Applying the Retarded Solutions of Electromagnetic Fields to Transmission Line RLGC Modeling

The RLGC model, and its variations, is one of the most common techniques to simulate transmission lines. The RLGC model uses circuit network elements consisting of Resistance R, Inductance L, Conductance G and Capacitance C (per unit length) to represent a small segment of the transmission line, and then cascades multiple segments to simulate the transmission line of arbitrary length. Typically, the parameters in RLGC model are extracted from the propagation constant and characteristic impedance of the transmission line which are found using numerical simulation methods. These resulting RLGC parameters for multi-GHz signaling are usually frequency-dependent. This paper introduces an analytical approach to extract RLGC parameters to simulate a transmission line, which results in a different model, the RLGC(p) model.


Introduction
Maxwell's Equations have been widely studied and used since their publication in 1861 by the Scottish physicist and mathematician James Clerk Maxwell.While Maxwell's Equations may appear in different forms, a common key aspect is that these equations specify the relationship of electric field intensity ( ⃑ ), magnetic field intensity ( ⃑ ), electric charge density (ρ) and electric current density (J ⃑ ) at a particular space-time [2] , ( ⃑ , ′).For example: Where ⃑ ( ⃑ , ′) = ⃑ ( ⃑ , ′) is the magnetic flux density at point ⃑ , time ′, and is the permeability of the medium.
To analyze electromagnetic field propagation we need to find the causal relationship between different space-times for the source ( ⃑ , ′) and the observation point ( ⃑, ) .Since electromagnetic fields propagate in the medium with a finite propagation speed, denoted as , there is a time difference in the relationship of electromagnetic fields at the source and the observation point.This phenomenon is known as time retardation and the time difference can be described as − ′ = | ⃑ − ⃑|/ .The retarded solution of electromagnetic fields can be derived from Maxwell's Equations with a Green's Function.The retarded solution for point source, which is also known as the Jefimenko's Equations [3] , can be found as: With a Fourier transform, this can be converted to Frequency Domain: The solution for other types of sources can be developed based on the solution for a point source.
In the above equations, the term is introduced due to consideration of the retardation, and the ratio can be used as an indicator for the impact of retardation.When the frequency is low, the electrical distance between source and observation point is small, ≪ 1, and the retarded solution of electromagnetic fields can be approximated by the static solution [1] .However, Feynman emphasized that the terms representing time retardation should not be omitted when they become significant [6] .Also, for discussion in future sections, it should be recalled that a phase shift in the frequency domain leads to a time retardation in the time domain [2] .

Theoretical Case Study
The retarded solution of electromagnetic fields provides an analytical way to solve the electromagnetic fields.In this section, we use it to solve the test case of a simplified transmission line.For this test case, we use a lossless thin straight line source along the direction to represent the transmission line and a flat PEC plane at distance ℎ away from the line source to represent the reference plane.We fill the entire space with uniform, lossless medium with ε = 4 and μ = in which the electromagnetic fields propagate at velocity, .As a simplification, the model is extended to infinity to avoid the need of handling any boundary condition., is assumed to be a sinusoidal wave where charge, , and attenuation, , are unity and frequency is 100 GHz (refer to Section 3.2 of reference [1]).We observe the electromagnetic fields on the surface of the PEC reference plane, for ℎ = 10 and ℎ = 100 , respectively.We plot the magnitude of the fields in color where positive peak is in red and negative peak is in blue.In addition to showing that the magnitude of field intensity at ⃑ is inversely proportional to its distance from ⃑ , these electric field intensity plots, Figure 3 to Figure 6, show the time retardation due to finite propagation velocity of fields.It is apparent that the peak of field intensity lags behind the peak of the source, and the delay is proportional to the distance, resulting in a "new moon" field pattern on the observation plane.This is more obvious with larger ℎ as the field has been spread out further when it arrives at the reference plane such as in Figures 3(b For the same study case, we can observe the field magnitude on the cross section of the model, as indicated in Figure 7.In this case, we set ℎ = 100 and set source frequency to 100 GHz and 10 GHz, respectively in Figure 8 and Figure 9.The direction of ⃑ (red arrows) and ⃑ (blue arrows) in both figures shows that the energy is propagating in the same direction for any point on the observation plane; however, the direction of fields are not uniform for 100 GHz.This is due to time retardation of the fields.The total field is a sum of the original fields directly from the line source and the reflected field from the PEC reference plane.The total field at a given observation location is dominated by a certain section of the line source due to time retardation and the dominating section is related to the distance of the observation location to line source.If the source is varying at a very high frequency, there is more variation in the dominating sections on the line source for the observation area, some of which segments may even be on different parity.This creates an irregular field pattern and opposite direction of the fields for 100 GHz source as shown in Figure 8. Conversely shown in Figure 9, if the source is varying slower there is minor variation in the dominating sections on the line source for the observation area; therefore, the total field is more uniform.Note that the direction of electric field intensity at the bottom of both Figure 8 and Figure 9 is perpendicular to the PEC as required.
Electromagnetic fields near the transmission line are the key to understanding the performance of the transmission line.As with field patterns in Figure 8 and Figure 9, there are unique characteristics for transmission line at high frequency.This can be studied analytically by using the retarded solution of the fields introduced in the previous section of this paper and by using the simulation program developed by the authors.

Simulation Case Study
In the previous section, it was demonstrated how to solve the electromagnetic fields near a transmission line using the retarded solution.In this section, these field solutions will be used to study the RLGC model which is a common technique for representing transmission line behavior.As in Section 2, and illustrated in Figure 7, a simplified representation of a transmission line is assumed where a lossless, thin, and uniform line source along the axis is placed above a flat, PEC plane.The medium is again assumed to be uniform and lossless with ε = 4 and μ = .Here, the height above the reference plane, h, is chosen to be 100 mils which is significant enough to invalidate the Classical Model (generally assumed to be one tenth of a wave length [10] ).
To simulate a transmission line, the propagation constant and characteristic impedance are needed.For a uniform transmission line along the direction, the voltage and current at a given location is ( ) = + and , where and are the voltage and current propagating in the + direction; and are the voltage and current propagating in the − direction.With this, the characteristic impedance is = + / + = − − / − , which is the ratio of voltage and current propagating along one direction.As shown in Figure 10, the Transmission Line is divided into multiple equal segments of ∆ in length, where and are found with the impedance and admittance of each segment: leads to a different result.From a physical point of view, is often referred to as the electrical resistance due to the transmission line material's conductivity, and is referred to as the electrical loss due to surrounding material.In a simplified model with assumptions of lossless transmission line as well as lossless surrounding material, resistance and conductance can be assigned as = 0 and = 0 .According to their definitions [7] , is the ratio of the total magnetic flux surrounding a unit segment (perpendicular to the segment and extending to ∞) to the current on such segment, and is the ratio of the charge on a unit segment over the voltage of such segment.Since the total magnetic flux and the voltage are integral products of the fields transmission line and such fields are solved by the retarded solution as demonstrated in Section 2, the and for the simplified transmission line model setup in Section 2 are: where is the radius of the transmission line, and are the starting and ending location of the transmission line, and = / .Additionally, we define (not to be confused with the resistance parameter, R) as the distance from the observation point to an integral segment, and as the distance from the observation point to the mirror image of the same segment with respect to the reference plane.This image line source is used to assure that fields on the PEC are orthogonal to that reference plane.
For and there is a common term in the solution: It has been shown [1,3] that this term becomes more important as a function of / according to the table below:  w) , where | | is the magnitude and (w) is the phase shift that accounts for retardation in the time domain.Furthermore, we can see from the four equations below that the classic RLGC model must be modified to account for time retardation effects as (w) becomes significantly greater than zero.
we focus mainly on comparing the accuracy of the results produced by several methods with each method configured to serve that purpose.It is beyond the scope of this paper to compare other aspects of each method such as computational efficiency and resources.

Discussion
From Figure 12 to Figure 19, it is shown that an RLGC(p) model is effective in representing the transmission line, its performance is in good agreement to the S parameters of a commercial 3D FEM field solver (Case <0>, Case <1> and Case <4>).Comparing Case <3> and Case <4> shows the differences between a classical RLGC model and a RLGC(p) model; in that, the classical RLGC model trends well but is not as accurate when compared to the RLGC(p) model.It should be emphasized that Case <1> and Case <3> are generated from the same field data of the 3D FEM field solver used in Case <0>.In Case <2>, the classical RLGC model from 2D field solver is incorrect for the propagation constant and characteristic impedance because time retardation was not taken into account.This tells us that ignoring time retardation at high frequency results in error.
Regarding the magnitude of S12 in Figure 14 and Figure 18, the insertion loss of the transmission line, Case <3> is drastically different from Case <0> because the 3D solver "corrects" the negative parameters, in this case the R parameter, for the classical RLGC model with the intention of preserving passivity.However, this "correction" disturbs the energy state as it eliminates the energy source, a negative R parameter, but preserves the energy consumer in G parameter.We cannot preserve the energy state by eliminating both the energy source and consumer in R and G parameter because it is effectively ignoring time retardation.As shown in Case <2>, this creates an incorrect propagation constant and characteristic impedance.The best approach is to make no "correction" as done in RLGC(p) model.As explained in Section 3, time retardation will create a complementary energy source and energy consumer, the passivity enforcement should avoid correcting such an energy source.

Conclusions
As demonstrated, the time retardation of electric and magnetic field propagation results in complex capacitance and inductance.The imaginary part of capacitance and inductance affect the RLGC parameters and cause the classical RLGC parameters at high frequencies to be inconsistent with their DC counterparts.A new RLGC(p) model is proposed to handle this inconsistency.It is necessary to use RLGC(p) model instead of the classical RLGC model to simulate a transmission line where retardation is significant.In this paper's transmission line example, the Field Retardation is considered significant, and the RLGC(p) model is shown to be in better agreement than the classical RLGC model to an FEM solution.Future work is ongoing to extend this model to multiple conductors; as well as, explore the limitations for extremely high frequencies.

Figure 1 :
Figure 1: Time Retardation of Electromagnetic Fields

Figure 11 :
Figure 11: RLGC transmission line model segment In a classical RLGC model, as shown in Figure 11, the impedance and admittance of each segment are represented by the Resistance R, Inductance L, Conductance G and Capacitance C, where = + , = + with the underlining assumption that R, L, G, C parameters are positive, real values.One key assumption to establish the cascaded segments as a good representation of the transmission line is to keep each segment small enough compared to the wavelength of the frequency of interest which is application dependent.To resolve this, the classical RLGC modeling technique introduces the per unit length R, L, G, C parameters, , , , , and multiplies the length of segment ∆ respectively to them to obtain R, L, G, C values for an application, that is = ∆ • , = ∆ • , = ∆ • , = ∆ • .In summary, for classical RLGC model: parameter= ( ) parameter= ( ) parameter= (Y) parameter= ( )

Figure 19 :
Figure 19: Phase of S12 from 88 GHz to 89 GHz

Table 1 :
Values of