Precision Slotted Line
The slotted line has long been recognized as a fundamental tool for measuring the dielectric properties of materials at high frequencies. In principle, the measuring technique is simple: fill a section of coaxial line with dielectric material, determine the propagation constant of the filled section of line from the phase and magnitude of the reflection introduced, as determined by measurement of standingwave ratio on the slotted line, and calculate the dielectric constant and loss tangent from the propagation constant.
For valid measurements, there are needed (1) an accurate slotted line, (2) sections of air line usable as sample holders, and (3) lowreflection, lowloss, coaxial connectors. These are all now available in the GR900 series, and the accompanying article tells how to use them.
The low and repeatable vswit and the low loss of the GR900 Precision Coaxial Connector make possible the use of GR900 equipment for the accurate determination of dielectric constant and loss tangent. No specialized dielectric measuring apparatus is necessary.
The measuring device is the Type 900LB Precision Slotted Line.1 The combination of a Type 900LZ Reference Air Line2 and a Type 900WNC Short Circuit makes a convenient sam
1 J. Zorzy, "Precision Coaxial Equipment — The 900 Series," General Radio Experimenter, November 1963.
'A. K. Sanderson. " Reference Air Lines for the (¡R1300 Series." General Radio Experimenter, January 1965.
3 For example, A. von Hippel. Dielectric \faterials and Application», Technology Press of MIT, 1954.
pie holder for solid dielectrics. The error introduced by the inclusion of the GR900 Connector between the sample and the point of measurement is negligible for most purposes.
The dimensions for a cylindrical sample of solid dielectric are shown in Figure 1. The total length of the sample may be made up of a number of pieces and may be equal to or less than the length of the sample holder. There should be no gaps between the individual pieces. The accuracy of the measurements will depend upon the precision with which the diameters are machined. A light press fit of the sample against the inner and outer conductors is desirable, but too tight a fit may damage the Type 900LZ Reference Air Line. For accurate losstangent measurements of a very lowloss material, the length of the sample should be selected by the procedure described below under Effect of Contact Resistance.
Standard lengths of Type 900LZ Reference Air Lines (5 cm, (i cm, 7.5 cm, 10 cm, 15 cm, 30 cm) will meet most needs. If other lengths are needed, they can be constructed from Type 09009508 rod, Type 09009509 tube, and Type 900AP Connector Kits.
THEORY
The measurements that will be considered here are those of nonmagnetic materials in a shortcircuited sample holder. Other types of measurements are described in various references.3
If a coaxial line containing a dielectric sample is shortcircuited at its far
NOTE.
DIMENSIONS IN INCHES. O. 5625
TIR = TOTAL INOlCATEO RUNOUT.
DIELECTRIC SAMPLE
DIELECTRIC SAMPLE
DIELECTRIC SAMPLE
DIELECTRIC SAMPLE
o. SAMPLE DIMENSIONS
Figure 1. Dimensions and installation of dielectric sample.
b. SAMPLE IN HOLDER
o. SAMPLE DIMENSIONS
b. SAMPLE IN HOLDER
end, the relationship between the propagation constant, y, of the dielectricfilled line and the standingwave ratio, S, and wavelength, in an attached airfilled section of the line is: 1 2ttX0
tan h yd yd j tan
2trd
Figure 1. Dimensions and installation of dielectric sample.
±1%. The simplified equations are:
(3d 2 ird where /3 = phase constant, and
where Xu = the distance from the face of the dielectric sample to the first voltage minimum in the airfilled line, d — the length of the sample, \0 = the wavelength in the airfilled line,
S = the standingwave ratio in the airfilled line.
This equation can be separated into its real and imaginary parts and, if tan 5 (the loss tangent) is less than 0.1, simplified with results accurate within
* T. W. Dakin and C. N. Works, "Microwave Dielectric .Measurements," Journal of Applied Physics, September 1947, p 789,
2vd S &d (I + tan2 0d)  tan &d where a = attenuation constant.
If the frequency and sample length are chosen so that Xa = 0, then tan (3d = 0 and &d = X^x, where Na is the number of half wavelengths in the sample. The equation for otd then becomes:
From a knowledge of a and 0 in the dielectricfilled line, the relative dielectric constant, er, and the loss tangent, tan <5, can be calculated. For the TEM mode in a coaxial line:
If or is small compared with /32, as is the case when tan 5 is less than 0.1, equations (5) and (6) simplify, for samples that are an integral number of halfwavelengths long, to
47t2
/n8 XqV
For small values of — (high standing
wave ratio) it is more accurate to determine —; by a widthofminimum o method rather than by direct measurement. ^ is related to the voltage at o
point X, a distance  from the mini mum by:
where 0
2 sin 6
7rAX Xo
If AX is the distance between points where the power is twice that at the minimum (3.0IdB points), then: 1 _ sin 9
For small values of 9, (<0.075 radian), sin 9—9 and cos 0 = 1 are close approximations. Then:
1 wAX
S X0
If the minimum is very narrow it may be desirable to use points more widely separated, such as the 10dB points. In that case, for small 9 1 TTAX
MEASURING PROCEDURE
(1) Insert the sample into the reference air line flush with one end and with no spaces between the pieces in the sample.
(2) Attach the Type 900WNC Short Circuit so that it is in contact with the sample, as shown in Figure lb.
(3) Connect the sample holder to the measuring setup (Figure 2), con
oetector  type dnt oetector  type dnt type 1267a power supply unix
* oscillator
874r22la patch cord
Figure 2. Setup for dielectric measurements.
type 900l8 slotteo line
type 1267a power supply unix
* oscillator
874r22la patch cord type 874fl lowpas s filter type 874  giol attenuator sisting of the Type 900LB Slotted Line, a Type DNT3 or DNT4 Heterodyne Detector, and an appropriate generator, such as a Gil Unit oscillator. A Type 874G Fixed Attenuator or an isolator should be used between the generator and the slotted line. An amplitudemodulated source and a standingwave indicator can be used if the frequency modulation of the source is kept very small.
(4) Adjust the frequency so that a voltage minimum occurs at the face of the dielectric sample, whether or not the sample completely fills the length of the holder. To do this, compare the positions of the minima first with the sample holder (containing the sample) on the Type 900LB Slotted Line and then with a Type 900WX Short Circuit on the slotted line. Then adjust the frequency until the proper relation exists between the minima. For example, if the sample completely fills the Type 900LZ Reference Air Line, the minimum position with a Type 900WX or WXC Short Circuit connected to the slotted line should be the same as when the sample is connected to the slotted line. If measurements must be made at a certain frequency, then it is necessary either to adjust the length of the sample or to use equations (2) and (3) with X0 not equal to zero.6
(5) Once the frequency is properly adjusted, proceed as follows: Record the position and width of the minimum at two places along the slotted line (one of them near the load end), preferably separated by 20 centimeters or more. Measure the width of minimum with the micrometer carriage drive. Count
* A. von Hippel gives charts of  and tables of ———
and suggests further references.
the number of half wavelengths between the two minima (distance between adjacent minima is X„/2). Then remove the sample from the holder, attach the empty sample holder to the line, and record the position and width of a minimum near the load end of the slotted line.
INTERPRETATION OF DATA
With the sample in place, the resulting width of minimum is determined by loss in the dielectric, loss in the sample holder, and loss in the slotted line up to the point of measurement. The width of minimum at a second point along the slotted line is increased by the loss in the slotted line between the two points. The width of minimum, with the sample holder empty, is determined by the losses in the sample holder and in the slotted line to the point of measurement. In order to determine the loss tangent of the dielectric, it is necessary to separate the dielectric loss from the other losses. Call AAT]s the width of minimum at position l\B with the sample in place, AA'2s the width of minimum at position ¡2s, and AA'i, the width of minimum at position lie, with the sample holder empty. Then the width of minimum due to loss in the dielectric is given by:
This width of minimum can be used in equation (12) or (14) to determine the loss tangent. The dielectric constant can be found from equation (7). If the approximate dielectric constant is unknown, then measurement at two frequencies will be necessar3r since Na will not be known.
Example: A Teflon* sample 15.00 centimeters long is measured. It is found that a voltage minimum occurs at the sample face when the frequency is adjusted so that X0 = 21.34.
. The dielectric constant is
(2(15.00)/' known to be approximately 2. There
2.024. Since the minimum is very narrow, the 10dB widthofminimum points are used. The width of minimum at lu = 21.34 is 0.1004 cm. The width of minimum at Z2i = 42.68 is 0.1518 cm. With the sample holder empty, the width of minimum at l\e = 17.01 is 0.0788 cm. Then the width due to losses in the sample is found from equation (15) as
0.00024.
Note that if a lossy material is measured (tan 5 >0.1), equations (2) and (3) are no longer valid and equation (1) must be solved."
FREQUENCY RANGE OF MEASUREMENT
The lowest frequency at which measurements can be made is determined by the dielectric constant of the material being measured and by the necessity that at least one minimum occur along
* DuPont trademark. ' Ibid,
• W. B. Westphal, "Techniques at Measuring the Permittivity and Permeability of Liquids and Solids in the Frequency Ratiite 3 c/s to 50 kMc/s," Technical Report No. S6, Laboratory for Insulation Research, M.I.TT, July 1950. (Out of print)
the slotted line so that its position and width can be measured. Type 900L10, L15, and L30 Precision Air Lines can be used between the sample holder and the slotted line to position a minimum on the slotted line at low frequencies. If these additional air lines are used they should be externally supported. Sample holders up to 66 centimeters long can be constructed for lowfrequency use. The sample can be made shorter than a halfwavelength and equations (2), (3), (5), and (6) used to determine the dielectric constant and loss tangent. With these methods, measurements can be made down to 50 MHz or even lower.
The upper frequency limit for the Type 900LB Slotted Line is 8.5 GHz, but special precautions should be taken
Vfr noted in the paragraph Existence of II igherOrder Modes.
, ERRORS
Sample Fit
One of the most common sources of error in dielectric measurements by the coaxial method is the presence of air gaps between the sample and the inner and outer conductors. Correction formulas based upon a uniform distribution of the air gap can be used, but, since the actual air gap will usually not be uniformly distributed, the gaps should be avoided for maximum accuracy. The corrections for uniform air gaps for tan 8 <0.1 are
Cr (coriect)
er (measured)
r (measured) Lj
tail ¿(measured) ^ 1 ¿r (correct) ^ 0 ' , k r r D* _u i D<
lJi Us
Do = inside diameter of sample, D3 = outside diameter of sample, and n>4 = 0.5625.
Meter Errors
If a 3dB width of minimum is used, meter indications on the GR Type 1216A Unit 1F Amplifier will, in general. cause negligible error when the upper part of the scale is used and when care is taken to tune the localoscillator frequency exactly for maximum output. A 10dB widthofminimum measurement may require that the if amplifier calibration be cheeked with a precision attenuator for greatest accuracy. As an example of the errors in losstangent measurements caused by poor if amplifier calibration, an error of 0.1 dB in a typical 3dB widthofminimum measurement will cause an error of 1.9% in tan 8. An error of 0.3 dB in a typical 10dB widthofminimum measurement, will result in a 3.9% error in tan 5.
Effect of Contact Resistance
Although the connector contact resistance is typically less than half a milliohm, a small part of the measured loss is due to this resistance. The magnitude of the error caused by this loss depends upon the relative current through the contact for each measurement and is significant only when very lowloss dielectrics are measured. If the current is the same when the sample is measured as when the empty sample holder is measured, the contact loss will have no effect on the accuracy of the tan 8 measurement. If the currents differ, there may be an error in tan 5 as large as 0.0001. The amount of loss due to the finite contact resistance in a given measurement is cos 20+1 Loss = — X maximum loss,
from a voltage minimum to the contact. Maximum loss occurs when a voltage minimum occurs at the contact. It is difficult to evaluate the maximum loss exactly because of its small value. The condition that the current be the same for both measurements (with and without sample) may be met by appropriate choice of length and frequency for a sample with a given dielectric constant. If the dielectric constant is unknown, it may be necessary first to measure dielectric constant and then to trim the sample to the proper length for accurate determination of loss. This is necessary only for very accurate measurements of the loss tangent of lowloss dielectrics. For lowloss materials, the current through the contacts will be of approximately the same magnitude with and without the sample in the holder when the frequency and length are so chosen that sample
where X, and X are integers and b is the length of the sample holder. Lengths
John F. Gilmore received his BSEE degree in 1961 and his MSEE in 1963, from Northeastern University. He was employed as a cooperative student at Alford Manufacturing Company and as an engineer with that company from 1961 to 1963. He joined the Microwave Group at General Radio as a development engineer in 1963 and is currently engaged in microwave circuit design.
that satisfy the relationship and the corresponding values of N„ can be determined from Figure 3 for a 15cm sample holder and from Figure 4 for a 30cm sample holder. Figure 4 shows only the most useful curve of a very large family of curves. As an example of the use of these curves, suppose that the loss tangent of a lowloss material with a dielectric constant of 2 is to be measured. If a sample 12.43 cm long is used in a 15cm sample holder, tan 5 can be measured with maximum accuracy at \0 = 17.60 cm, 11.73 cm, 8.80 cm, 7.05 cm, 5.87 cm, and 5.03 cm, corresponding to Ns = 2, 3, 4, 5,
G, 7. If, instead, a sample 13.54 cm long were chosen, tan 8 could be measured with maximum accuracy only at = 5.48 cm, Na = 7.
Existence of HigherOrder Modes
At frequencies higher than —y= GHz,
"V e higherorder modes, particularly the TEu mode, can be excited by axial dissymmetries in the dielectric material. While the airfilled section of line between the sample and the point of measurement acts as a filter for these higherorder modes, in some instances coupling between the TEM and TE modes may be great enough to produce an error in measurement. Measurements above this frequency, therefore, should be made at small (say 10%) frequency increments and compared
with measurements below —r. GHz,
Vt in order that anomalous results can be detected.
 \  
\  
  
  
Figure 4. Lengths of samples for a Type 900LZ30 Reference Air Line for minimum tan 6 error in lowloss dielectric measurements. DIELECTRIC CONSTANT Figure 4. Lengths of samples for a Type 900LZ30 Reference Air Line for minimum tan 6 error in lowloss dielectric measurements. 2.80 2.70 2.60 2.50 lu 2.30

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