The Measurement of the Characteristic Impedance of ...

The Measurement of the Characteristic Impedance of Transmission Lines using Nanoscale Resistive Films Jimmy G. M. Yip1, M-H John Lee2, Nick M. Ridler1, and Richard J. Collier2 1 2

National Physical Laboratory, Teddington, UK.

Microelectronics Research Centre, Cavendish Laboratory, University of Cambridge, UK.

Abstract — The value of the characteristic impedance, Z 0 , is well known for most transmission lines either from mathematical formulations or computer modelling [1]. So much so, that when an impedance, Z L , is evaluated, a value of Z 0 is assumed and only a measurement of the reflection coefficient, ρ , is needed. However, there are instances where an experimental value of Z 0 is required, for example in the cases of some lines with attenuation or exotic lines where mathematical equations are not available and where computer models are not rigorous. This paper shows that by reversing the procedure described above, a value Z 0 can be found from a known value of Z L and a measurement of ρ . In order to demonstrate the validity of this technique, some well-characterised transmission lines were measured at the National Physical Laboratory (NPL), using nanoscale sheet resistances to give Z L , and the results are given in this paper. The technique can now be extended to examine those instances mentioned above.

I. THEORY AND MEASUREMENT The construction of a known value of Z L from a sheet resistance of value, R , is made from a nanoscale resistive film on a thin dielectric substrate, where the film is one whose electrical thickness is much less than the skin depth. The value of R is proportional to the resistivity of the film which in turn is usually constant over a wide frequency range (0 – 100 THz) [2]. So this resistance can easily be measured at any convenient frequency, even at DC, thus making a transferable standard. Figure 1 shows a thin resistive film sandwiched between two waveguide sections illuminated by the principal waveguide mode. The power levels were kept sufficiently low to avoid any heating of the resistive film and thus a change in resistance, the dielectric substrate acts as a heat sink. Under these conditions, the surface electric field is consistent throughout the film, whereas the surface magnetic field is not. As a direct result of this consistent electric field, the transmission coefficient through the film, τ , is related to the reflection coefficient, ρ A , by [3]:

τ = 1+ ρA

(1)

An input impedance, Z in , A , at point A consists then essentially of the known sheet resistance, in parallel with the unknown Z 0 and Z 1 , where Z 0 is the characteristic impedance of the waveguide and Z 1 represents the dielectric loss as well as leakage through the gap between the waveguide flanges resulting from the presence of the thin film and substrate, and is given by:

Z in , A =

R(Z 0 + Z 1 ) R + (Z 0 + Z 1 )

(2)

So from equations 1 and 2, Z 0 can be found from any of the following:

Z0 =

Z in , A (1 − ρ A ) 1+ ρA

=

Z in , A (2 − τ )

τ

=

Z in , A (1 − ρ A )

τ

(3)

Equation 3 suggests that for every value of R it should be possible to measure ρ A , or τ , or both, and hence find Z 0 . However certain values of R – zero and infinity – cannot be used as they give, from the first part of equation 2, Z in , A as a Thin Film

Substrate

Rectangular Waveguide (a)

Z1 B

A

Z 0 Z in , A

Z0

R

τ

ρA (b)

Fig 1. (a) Resistive film sandwiched waveguides. (b) Equivalent circuit.

between

two

short circuit and a matched load respectively. From equation 2 and 3, the reflection coefficient at point A, ρ A , can be re-written as:

ρA =

Z in , A − Z 0 Z in , A + Z 0

=

− Z 0 + n (R − Z 0 ) 2 R + Z 0 + n(R + Z 0 )

an in-line four-point probe. For a semi-infinite thin film resting on an insulating support, the sheet resistance, R, is given by [4]:

Z1 . Z0 The measurements were made using metallic waveguide at millimetre wave frequencies (75 – 110 GHz) where Z 0 is dispersive and may be complex depending on the level of attenuation. Since the dielectric substrate had a larger area than the metallic waveguide, there was a gap of about 100 µm between the waveguide flanges. The thin resistive film made good contact with the waveguide walls and the effect of the gap, Z 1 , was removed from the measurement by comparing two different thin films, 30 nm and 70 nm. Eliminating n from equation 4, the value of Z 0 can be re-expressed in terms of these two measurements using:

2 R1 R2 (ρ1 − ρ 2 ) (R1 − R2 )(1 + ρ1 )(1 + ρ 2 )

V log e (2 ) I

(6)

π

where

is a general geometric factor (geometric log e (2 ) factors for non-semi-infinite samples are listed in [5]), and V and I are the voltage and current, respectively. In this case, both V and I were measured using a digital multimeter with an assumed worst-case uncertainty of 0.1 %. The sheet resistance measurements were also confirmed using a microwave technique at 130 GHz [6]. The measurements of the reflection coefficients at microwave frequencies were made using NPL’s primary national standard microwave impedance measurement system [7]. This system uses the Through-Reflect-Line technique [8] to calibrate a Vector Network Analyser. In rectangular waveguide, the Line standard consists of a section of precision line that is approximately a quarterwavelength at the mid-band frequency. In the case of the WR10 waveguide size (nominal frequency range 75 GHz to 110 GHz), this corresponds to a line length of approximately 1 mm. These results are shown in figure 2.

where n =

Z0 =

π

R=

(4)

(5)

where R1 , R2 , ρ1 , and ρ 2 are the corresponding sheet resistances and reflection coefficients of thin film 1 and 2. The DC sheet resistance of R1 and R2 were measured using

Characteristic Impedance, Z0, of Rectangular Waveguide 650 Theoretical values Measured values

Z0 (ohms)

600

550

500

450

400 75

80

85

90

95

100

105

110

Frequency (GHz)

Fig 2. The characteristic impedance of rectangular waveguide: theoretical and measured values against frequency. The theoretical value [9, 10] assumed no attenuation due to metallic loss, and the imaginary parts of the reflection coefficients were neglected because they were small.

II. CONCLUSIONS A practical method of measuring the characteristic impedance of transmission lines is described and applied to metallic rectangular waveguide at millimetre wave frequencies to demonstrate its validity. This paper demonstrates a method that can now be applied to other exotic transmission lines where the existing knowledge of their characteristic impedances is incomplete. III. ACKNOWLEDGEMENT The work described in this paper was funded by the National Measurement System Directorate of the UK government’s Department of Trade and Industry. © Crown Copyright 2005. Reproduced by permission of the Controller of HMSO. IV. REFERENCES [1] WADELL, B.C.: “Transmission Line Design Handbook”, Artech House, 1991. [2] RAMO, S., WHINNERY, J. R., and VAN DUZER, T.: “Fields and Waves in Communication Electronics”, Third Edition, John Wiley & Sons, 1993, Chapter 13. [3] Reference [2], Chapter 6. [4] MAISSEL, L I. and GLANG, R.: “Handbook of Thin Film Technology”, McGraw-Hill Book Company, 1970. [5] LOGAN, M. A.: “An AC Bridge for Semiconductor Resistivity Measurements Using a Four-Point Probe”, Bell System Tech. J., 40, pp. 885-919, 1961. [6] LEE, M-H J. and COLLIER, R. J.: “The Sheet Resistance of Think Metallic Films and Stripes at both DC and 130 GHz”, Microelectronic Engineering, 73-74, pp. 916-919, April 2004. [7] RIDLER, N. M.: “A review of existing national measurement standards for RF and microwave impedance parameters in the UK”, IEE Colloq Digest No 99/008, pp. 6/1 – 6/6, February 1999. [8] ENGEN, G. F. and HOER, C. A.: “Thru-Reflect-Line: An Improved Technique for Calibrating the Dual Six-port Automatic Network Analyser”, IEEE Trans, MTT - 27(12), pp. 987-993, December 1979. [9] MARCUVITZ, N.: “Waveguide Handbook”, IEE, 1993. [10] Reference [2], Chapter 8.