IDEAL TRANSMISSION-LINE FUNDAMENTALS
3.2 WAVE PROPAGATION ON LOSS-FREE TRANSMISSION LINES Transmission lines are designed to guide electromagnetic waves from one point
3.2.1 Electric and Magnetic Fields on a Transmission Line
On a transmission line, information is transferred from one component to the other by guiding electromagnetic energy from point A to point B. To gain an intuitive understanding of how a signal propagates on a transmission line, we
must understand how the electric and magneticfield patterns are distributed. To do this, wefirst derive the boundary conditions of the electricfield at the interface between a dielectric and a perfect conductor, which allows us to draw the electric field patterns for typical transmission line cross sections in a uniform dielectric.
Next, we explore how the electricfield behaves at a dielectric boundary, such as the interface between the air and the board material in a microstrip transmission line. Finally, we use the relationships for a TEM wave derived in Chapter 2 to obtain the magneticfield pattern from the electricfield.
When a voltage is applied between the signal conductor and the reference plane on one end of a transmission line, an electric field is established as described in Section 2.4. The voltage is calculated by the line integral of the electricfield between the signal conductor and the reference plane:
v = − b
a
E·dl (3-1)
As discussed in Section 2.3.2, once the electricfield is established, the properties of the magneticfield can be calculated using Faraday’s and Amp`ere’s laws (2-1) and (2-2). From Section 2.3.2, the relationship between the electric and magnetic fields was shown always to remain orthogonal:
ay
∂Ex
∂z = −µ∂Hy
∂t
(2-29)
ax
ε∂Ex
∂t = −∂Hy
∂z
(2-30) Note that since we are applying a voltage source to generate the electricfield, and voltage is defined in terms of the work done per unit charge, equation (2-58) implies that charges must be present on both the signal and the reference con- ductors:
W q
a→b
=v(b)−v(a)= − b a
E·dl (2-58)
Furthermore, Gauss’s law (∇ ·εE=ρ) states that the divergence of εE is nonzero and equal to the charge density, which implies that the sources of the electric field are the electrical charges. Equivalently, if the electric field terminates abruptly, the termination must be an electric charge. To understand how the electric field behaves at the interface between the dielectric and a conducting surface, we use the integral form of Gauss’s law to calculate the boundary conditions:
S
εE·ds=
V
ρdV =Qenc (2-59)
Dielectric (Region 1)
Conductor (Region 2) h
A
n n
Figure 3-3 Surface used to calculate the boundary conditions at the dielectric–
conductor interface.
First, we must choose a surface to integrate and compare the normal components ofεE on both sides of the conductor–dielectric boundary. A convenient surface is a cylinder, as depicted in Figure 3-3. Since we are observing the behavior of fields at the surface, the height (h) of the cylinder can be made infinitely small, reducing (2-59) to
S
εE·ds=
S
ε1E1·ds1+
S
ε2E2·ds2=(n·ε1E1)A+(−n·ε2E2)A=ρA (3-2) whereε1is the dielectric permittivity of the dielectric in region 1 andε2describes the dielectric permittivity of region 2, which in this example is a perfect con- ductor. To interpret this equation, recall the discussion in Section 2.7.1, where it was shown that the electricfield (E2 in this case) must be zero inside a perfect conductor. Setting E2=0 allows us to simplify (3-2) for the special case of a boundary between a dielectric and a perfect conductor:
n·εE=ρ C/m2 (3-3)
Equation (3-3) means that the electricfield must emanate normal from and ter- minate normal to the conductor surface. Since equations (2-29) and (2-30) say that the magneticfield must be orthogonal to the electric field, we can conclude that the magnetic field must be tangential to the conductor surface. These two rules make it easy to visualize electric and magneticfields on transmission-line structures with perfect electrical conductors. Simply draw the electricfield lines so that they are always perpendicular to the conductor surface, emanating from the high and terminating in the low potential conductor, and then draw the mag- netic field lines so that they are always perpendicular to the electric field lines
Electric field Magnetic field
(a)
(b)
Figure 3-4 Electric and magneticfield patterns for homogeneous dielectrics in (a) a stripline and (b) a microstrip.
er= 1 er> 1
Figure 3-5 How the electricfield behaves at a dielectric boundary.
and tangential to the conductor surfaces. If we assume that the voltage is applied with the positive value on the signal conductors as shown in Figure 3-4, we can draw the electric and magneticfields for various transmission-line structures assuming a homogeneous dielectric.
When the dielectric is not homogeneous, as is almost always the case with microstrip transmission lines, thefield lines are distorted as they cross dielectric boundaries. Figure 3-5 illustrates howthe electricfield lines are bent away from the normal to the dielectric boundary when the relative dielectric permittivity on the top half of the structures is smaller than that on the bottom.
To understand why thefield lines are distorted, we must examine how elec- trostatic fields behave at the boundaries between two dielectric regions. First,
we can recycle equation (3-2) to derive the boundary conditions of the normal components of the electricfield at the interface between two dielectrics:
(n·ε1E1)−(n·ε2E2)=ρ (3-4) If the surface charge density between the dielectric layers is assumed to be zero (usually, a valid assumption), the relationships between the electricfields in both regions is described by
n·ε1E1= n·ε2E2 (3-5) Equation (3-5) means thatthe normal component of the electricfield is not con- tinuous across a dielectric boundary.
In Section 2.7.2 it was mentioned that the tangential component of the electric field must remain continuous across a dielectric boundary. This can be shown with the integral form of Faraday’s law for the electrostatic case:
l
E·dl =0 (3-6)
If we integrate (3-6) around a closed differential contour that encompasses the dielectric boundary such as that shown in Figure 3-6, we can calculate the tan- gential components of the electricfield:
l
E·dl = b
a
E·dl+ c
b
E·dl+ d
c
E·dl+ a
d
E·dl (3-7) Since we are considering the behavior at the surface (h→0), segmentsda and bc can be eliminated. Furthermore, the tangential segments ab andcd are equal but opposite, which means that (3-7) can be simplified to
(E1t−E2t)l=0→E1t =E2t (3-8) Equation (3-8) means that the tangential components of the electricfield across a dielectric boundary must remain continuous.
Assume that an electric field E1 is incident on a boundary between two dielectrics as shown in Figure 3-7. The change in the orientation of the electric
a e1 e2
b
d c
∆l
∆h
Figure 3-6 Differential contour encompassing a dielectric boundary.
e1E1
e1 e2
q1
q2 e2E1
Figure 3-7 Change in the orientation of an electricflux line at a dielectric boundary.
flux lines across the interface can be calculated by using the boundary conditions for the normal and tangential components of the electric field derived in (3-5) and (3-8). Whenθ1=0, the boundary conditions of (3-5) apply. Consequently, we need a function ofθ1that will satisfy (3-5) when the electricfield is normal to the boundary. Since cos(0)=1, the following equation will satisfy the boundary conditions for the normal components of the electricfield:
ε1E1cosθ1=ε2E2cosθ2 (3-9) Similarly, when θ1=90◦, the boundary conditions of (3-8) apply, and since sin(90◦)=1, the following equation will satisfy the boundary conditions for the tangential components of the electricfield:
E1sinθ1=E2sinθ2 (3-10) IfE1is calculated from (3-10) and substituted into (3-9), the change in orientation that the electricfields experience at a dielectric boundary can be calculated:
θ2=tan−1 ε2
ε1tanθ1
(3-11) Equation (3-11) means that the field lines will be bent farther away from the normal to the dielectric interface in the medium with the higher permittivity.