Telegrapher equations for arbitrary frequencies and modes: Radiation of an infinite, lossless transmission line

Authors


Abstract

[1] Maxwell's equations for an infinite, lossless transmission line above a perfectly conducting ground are transformed into telegrapher equations with new generalized per-unit-length parameters of the conductor. These new line parameters are complex-valued, frequency-dependent, and contain the radiation resistance. Their explicit expressions depend on the chosen gauge, but there is also a gauge-independent representation for them. In the quasi-static approach of the Maxwell-Telegrapher equations the line parameters become real-valued, and radiation is absent. A Poynting vector analysis leads to a deeper physical understanding and interpretation of the new parameters.

1. Introduction

[2] The electromagnetic (EM) interaction of multiconductor transmission line structures with high frequency EM sources (up to several GHz) becomes an increasing topic of current research. This is due to a rapid development in the information and communication technology and the accompanying necessity to guarantee a smooth EM operation of all connected devices and systems. Since radiation phenomena occur more frequently and lead to EMC-relevant perturbation effects, they have to be included in the EM analysis of electrical and electronic systems. In particular, the effective simulation of new systems in the design phase becomes a cost-saving factor. There, the demand for numerical programs which can efficiently calculate the interaction of complex EM systems with high frequency fields is one resulting consequence. In this context the use of the telegrapher equations for nonuniform multiconductor transmission lines [Nitsch and Gronwald, 1999; Baum and Steinmetz, 2003] seems to be an adequate means. They have to be, however, extended to become valid for arbitrary frequencies and modes. This was done by Haase and Nitsch [2001]. Different from their approach, in the present paper we deal with a simple line configuration, an infinite, uniform transmission line above a perfectly conducting ground, and show that the Maxwell equations for this line can be cast into the form of the telegrapher equations, by keeping the source fixed but changing the classical line parameters to generalized, complex-valued ones.

[3] In section 2 we calculate the new line parameters in a gauge-independent way, using the Helmholtz decomposition [Cohen-Tannoudji et al., 1997] for the electric field. Their relation to the radiation resistance is established on the basis of a Poynting vector analysis for the radiating infinite line (section 4). We also perform a quasi-static approach for the infinite line (section 3) and obtain solutions without radiation fields. In particular, the corresponding parameters are real.

[4] Our generalized description of transmission lines can be extended to include multiconductor lines of finite length with losses [Haase at al., 2004]. Then, when incorporated into an existing field-theoretical computer program for complex systems as a module for very efficient calculations of linear structures, simulations of electronic systems in the GHz-regime become essentially faster. The present paper, in a first step, gives insight into new physical phenomena which are connected and inherent in the new parameters.

2. Telegrapher Equations in the Coulomb Gauge

[5] In this section we rely on the Helmholtz-decomposition of vector fields equation image(equation image) into a longitudinal part, equation image(equation image), and a transverse part, equation image(equation image), such that [Cohen-Tannoudji et al., 1997; Batygin and Toptygin, 1978, p.191]

equation image

and

equation image
equation image

holds in real space. In Fourier space these equations read after a (three-dimensional) spatial Fourier transform

equation image
equation image

as follows:

equation image

and

equation image

At all points equation image, in the reciprocal space, equation image is obtained by projection of equation image onto the unit vector equation image in the direction equation image. Thus one has:

equation image

and

equation image

Observe that in reciprocal space (equation image - space) the relationship between a vector field equation image and its longitudinal and transversal components is of local nature, whereas in the real space their relationship is not local: equation image, e.g., depends on values equation image of equation image at all other points equation image.

[6] After these introductory remarks, we turn to the Maxwell equations in frequency space where we indicate the decomposition of the electric field into its longitudinal and transverse parts:

equation image
equation image
equation image
equation image

As usual we introduce the potentials, ϕ(equation image) and equation image(equation image), and choose the Coulomb gauge (indicated by the index C)

equation image

to simplify the solution of Maxwell's equations.

[7] Then the fields can be expressed in terms of the potentials:

equation image

Inserting these equations into the Maxwell equations delivers two nontrivial equations:

equation image

with

equation image

and

equation image

Transformation of (11) and (13) into the reciprocal Fourier space yields

equation image
equation image

For the current density equation image(equation image) we use a simple distribution only along the z-axis

equation image

where equation image stems from the incident plane wave and is connected with the wave number k via the angle of incidence θ (we consider an excitation by a vertically polarized plane wave; see Figure 1). (In this paper we only consider the excitation of the wire by a plane wave. The inclusion of fixed sources (δ-sources) along the wire is possible and has been treated in Nitsch and Tkachenko [2004].).

equation image

Thus we easily find the Fourier transform of this current density

equation image

From this we derive with the aid of the continuity equation the charge density ρ(equation image) and get for ϕC (equation image):

equation image

Now we explicitly know all sources in equations (14) and (15) and can perform the back-transformation into the local space. These calculations become a little bit lengthy and cumbersome and also need some observations concerning the integration contour in the complex plane. At this place we only can present the results. For ϕC (equation image) we obtain [see also Abramowitz and Stegun, 1965, p. 376]

equation image

with η0: = equation image, ρ = (x2 + y2)1/2 distance perpendicular to the line, and K0(z) as a modified Bessel function. For equation image (equation image) we find after a longer calculation

equation image

Here H0(2) denotes a special Hankel function [see, e.g., Abramowitz and Stegun, 1965, p. 360]. In our approximation we always have k2 > equation image. Therefore the first row in equation (21) applies.

Figure 1.

Geometry of the problem.

[8] Since for our further calculations we only need the z-component, Az, of the vector potential, it is sufficient to evaluate

equation image

In order to get the total z-component of the vector potential we have to form the sum of the z-component in equation (21) and of equation (22). Also we must extend our above results to a wire above a perfectly conducting ground by using the mirror principle. Then we obtain, instead of equations (20)–(22):

equation image

and

equation image

Here we have used the abbreviations:

equation image

and

equation image

where ρ1 = ((xh)2 + y2)1/2 and ρ2 = ((x + h)2 + y2)1/2 are distances to an observation point perpendicular to the wire and its image, respectively.

[9] Now we are prepared to express explicitly the z-component of the electric field in terms of Ez and Ez. Choosing for the local coordinates ρ1 and ρ2 the surface of the wire, i.e., ρ1 = a and ρ2 = 2h (note that the thin-wire approximation is used), then we obtain

equation image
equation image

The capital letter G-functions replace those from equations (23) and (24) where we have fixed the local values ρ1 and ρ2 on the surface of the wire.

[10] Note that equation image and equation image are gauge independent quantities. Therefore their representation in equations (27) and (28) does not depend on our choice of the Coulomb gauge. Also in the Lorenz gauge we would obtain the same result.

[11] Next we want to correlate the longitudinal and transverse electric field to the line parameters per-unit-length relying on the representation of the differential-power density by the induced-EMF (IEMF) method [Miller, 1999]:

equation image

Insertion of the fields (equations (27) and (28)) and the current

equation image

into equation (29) yields:

equation image

with

equation image

and

equation image

with

equation image

We may denote CC as a generalized capacitance per-unit-length and LC as generalized inductance per-unit-length of our lossless conductor above ground. The quantity CC is a pure real quantity which does not depend on k. This is not quite surprising since it was calculated from an instantaneous field. LC is complex-valued, and we will show that the imaginary part of it is correlated with the radiation resistance. An expansion of LC and CC for small arguments, i.e., ka ≪ 1 and 2kh ≪ 1, leads to the well-known classical static transmission line parameters

equation image

Also the case of grazing (θ = 0) incidence is of interest. Then the parameters become inverse to each other and assume almost quasi-static values like for a TEM-mode excitation [Reibiger, 2003]:

equation image

In concluding this section we justify the notation of LC and CC as line parameters showing the close analogy to the classical transmission line equations. We refer to equations (23) and (24) where we have chosen the local values on the surface of the wire:

equation image
equation image

The derivation of ϕC(z) w.r.t. z yields (use equations (30) and (37))

equation image

the first telegrapher equation. The second telegrapher equation is derived from the boundary condition:

equation image

or, equivalently,

equation image

Insertion of (38) into (41) gives

equation image

Thus we have proven that the Maxwell equations can be transformed into telegrapher equations with generalized line parameters. Therefore all known solution procedures for the telegrapher equations can also be applied to solve equations (39) and (42).

[12] It is of interest to investigate the line parameters, when the height h of the wire substantially increases in comparison to the field wavelength λ. Using the known asymptotes of the modified Bessel functions and of the Hankel functions [Abramowitz and Stegun, 1965, pp. 364 and 378], we obtain from (32) and (34), respectively

equation image

and

equation image

Combining these two asymptotic parameters to a distributed impedance Z′ according to equations (31) and (33)

equation image

one gets the well-known solution of the diffraction problem for a plane wave impinging on a cylindrical wire in free space [Batygin and Toptygin, 1978, p. 108; Ufimtsev, 1971, chapter 7]. Thus, even for one wire in free space, it is possible to introduce generalized transmission line parameters.

3. Maxwell's Equations Without the Transverse Displacement Current: Quasi-Static Approach

[13] This section is devoted to the investigation of the influence of the transverse part of the displacement current on the solutions for the electromagnetic potentials and the line parameters. Frequently, in the literature, the cancellation of the transverse part of the displacement current density in Maxwell's equations is called “the quasi-static approximation” and the parameters G0 (equation image) are called “quasi-static parameters” [Reibiger, 2003; Mathis, 2001]. In the course of this section we will derive the quasi-static solutions and the corresponding line parameters and establish their relation to G0 (equation image).

[14] In equations (5)–(8) we only modify Ampere's law by suppression of the transverse part of the displacement current density

equation image

All other equations remain unchanged, and instead of equation (11) we now obtain

equation image

The Fourier transform of this equation into the reciprocal space reads

equation image

Now we solve this equation in analogy to the previous case and get for the vector potential components

equation image
equation image

The solution for the scalar potential ϕC(equation image) remains the same as in equation (20). Therefore also the longitudinal field Ez does not change (see equation (27)). Only Ez is modified and reads

equation image

and for the wire above perfectly conducting ground becomes:

equation image

with

equation image

On the surface of the wire we have ρ1 = a and ρ2 = 2h, and equation image is replaced by equation image(equation image, a, h). Analogous steps to equations (29) to (34) now lead to the quasi-static line parameters:

equation image

(unchanged, like in (32)) and

equation image

In Figure 2 we display a simple example for the different line parameters.

Figure 2.

Frequency dependency of generalized line parameters (h = 0.5 m, a = 0.001 m, θ = 45°). (a) Capacitance per-unit-length; (b) inductance per-unit-length.

[15] Obviously, the quasi-static inductance per-unit-length is quite different from the result (34) and in particular from equation (36). In the quasi-static approximation the line parameters are not inverse to each other. This, however, happens in the case of small arguments, i.e., equation imagea ≪ 1 and equation image2h ≪ 1. Then equation image(equation image, a, h) ≅ 0 and we have the same result as in equation (36). Note that the quasi-static parameters are real functions, and therefore we do not have radiation losses.

4. Poynting Vector of an Infinite Line

[16] In the previous section we have derived new, generalized line parameters which occur in the generalized transmission line equations (41) and (42). Our interpretation of the line parameters as generalized per-unit-length capacitance and inductance was based on the differential-power density representation with the aid of an impedance function and the square of the current magnitude (compare equation (29)). It was mentioned that the real part of this per-unit-length impedance equals the radiation resistance. The proof of this statement is the subject of this section and will be performed in the Lorenz gauge. The use of the potentials in this gauge will simplify the following calculations considerably.

[17] From the electromagnetic potentials [Nitsch and Tkachenko, 2004] in the Lorenz gauge

equation image

and

equation image

with

equation image

we calculate the magnetic field equation image and the electric field equation image, respectively:

equation image

The electric field is represented in two components: the parallel component to the conductor

equation image

and the transverse part to the conductor

equation image

Here we have introduced the unit vectors equation imageequation image and equation imageequation image.

[18] As usual, we now calculate the Poynting vector

equation image

and decompose it into two summands. The first term, equation image, represents the power density perpendicular to the conductor, the second term, Szequation image, is the power density which is conducted along the wire. For equation image we find

equation image

Integration over the area of the wire surface yields

equation image

Comparison of this expression with the expression for Z′(jω) (see equations (29), (31), and (33))

equation image

leads us to equation

equation image

where the factor two in the denominator stems from the time averaging procedure for the Poynting vector. Obviously, the real part of Z′ represents the radiation resistance, whereas its imaginary part is related to the stored energy in the near fields of the conductor.

[19] The transported energy along the wire direction can be evaluated using the power density

equation image

We recognize that this function is real and positive for equation image > 0, and also constant along the z-axis. Thus no energy is stored in this direction, almost all is led in the close neighborhood along the conductor.

5. Conclusion

[20] We have shown that the Maxwell equations of an ideal conductor above a perfectly conducting ground can be cast into the form of telegrapher equations with generalized line parameters per-unit-length. The representation of these new line parameters turned out to be gauge-dependent. However, a definition of the generalized line parameters with respect to a Helmholtz decomposition of the electric field yields gauge-independent expressions for them. Since the Coulomb gauge is compatible with such a decomposition, the results obtained in this gauge for the parameters directly correspond to Z and Z, respectively. The imaginary part of the new inductance per-unit-length, Im(ωLC), equals the radiation resistance of the infinite line. This proof was given by a Poynting vector analysis (compare equation (66)).

[21] From classical transmission line theory it is known that for lossless lines (where only the TEM mode is assumed to propagate) the line parameters are inverse to each other. More precisely, one has [Reibiger, 2003]

equation image

with so-called “static” or “quasi-static” parameters. In our new approach we only obtain this result in the low-frequency approximation (see equation (35)) or in case of a grazing incidence of the exciting plane wave (see equation (36)).

[22] Special emphasis was laid on the quasi-static approach for the fields and the new line parameters. As expected, due to the missing retardation of the fields, the line parameters turn out to be real, but do not fulfil equation (68).

[23] Note that the line parameters also depend on equation image, a parameter which occurs in the representation of the translation group. (The translation group is a symmetry group for the straight wire.) It might be desirable to describe the properties of the line solely by its geometrical and inherent physical parameters, independent of its excitation. It is possible to meet these requirements. A corresponding theory for the multiconductor lines of arbitrary configuration which interact with very high frequency sources has been established by Haase and Nitsch [2001] and [2003] and Haase et al. [2003]. This leads to an iterative solution procedure of the Telegrapher equations during the course of which the sources have to be redefined at any iteration step. In our present representation the sources are kept fixed and the parameters have to be adjusted to them.

[24] The extension of the presented theory to multiconductor lines with losses and infinite length is straightforward. For finite multiconductor lines the theory is expected to become more involved and will to be the subject of our future investigations [see also Nitsch and Tkachenko, 2004].

Acknowledgments

[25] The authors would like to thank Dr. C. E. Baum and Prof. G. Wollenberg for helpful discussions and Prof. A. Reibiger for his constant interest in this work. This work was sponsored by the Deutsche Forschungsgemeinschaft DFG under contract FOR 417.

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