction factor g as discussed below.
Through reorganizing (11), we can achieve that
With (9), it is found that
In this study, a power ratio
between
and
is defined as
, then we can get
Combining (15) with (8), a new correction factor g is proposed in this study for evaluating the waveguide microcalorimeter as follows,
Note that the significant uncertainty portion involving
and
as underbraced in Equation (12) has been eliminated in Equation (16); however, with the introduction of another factor determined by power ratio
and equivalence factor
which may be under control better. This proposed solution in (16) theoretically may offer a smaller combined uncertainty. The equivalence factor
that considers the thermal paths which are different comparing to those from the thermistor mount to the thermopile approximates to be 0.5 (as representative of all the microcalorimeters in [
7]). This is because the relative heating effectiveness through the TIS changes linearly from a value of approximately one at the mount flange to almost zero at the far end as discussed in [
7]. As a result, only half of the heating in the TIS is measured by the thermopile.
Therefore, proper determination of the power ratio between (the power dissipated at the foil short) and (the power dissipated at the TIS) in “Foil Short” measurements becomes very important for evaluating the system constant of a waveguide microcalorimeter, and thereby its correction factor g for calibrating the thermistor mounts. In the following section, we propose to apply the electromagnetic field theory analysis to determine this power ratio theoretically in this study.
4. Mathematical Modeling Through Electromagnetic Field Theory Analysis
The properties of waveguides in support of wave propagation and mode are characterized by the presence of longitudinal magnetic or electric field components, and can be derived by electromagnetic field theory analysis ([
15] Chapter 3).
In a rectangular waveguide, the dominant wave propagating inside is the TE
1,0 mode. In the following analysis, it is assumed that that both the waveguide walls and the foil short have high conductivity
and small skin depth
resulting in small losses (almost lossless, with attenuation constant α ≈ 0), which do not appreciably perturb the TE
1,0 mode fields. For the incident wave in +
z direction with a peak amplitude level of
A, and with a foil short at
z = 0 along the rectangular waveguide (
) as shown in
Figure 3, the transverse field components are
where
is the cutoff wave number,
is the phase constant, and
is the wave impedance, as follows,
Here,
is the guide wavelength and equal to,
for the wavelength
in TE
1,0 mode.
is the voltage reflection coefficient at
z = 0 (approximately 1 for the foil short) as shown in
Figure 3. The incident power
(at
z = 0) is
For convenient in calculation, Equation (17) can be reformatted as [
16]
Note that the dissipated power
at a wall surface with surface resistance
is
where the surface current density
is given by
Therefore, for the broad wall (
by
) as shown in
Figure 3, the magnitudes of the
and
component of the current densities are
The magnitudes of the current density in the narrow wall (
by
) is
The magnitudes of the current density in the foil short (
by
) is
According to (20), the power
dissipated in the two broad walls is
Similarly, the power
dissipated in the two narrow walls is
and the power
dissipated in the foil short is
Here, it needs to be highlighted that final expression in (29) is achieved with the elimination of the surface resistance at both the denominator and numerator. This elimination/simplification is valid only under the assumption that the TIS and the foil short share the same (or approximately the same) electrical characteristics such as conductivity and skin depth , and if the metal thickness is higher than the skin depth for both the TIS and the foil short.
In practice, these requirements could be achieved during the fabrication of TIS and foil short using the same metal material with enough thickness and with same surface treatment. Together with Equation (16), the correction factor g can be determined properly using (29). In the next section, its performance will be compared with conventional method with detailed discussion.
6. Conclusions
In this paper, a new method for determining the correction factor g of a waveguide microcalorimeter was reported, using the electromagnetic field theory to analysis the effect of waveguide TIS in “foil short” measurement mode. The proposed method determines the contribution of the power dissipated within the TIS into the correction factor g, in term of the physical dimensions of the TIS.
The proposed method has been implemented to evaluate a newly fabricated WR-15 microcalorimeter at the NIM, China. The estimated correction factor g of the microcalorimeter using the proposed method has been compared against the conventional method, and good agreements have been observed. To further evaluate its performance, the proposed method with the newly fabricated WR-15 microcalorimeter has been evaluated in an informal international comparison of WR-15 (50 to 75 GHz) power measurements with the NIST of USA, the PTB of Germany and the NMC of Singapore, where good equivalence has been observed.