A Simple Wideband Transmission Line Model

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1 A Simple Wideband Transmission Line Model Prepared by F. M. Tesche Holcombe Dept. of Electrical and Computer Engineering College of Engineering & Science 337 Fluor Daniel Building Box Clemson, SC Submitted to IEEE Transactions on Electromagnetic Compatibility June 5, 26 Abstract This paper discusses a simple model for approximating the per-unit-length parameters of a lossy cable in such a way so as to provide a smooth transition from low frequencies to high frequencies. Using Schelkunoff s classical expressions for the transmission line parameters of a coaxial line, the simple model is postulated and used to provide approximate responses that can be compared with the rigorous solutions. This approximate model is shown to be accurate, and offers an alternative to evaluating the Bessel function expressions for the line parameters. Work supported under the provisions of AFOSR MURI Grant F

2 A Simple Wideband Transmission Line Model 1. Introduction In a publication in 1934 Schelkunoff developed a rigorous analysis (within the limitations of TEM transmission line theory) of a lossy coaxial cable [1]. In that paper he developed analytical expressions for the per-unit-length (PUL) impedances of the inner and outer conductors that must be added to the PUL length inductive impedance of the coaxial line. When taken with the PUL admittance of the line, which results from the PUL line capacitance (and neglecting internal dielectric losses), these impedance and admittance parameters form the basis of a transmission line model that is accurate over a wide range of frequencies and electrical conductivities. Schelkunoff s expressions for the PUL conductor impedances involve various ratios of modified Bessel functions of complex argument and these are very easy to evaluate with present-day computing platforms like Matlab or MathCAD. However, in the 193 s the evaluation of these functions posed a problem, especially if a wideband solution were desired for the construction of a transient solution for the line. As a consequence, various low- and high-frequency approximations to these impedances have been derived. [1], [2]. These approximations to the PUL wire impedances are very useful for single frequency or narrow band models; however for wideband calculations that span both frequency limits, there is a problem in seamlessly transitioning from one frequency regime to another. In the present paper, a simple model for the lossy cable is suggested that provides a smooth transition from the low frequency to the high frequency approximation. This model is shown to be very accurate, and offers an alternative to evaluating the Bessel function expressions for these line parameters. 2. The Schelkunoff Model To illustrate this line model, consider a simple coaxial transmission line having a crosssection shown in Figure 1. The inner conductor has a radius a and the outer conductor (shield) has an inner radius b and thickness. Note that the outer radius of the shield is c = b +. Each conductor is assumed to be lossy with the same electrical conductivity σ and the cable interior is filled with a lossless dielectric having a relative permittivity ε rel. In the development to follow, the magnetic permeability is assumed to be that of free space, µ. 1

3 outer shield conductivity σ radius a conductivity σ b c Z' a Z' b L' C' inner dielectric ε rel thickness Figure 1. Cross-section of the coaxial transmission line and its per-unit-length equivalent circuit. For the lossless coaxial line Paul [3] provides the per-unit-length line inductance and capacitance parameters L' and C' as µ o r a L = ln 2π r rel b ln ( ra / rb) 2πε ε C =. (1) When the conductors of the coax are finitely conducting, there will be additional perunit-length impedance elements in the transmission line model that take into account both the magnetic flux penetration into the conductors and the resistive loss. For the inner conductor with radius a, the per-unit-length impedance derived by Schelkunoff [1] is Z η I ( γ ca) ( γ ) a = 2πa I1 ca, (2) where I and I 1 are modified Bessel functions. The term η is the wave impedance in the lossy conductor, and if the displacement current in the conductor is neglected this term is jωµ η. (3) σ The term γ in Eq.(2) is the propagation constant in the conductor and is given as γ jωσµ (4) c The per-unit-length impedance of the outer shield (for the return current path lying inside the shield) is derived by Schelkunoff as 2

4 Z ( γc ) ( γc ) + ( γc ) ( γc ) ( ) ( ) ( ) ( ) η I b K c K b I c 1 1 b = 2πb I1 γcc K1 γcb I1 γcb K1 γcc (5) where K and K 1 are the modified Bessel functions of the second kind, and c is the outer radius of the shield c = b +. The inductance and capacitance parameters of Eq.(1) and the conductor impedances of Eqs.(2) and (5) provide the total PUL coaxial line impedance and admittance as Z = jωl + Z + Z and Y = jωc. (6) Using these parameters, the transmission line propagation constant is and the characteristic impedance of the line is a b γ = Z ' Y ' (7) Z c Z ' =. (8) Y ' 3. The Approximate Model 3.1 High Frequencies While the impedances in Eqs.(2) and (5) can be easily evaluated numerically, it is advantageous to develop approximate expressions for these terms. For high frequencies where γ ca > 5 and γ cb > 5, Eq.(21) becomes 1 jωµ 1+ j ωµ Z ahf = 2πa σ 2πa 2σ R + jωl ahf ahf (high frequency). (9) In this last equation, the PUL resistance and inductance are both frequency dependent with values and 1 ωµ 2πa 2σ R = (1a) ahf 1 µ 2πa 2ωσ L =. (1b) ahf 3

5 For the coax outer conductor, an expression similar to Eq.(9) results from the high frequency simplification to Eq.(5), with the radius a being replaced by b. 3.2 Low Frequencies At low frequencies where γ ca < 1 and γ cb < 1 the impedances Z a and Z b take on limiting values that relate to the dc resistance and the inductance arising from the internal magnetic flux in conductors. Thus, Z R + jωl and Z R + jωl (low frequency) (11) alf adc aint blf bdc bint where R' a dc and R' b dc are given by 1 1 R and R adc 2 bdc πa σ 2πb σ (Ω/m). (12) L' a int and L' b int are the per-unit-length internal inductances of the inner and outer coaxial conductors. These are given in [2] as µ µ c ln( c/ b) b 3c L a int = and L bint = + ( H / m) π 2 π ( c b ) 4( c b ) (13) 3.3 The Approximate Wideband Model A simple wideband model for the coaxial line that incorporates both the low and high frequency asymptotic behavior is shown in Figure 2. This is a circuit model for the impedance element Z' a for the inner conductor in Figure 1. With the appropriate circuit parameters from Eqs.(1a, 1b, 12 and 13) this circuit provides for the lossy effects of the internal conductor. For this circuit, the impedance Z' a is Z ( + ω ) ω ( ) jωl R j L (14) + + a int a hf a hf a = R adc + Rahf j La hf La int The same circuit with different parametric values is used for the outer conductor to represent Z' b. When these two circuits are combined in series with the line inductance, as shown in Figure 1 and expressed in Eq.(6), the wideband model of the line is formed. 4

6 Z' a L' a int R' a dc R' a hf L' a hf Figure 2. A simple model for the per-unit-length model for the inner conductor of the coaxial transmission line. 4. Comparison of the Models To illustrate the simplified wide-band cable model and compare it with Schelkunoff s model, we consider the coaxial line with the following parameters: a = 2.5 mm, b = mm, c = mm (or = c b =.6 mm), and ε rel = 2.5. These parameters provide a high frequency characteristic impedance of 5 Ω. To provide a wide range of conductivities for this comparison, we consider the following materials comprising the inner and outer conductors of the line: Copper: σ = S/m Cast iron: σ = S/m Tellurium: σ = S/m Sea water: σ = 5 S/m The last material, sea water, is perhaps not appropriate for consideration as a conductor in a transmission line as the validity of the TEM assumption is open to question for highly lossy conductors. Nevertheless, it does permit the comparison of the Schelkunoff and approximate line models in the high loss regime. 4.1 Frequency Domain Comparisons Figure 3 presents the real part of the PUL line impedance Z' using the exact Schelkunoff model (dotted line) and the approximate model (solid line) for the four different conductivities. The transition from the constant low frequency resistance to the frequency dependent resistance arising from the skin depth in the conductors is clear and the approximate model does a good job in representing the overall behavior of the resistance. From the total line impedance, we can extract the total effective PUL inductance as L' tot = Im(Z')/ω, which arises from the normal line inductance L' in Eq.(1) and the reactive components of Z a and Zb. Figure 4 presents plots of this inductance for the two models and the different conductivities. 5

7 A close examination of the results in Figures 3 and 4 shows that the maximum error in the total inductance of the line is less than 5% over the entire frequency range, and is independent of the electrical conductivity, although the frequency for which this maximum error occurs is frequency dependent. For the PUL line resistance, the maximum error is about 18%, a fact that is masked in Figure 3 by the log scale Sea water R' (Ohms/m) Tellurium Cast iron.1.1 Copper Frequency (Hz) Exact Approximate Figure 3. Plot of the real part of the total per-unit-length impedance of the coaxial line. 6

8 Copper Cast iron Sea water Tellurium L'tot (H/m) Frequency (Hz) Exact Approximate Figure 4. Plot of the total per-unit-length inductance, computed from the imaginary part of the total impedance of the coaxial line. As noted in Eqs.(7) and (8), the PUL line parameters are used to define the complex propagation constant of the line and the complex characteristic impedance. To illustrate the comparison of the approximate and exact results for these line constants, Figure 5 presents the magnitude of the propagation constant for the approximate and exact line models. Similarily, Figure 6 presents plots of the magnitudes of the characteristic impedance. A comparison of the numerical values for γ and Zc provides the interesting result that the maximum error in both of these quantities is about 2.2 %, which is lower than the maximum errors found in the per-unit-length resistance and inductance parameters 7

9 gamma (1/m) Sea water Tellurium Cast iron Copper Frequency (Hz) Exact Approximate Figure 5. Plots of the magnitude of the transmission line propagation constant for the approximate and exact line models. 8

10 Sea water Tellurium Zc (Ohms) Cast iron 1 Copper Frequency (Hz) Exact Approximate Figure 6. Plots of the magnitude of the transmission line characteristic impedance for the approximate and exact line models. 4.2 Time Domain Comparisons It is also useful to compare the exact and approximate line models when they are applied to transient problems. For this example a 1-meter length of coaxial line having the geometry defined previous is assumed to be excited by a voltage pulse function with an amplitude of 1-volt and a duration of 5 ns. As shown in Figure 7, this source is located at position x s =.2 m from the left end (node #1). The terminating loads on the line are assumed to be resistive with values R L1 = 1 Ω and R L2 = 1 Ω. I 1 V - s + 1 Ω Z c, γ + V 1 + V 2 1 Ω - - x s =.2 m L= 1 m x Node #1 Voltage Excitation Node #2 Figure 7. Geometry of the voltage-excited transmission line. 9 I 2

11 With the line s characteristic impedance and propagation constant, in either the exact or approximate form, transmission line theory can be used to compute the spectral responses of the load voltages at each end of the line [4]. Once this is accomplished, the spectra can be transformed into the time domain using a numerical Fourier transform. Figure 8 presents transient plots of the load voltages V 1 and V 2 at each end of the line for the four different conductor conductivities and the exact and approximate line models. As can be noted in these plots, there are some very slight differences in the results for the lower conducting cases, but aside from these differences, the responses are virtually identical. We conclude that while the transmission line s approximate parameters may differ slightly (e.g, 2.2%) from the exact values for some frequencies, the overall transient responses are not very sensitive to these deviations in the parameters. 1

12 .2 Load #2.2 Load #2 Load Voltage (V).2.4 Load Voltage (V) Load #1 Load # Time (ns) Load #1 Approximate Load #1 Exact Load #2 Approximate Load #2 Exact Time (ns) Load #1 Approximate Load #1 Exact Load #2 Approximate Load #2 Exact a) Copper b) Cast iron.2 Load #2.1 Load #2.1 Load Voltage (V).2.4 Load Voltage (V) Load #1.5.6 Load # Time (ns) Load #1 Approximate Load #1 Exact Load #2 Approximate Load #2 Exact Time (ns) Load #1 Approximate Load #1 Exact Load #2 Approximate Load #2 Exact c) Tellurium d) Sea water Figure 8. Plots of the transient load voltages at the ends of a 1-meter transmission line for the exact and approximate per-unit-length parameters. 5. Conclusion This paper has described a simple model for a lossy coaxial transmission line that provides both the low-frequency and high-frequency line parameters. This model uses two equivalent circuits, one for the inner conductor and another for the outer conductor (shield) of the line, together with the normal per-unit-length inductance of the line. Each of these circuits 11

13 contains resistance and inductance elements that represent the low- and high-frequency behavior of the line, with the circuit providing a smooth transition between the two frequency regimes. While this model has been discussed in the context of a coaxial line, its results can also be applied to the case of a line consisting of two thin parallel cylindrical conductors, as long as the conductors are not too close together and the currents flowing in the conductors are approximately axially symmetric. 6. References 1. Schelkunoff, S. A., The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields, Bell System Technical Journal, 1934, pp Ramo, Whinnery and Van Duzer, Fields and Waves in Communication Electronics, John Wiley, Paul, C. R., Analysis of Multiconductor Transmission Lines, John Wiley & Sons, 1994, New York. 4. Tesche, F. M., M. V. Ianoz and T. Karlsson, "EMC Analysis Methods and Computational Models," Wiley-Interscience Publication, John Wiley & Sons. Inc., New York,

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