Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home   |   Pavements Home
Report
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-08-035
Date: March 2008

LTPP Computed Parameter: Moisture Content

Appendix A. Transmission Line Equation

The new approach for calculating dielectric constant in this project utilizes the transmission line equation (TLE). The following describes the basic theories and concepts of electromagnetics and the TLE.

Maxwell's Equations

In the study of electromagnetics, the four vector quantities called electromagnetic fields, which are functions of space and time, are involved: (2)

E= electric field strength (volts per meter, V/m)
D= electric flux density (coulombs per square meter, C/m2)
H= magnetic field strengths (amperes per meter, Am/m)
B= magnetic flux density (webers per square meter, wb/m2)

The fundamental theory of electromagnetic fields is based on Maxwell's equations governing the fields E, D, H, and B:

Equation 51. Equation. the curl of E equals minus the partial derivative of B with respect to t. (51)

Equation 52. Equation. the curl of H equals the sum of J and the partial derivative of D with respect to t. (52)

Equation 53. Equation. the divergence of B equals 0. (53)

Equation 54. Equation. the divergence of D equals rho sub v. (54)

Where:

J = electric current density (Am/m2)
?v = electric charge density (C/m3)

J and ?v are the sources generating the electromagnetic field. The equations express the physical laws governing the E, D, H, and B fields and the sources J and rv at every point in space and at all times.

In order to understand concepts of Maxwell's equations, some definitions and vector identities are described. The symbol ? in Maxwell's equations represents a vector partial-differentiation operator as following,

Equation 55. Equation. the gradient equals the sum of the partial derivative of unit vector x with respect to x, the partial derivative of unit vector y with respect to y, and the partial derivative of unit vector z with respect to z. (55)

Where = unit vectors along the x, y, and z axes

If A and B are vectors, the operation ? x A is called the curl of A, and the operation ? x B is called the divergence of B. The former is a vector and the latter is a scalar. In addition, if f (x, y, z) is a scalar function of the coordinates, the operation ?f is called the gradient of f. The operator as a vector is only permissible in rectangular coordinates.

Some useful vector identities are as follows: (12)

Equation 56.  Equation.  the curl of the product of curl of A is defined as the subtraction of the squared gradient multiplied by A from the divergence of the product of divergence of A. (56)

Equation 57.  Equation.  the divergence of the product of curl of A is defined as 0. (57)

Equation 58.  Equation.  the curl of the product of gradient multiplied by phi is defined as 0. (58)

Equation 59.  Equation.  the divergence of the product of A multiplied by B is defined as the subtraction of the product of A multiplied by the product of curl of B from the product of B multiplied by the product of curl of A. (59)

Where: Equation 60.  Equation.  Divergence squared is equal to delta squared over delta times x squared plus y hat times delta squared over delta y squared plus z hat times delta over delta times z squared. (60)

Conservation Law of Electric Charge

The Maxwell equation (55) can be presented using the vector identity (57) and multiplying both sides by ? as follows:

Equation 61.  Equation.  the sum of the divergence of J and the partial derivative of the divergence of D with respect to t equals the divergence of the product of curl of H, which equals 0. (61)

Being replaced with equation 54, the conservation law for current and charge densities is defined as the following:

Equation 62.  Equation.  the sum of the divergence of J and the partial derivative of rho sub V with respect to t equals 0. (62)

The conservation law means that the rate of transfer of electric charge out of any differential volume is equal to the rate of decrease of total electric charge in that volume. This law is also known as the continuity law of electric charge. In fact, to solve electromagnetic field problems, it is essential to assume that the sources J and rv are given and satisfy the continuity equation. (12)

Constitutive Relations

Constitutive relations can provide physical information for the environment in which electromagnetic fields occur, such as free space, water, or composite media. Also, they can characterize a simple medium mathematically with a permittivity, e, and a permeability, µ, as follows:

Equation 63.  Equation.  D equals epsilon multiplied by E. (63)

Equation 64.  Equation.  B equals mu multiplied by H. (64)

For free space such as air, µ = µ0 = 4p10-7 H/m and e = e0 = 8.85 x 10-12 F/m

Maxwell's Equations for Time-Harmonic Fields

Time-harmonic data is the large class of physical quantities that vary periodically with time. While physical quantities are usually described mathematically by real variables of space and time and by vector quantities, the time-harmonic real quantities are represented by complex variables. (12) A time-harmonic real physical quantity V(t) that varies sinusoidally with time can be expressed as follows:

Equation 65.  Equation.  V at t equals the product of V sub 0 multiplied by the cosine of the sum of omega multiplied by t and phi. (65)

Where:

V0= amplitude,
? = angular frequency ( = 2pf )
f= frequency of V(t)
t= time
f= phase of V(t)

Figure 33 illustrates V(t) as a function of time t.

Figure 33.  Graph.  Time-harmonic function V(t).  The graph shows the time-harmonic real physical quantity as a function of time.  Time is on the horizontal axis and the time-harmonic real physical quantity is on the vertical axis.  A sinusoidal line is on the graph.
Figure 33. Graph. Time-harmonic function V(t). (12)

The V(t) can be expressed by using the symbol of Re{ }, which means taking the real part of the quantity in the brace as follows:

Equation 66.  Equation.  V at t equals Re of the product of V multiplied by e to the power j, omega, and t, which equals Re of the product of V sub 0 multiplied by the product of e to the power j and phi multiplied by the product of e to the power j, omega, and t. (66)

Hence, the derivation with respect to time can be expressed as

Equation 67.  Equation.  the partial derivative of V at t with respect to t equals the product of minus omega multiplied by V sub 0 multiplied by the sine of the sum of omega multiplied by t and phi, which equals Re of the product of the following: j times omega times V sub 0 times e to the power j and phi times e to the power j, omega, and t. (67)

So,

Equation 68.  Equation.  the partial derivative of V at t with respect to t is true if the product of j multiplied by omega multiplied by V is true and the partial derivative of V at t with respect to t is false if the product of j multiplied by omega multiplied by V is false. (68)

As shown in equation 67, the time derivative Derivative slash derivativet can be replaced by j?in the complex representation of time-harmonic quantities.

Maxwell's equations can be expressed with respect to the complex representations for the time-harmonic quantities as follows:

Equation 69.  Equation.  the curl of E equals minus the partial derivative of B with respect to t, which equals minus the product of j multiplied by omega multiplied by B. (69)

Equation 70.  Equation.  the curl of H equals the sum of J and the partial derivative of D with respect to t, which equals the sum of J and the product of j multiplied by omega multiplied by D. (70)

Equation 71.  Equation.  the divergence of B equals 0. (71)

Equation 72.  Equation.  the divergence of D equals rho sub v. (72)

Uniform Plane Waves in Free space

Given that electromagnetic fields are generated in free space by source J and ?v in a localized region, then, for electromagnetic fields outside the region, J and ?v are equal to zero and Maxwell's equation can be expressed with free space constitutive relations of equations 63 and 64 as the following: (11)

Equation 73.  Equation.  the curl of E equals the product of minus j multiplied by omega multiplied by B, which equals the product of the following: minus j times omega times mu sub 0 times H. (73)

Equation 74.  Equation.  the curl of H equals the product of j multiplied by omega multiplied by D, which equals the product of the following: j times omega times epsilon sub 0 times E. (74)

Equation 75.  Equation.  the divergence of B equals the divergence of H, which equals 0. (75)

Equation 76.  Equation.  the divergence of D equals the divergence of E, which equals 0. (76)

By taking the curl of (73) and substituting (74), the following can be obtained:

Equation 77.  Equation.  the curl of the product of curl of E equals the product of squared omega multiplied by mu sub 0 multiplied by epsilon sub 0 multiplied by E. (77)

The wave equation for E can be obtained with regard to vector identity (56) and equation 74 as follows:

Equation 78.  Equation.  the sum of the product of squared gradient multiplied by E and the product of squared omega multiplied by mu sub 0 multiplied by epsilon sub 0 multiplied by E equals 0. (78)

The wave equation (78) is a vector second-order differential equation. The simple solution is expressed as follows;

Equation 79.  Equation.  E equals the product of the following: unit vector x times E sub 0 times e to the power minus j, k, and z. (79)

From equations 78 and 79, the following is obtained;

Equation 80.  Equation.  squared k equals the product of squared omega multiplied by mu sub 0 multiplied by epsilon sub 0. (80)

The magnetic field H of the wave can be determined from equation 73 or 74:

Equation 81.  Equation.  H equals the product of the following: unit vector y times the square root of the product of epsilon sub 0 divided by mu sub 0 times E sub 0 times e to the power minus j, k, and z. (81)

In equation 81, the factor Square root epsilon sub zero slash Mu sub zero is known as the intrinsic impedance of free space,

Equation 82.  Equation.  eta equals the square root of the product of mu sub 0 divided by epsilon sub 0. (82)

The wave has the electric field E in the Unit vector X-direction and the magnetic field H in the Unit vector Y-direction and propagates in the Unit vector Z-direction. Figure 34 shows the velocity of propagation with time in a sinusoidal wave.

Figure 34.  Graph. Electric field as a function of z direction at different times.  The graph shows the wave velocity of propagation with time in a sinusoidal wave.  The coordinate z is on the horizontal axis and the electric field of a uniform plane wave is on the vertical axis.  A sinusoidal line is on the graph.
Figure 34. Graph. Electric field as a function of z direction at different times.(12)

Therefore, the velocity of light in free space becomes:

Equation 83.  Equation.  v equals omega divided by k, which equals 1 divided by the square root of the product of mu sub 0 multiplied by epsilon sub 0. (83)

Where:

?= angular frequency
k= propagation constant

Transmission Line Equation of Coaxial Transmission Line

In the case that electromagnetic waves propagate in free space, the path of the wave is straight, and the intensity is uniform on the transverse plane. However, if the wave is guided along a curved and limited path, the wave is not uniform on the transverse plane and the intensity is limited to a finite cross-section. The finite structure transmitting electromagnetic waves is called a transmission line or waveguide. The wave can be transmitted along different types of waveguides: parallel-plate waveguides, rectangular waveguides, and coaxial lines. This study considers the coaxial lines, which are involved in TDR.

Coaxial Lines

The most commonly used transmission line to guide the electromagnetic wave is the coaxial line. The coaxial line consists of inner and outer conductors and an inner dielectric insulator. As shown in figure 35, a coaxial line has an inner conductor of radius, a, and an outer conductor of inner radius, b, insulated by a dielectric layer of permittivity, e. Figure 36 presents the cylindrical coordinate system for the solution inside coaxial lines.

Figure 35.  Diagram.  Coaxial line.  Sectional diagram of coaxial line with inner conductor of radius a and an outer conductor of inner radius b.  The three-dimensional Cartesian coordinates are on the diagram.
Figure 35. Diagram. Coaxial line. (11)

Figure 36.  Graph.  Cylindrical coordinate system.  The graph depicts the cylindrical coordinate system with x, y, and z axes.  Points A, B, and C are on x-y plane, the x-axis, and y-axis, respectively.  Rho is labeled on the graph as the length 0A.  Phi is labeled on the graph as the angle between 0A and the x-axis.  Z is labeled on the graph as the distance from the x-y plane.  Point A in space is represented by three coordinates of rho, phi, and Z.
Figure 36. Cylindrical coordinate system. (12)

In the cylindrical coordinate system, coordinate r is the distance from the z-axis or length 0A, f is the angle between 0A and the x-axis, and z represents the distance from the x-y plane. The three coordinates, r, f, and z represent the point P and are expressed in terms of unit vectors, Rho hut,Pie hut and Unit vector Z.

Transverse Electric and Magnetic (TEM) Mode in a Coaxial Line

In order to explain the fundamental mode on the coaxial line, it is necessary to consider the case where the inner radius, a, is close to the outer radius, b. When the coaxial line is cut along the x-y plane and unfolded into a parallel strip, the line can be illustrated as figure 33:

7

Figure 37.  Diagram.  Coaxial line developed into a parallel-plate waveguide.  The diagram shows the coaxial line which is cut along the x-axis and unfold into a parallel strip.
Figure 37. Diagram. Coaxial line developed into a parallel-plate waveguide. (12)

From Figure 37, it is realized that the wave has the electric field E in the Rho hut-direction and the magnetic field H in the Pie hut-direction and propagates in the Unit vector Z-direction. Therefore, E and H can be expressed as follows:

Equation 84.  Equation.  E equals e to the power minus j, k, and z multiplied by unit vector rho multiplied by the product of V sub 0 divided by rho. (84)

Equation 85.  Equation.  H equals e to the power minus j, k, and z multiplied by unit vector phi multiplied by the product of V sub 0 divided by the product of eta multiplied by rho. (85)

Where:

k= propagation constant, Propagation constant
?= intrinsic impedance, Intrinsic Impedance

Since the E and H are transverse to the direction of wave propagation, the set of equations 84 and 85 is called the transverse electromagnetic mode (TEM) of the coaxial line.

Transformation Rules for Transmission Lines

The following rules are for transforming the field quantities into network parameters. (12)

Rule 1.Equation 86.  Equation.  V at z equals alpha sub 1 multiplied by the integral over C sub t with respect to s multiplied by E. (86)

Where:

a1= proportional constant
Ct= integration path transverse to z

Rule 2.Equation 87.  Equation.  I at z equals alpha sub 2 multiplied by the integral around C sub zero with respect to s multiplied by H. (87)

Where:

a2= proportional constant
C0= closed contour of integration

The power relationship must hold:

Rule 3.Equation 88.  Equation.  One half Re multiplied by the product of V at z and I at z equals the integral over A with respect to A multiplied by the product of the unit vector of z, one half Re, E, and H. (88)

Where A = cross-sectional area of the line or waveguide

Transmission Line Equation

The electric and magnetic fields E and H for a coaxial line in the TEM mode are:

Equation 89.  Equation.  E sub rho equals e to the power minus j, k, and z multiplied by the product of V sub 0 divided by rho. (89)

Equation 90.  Equation.  H sub phi equals e to the power minus j, k, and z multiplied by the product of V sub 0 divided by the product of eta multiplied by rho. (90)

By applying the field equations to the transformation rule, the following equations can be defined as:

Equation 91.  Equation.  V at z equals alpha sub 1 multiplied by the integral from b to a with respect to rho multiplied by E sub rho, which equals the product of the following: alpha sub 1 times V sub 0 times the natural log of the product of b divided by a times e to the power minus j, k, and z. (91)

Equation 92.  Equation.  I at z equals alpha sub 2 multiplied by the contour integral around C sub 0 of rho with respect to phi multiplied by H sub phi, which equals the product of the following: alpha sub 2 times the product of 2 multiplied by pi multiplied by V sub 0 divided by eta times e to the power minus j, k, and z. (92)

Where:

a1, a2 = calibration constants
V0= applied voltage
a, b= inside and outside coaxial transmission line diameters (figure 13)

If the calibration constants are one (a1= a2 = 1), equations 91 and 92 become:

Equation 93.  Equation.  E sub rho equals the multiplication of the product of V at z divided by the natural log of the product of b divided by a by the product of 1 divided by rho. (93)

EEquation 94.  Equation.  H sub rho equals the multiplication of the product of I at z divided by the product of 2 multiplied by pi by the product of 1 divided by rho. (94)

Maxwell's equations for electric and magnetic fields can be cast in the standard form of TLEs in terms of voltage and current, V and I, by using cylindrical coordinates. Maxwell's two curl equations are defined as the following TLEs:

Equation 95.  Equation.  the derivative of V with respect to z equals the product of the following: minus j times omega times mu times the product of I divided by the product of 2 multiplied by pi times the natural log of the product of b divided by a. (95)

Equation 96.  Equation.  the derivative of I with respect to z equals the product of the following: minus j times omega times epsilon times the product of V divided by the natural log of the product of b divided by a times 2 times pi. (96)

By eliminating I from equation 95, a wave equation for the voltage V can be obtained as follows:

Equation 97.  Equation.  the squared derivative of V with respect to z equals the product of the following: minus squared omega times mu times epsilon times V. (97)

V has two solutions of epsilon superscript negative jomega square root mu epsilon and epsilon superscript positive jomega square root mu epsilon. Each solution has an integration constant as a multiplier. V can be expressed by introducing two constants, V+ and V-, as:

Equation 98.  Equation.  V equals the sum of V sub plus multiplied by the exponential to the power minus j, k, and z and V sub minus multiplied by the exponential to the power j, k, and z (98)

Where Kappa equals omega square root mu epsilon

The amplitude of V+ represents a wave traveling in the positive z-direction and the amplitude of V- represents a wave traveling in the negative z-direction.

 

< PreviousContentsNext >>

The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration