Lebanese University Faculty of
Engineering
Electromagnetism Research Assignment
Boundary reflection of uniform plane
waves and transmission lines
Presented By
Charbel E. Abi Daher
Presented To
Dr. Hadi Jardak
April 29, 2022
Abstract
Electromagnetic waves are the main means of wireless and wired telecommu-
nications and electric power transmission, without any of those, the modern day
human would be almost incapable of living as just about any essential service re-
lies upon one (or really, all) of these. For this reason we shall have a through
study of the reflection of transverse electromagnetic waves (TEM waves) on various
boundaries. We shall also establish the analogy between uniform plane waves and
waves propagating in transmission lines. And we finally study the meaning of the
standing wave ratio. Throughout all of that we shall try to derive the most general
equations that would be applicable in both lossy and lossless media without using
any approximations.
Contents
1 Preliminaries: Uniform Plane Waves 1
2 Normal incidence on a single boundary 1
2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 Existence of a reflected wave . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Reflection and transmission coefficients . . . . . . . . . . . . . . . . . 3
3 Normal incidence onto a perfect conductor and the standing wave 4
3.1 The standing wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Power transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 General case of normal incidence 6
4.1 Average power densities . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 The standing wave ratio . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 Transmission lines 8
5.1 Analogy with waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 Input impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Meaning of the standing wave ratio 10
6.1 In uniform plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.2 In transmission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
A Maxwell’s equations 13
List of Figures
1 Illustration of a transverse electromagnetic wave . . . . . . . . . . . 1
2 Single boundary incidence . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Integration surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4 Corrected case of single boundary normal incidence . . . . . . . . . . 3
5 Illustration of the oscillation of a standing wave with respect to time 5
6 Electric and magnetic field intensities, and electric current density
distributions in a lossy earth [1: p200] . . . . . . . . . . . . . . . . . 7
7 General representation of a transmission line . . . . . . . . . . . . . 8
8 Table summarizing the analogy between transmission lines and uni-
form plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
9 Different standing waves on a transmission line. SWR = 4, 2, 9. . . 10
1 Preliminaries: Uniform Plane Waves
The simplest form of an electromagnetic wave is a uniform plane wave which
propagates in a single given (fixed) direction. Such waves may propagate in many
modes, but we restrict our study in this document to transverse electromagnetic
waves (also known as TEM waves) where the electric and magnetic fields vary in
planes which are transverse (perpendicular) to the propagation direction.
Figure 1: Illustration of a transverse electromagnetic wave
The equations of these waves are found by solving Maxwell’s equations (Ap-
pendix A.) We shall recall them to be, in the time domain:
Ex = Ex0 e−αz cos(ωt − βz) (1)
Ex Ex0 e−αz
Hy = = cos(ωt − βz − θη ) (2)
η ηm
and in phasor notation:
n o
Exs = Re Ex0 e−γz = Re Ex0 e−αz e−jβz ejωt
(3)
Ex0 e−αz −j(βz+θη ) jωt
Exs
Hys = = Re e e (4)
η ηm
Where:
α : attenuation constant
β : phase constant
p
γ : propagation constant: γ = α + jβ = (σ + jωε)(jωµ)
η : intrinsic impedance, used as η = ηm ejθη
σ : conductivity of the medium
µ : permeability of the medium
ε : permitivity of the medium
2 Normal incidence on a single boundary
2.1 Boundary conditions
Let us consider a single boundary placed at z = 0 separating 2 random me-
dia and an incident TEM wave (in the same conventions defined in the previous
paragraph.)
1
Region 1 (z < 0) Region 2 (z > 0)
µ1 , ε1 , σ1 µ2 , ε2 , σ2
- -
E1+ , H1+ incident E2+ , H2+ transmitted
Figure 2: Single boundary incidence
For each of the electric and magnetic field, there are 2 boundary conditions: one
for the component that is normal to the boundary, and another for the component
that is tangential to the surface. However, since the boundary has the equation
→
− →
−
z = 0 and E is in the ⃗i direction and H is in the ⃗j direction, they both only have
a tangential component, so we end up with 2 boundary conditions.
Let’s first
I find the Zboundary condition for E, using Maxwell’s first equation we
− →
→ − → −
− →
have that E · dl = − ∂∂tB dS, and by taking the following surface:
S
Figure 3: Integration surface
And here, if we takeIthe thickness of the rectangle to be arbitrarily small then
− →
→ −
dS → 0 and we have E · dl = 0. Furthermore, the horizontal parts of the
boundary will have a very minimal contribution to dl and so we end up with the
boundary condition:
Et1 = Et2 (5)
This means that the electric field is continuous acrossIthe boundary. Z
− →
→ − −
→
∂D −
→
Similarly, we can use Maxwell’s second equation H · dl = I + − ∂t dS
S
while setting I = 0 (since there are no charges here) to arrive at the boundary
→
−
condition for H :
Ht1 = Ht2 (6)
2.2 Existence of a reflected wave
Now, taking a closer look at figure 2, and at the boundary conditions we just
derived, we notice something odd. By plugging equation (4) in equation (6) we get:
E1 E2
=
η1 η2
Which only holds in the case of η1 = η2 , meaning the 2 mediums are essentially
the same material, which is only the trivial case and not what we’re interested in
2
studying. This leads us to the conclusion that there must be a reflected wave in
medium 1 for the equations to be consistent. We shall call E1+ the incident electric
field of the wave, E1− that of the reflected wave, and E2+ that of the transmitted
wave, and do the same for H, as is shown in the following updated figure:
Region 1 (z < 0) Region 2 (z > 0)
µ1 , ε1 , σ1 µ2 , ε2 , σ2
-
E1+ , H1+ incident
-
E2+ , H2+ transmitted
E1− , H1− reflected
Figure 4: Corrected case of single boundary normal incidence
We denote now the new boundary conditions:
E1+ + E1− = E2+ (7)
H1+ + H1− = H2+ (8)
Where E1− is in algebraic value.
2.3 Reflection and transmission coefficients
We then turn to define two very important coefficients that we will use in our
calculations, the reflection coefficient
−
Ex10
η2 − η1
Γ= + = (9)
Ex10
η2 + η1
And the transmission coefficient:
+
Ex20
2η2
T= + = (10)
Ex10
η2 + η1
We also explicitly write the general equations of the electric and magnetic fields
that follow directly from equations (7) to (10), which shall be used in the remainder
of the document when necessary:
+
+ + −α1 z −jβ1 z Ex1
+ 0 −α1 z −jβ1 z
Exs1
= Ex1 e e Hys1
e= e (11)
0
η1
+
− + −α1 z +jβ1 z − ΓEx1 0 −α1 z +jβ1 z
Exs 1
= ΓE x10 e e and H ys1 = − e e (12)
η1
+
+ + −α2 z −jβ2 z TEx1 0 −α2 z −jβ2 z
Exs 2
= TE x10 e e H ys 2 = e e (13)
η2
And we note that while E1+ had the following equation Exs + = Re E + e−γz
1 x10
which means that it is a real-valued function, E1− has the equation Exs − = ΓE +
1 x10
which makes it a complex-valued function (because η may be complex and so Γ
may be complex too); physically, this indicates the possibility of the reflected wave
being phase-shifted with the incident wave.
3
3 Normal incidence onto a perfect conductor
and the standing wave
In order to better understand the effect of the reflected wave in normal incidence,
let’s first consider the case where a TEM wave traveling in a perfect dielectric
(σ1 = 0, γ1 = 0 + jβ1 ) hits the boundary between this dielectric and a perfect
conductor (σ2 → ∞.) We see from the expression of the skin depth in conductors:
1
δ=√ that when limσ→∞ δ = 0 and it follows that:
πf µσ
lim β2 = ∞
σ2 →∞
lim α2 = ∞
σ2 →∞
lim η2 = ∞
σ2 →∞
lim Ex2s = 0
σ2 →∞
lim Hx2s = 0
σ2 →∞
And so we conclude that time-varying fields cannot exist in perfect conductors.
This also yields Γ = −1 and T = 0 (another confirmation of the non-existence of a
transmitted wave.) Using all the previous information, we get the set of equations
for the incident and reflected waves:
+
+ + −jβ1 z + Ex1 0 −jβ1 z
Exs1
= Ex1 e Hys1
= e (14)
0
η1
E+
−
Exs 1
+ +jβ1 z
= −Ex1 e and −
Hys1
= x10 e+jβ1 z (15)
0
η1
+
Exs 2
=0 Hys2 =0 (16)
From here, we are interested in the total electric and magnetic field in medium
1, which are:
Exs1 = Ex+1 + Ex−1
+ −jβ1 z + +jβ1 z
= Ex1 0
e − Ex1 0
e
+ −jβ1 z +jβ1 z
= Ex10 e −e
+
= Ex1 (−2j) sin(β1 z)
0
Ex1 = Re Exs1 ejωt
+
= 2Ex10
sin(β1 z) sin(ωt) (17)
4
And:
Hys1 = Hy+1 + Hy−1
+ +
Ex1 0 −jβ1 z
Ex1 0 +jβ1 z
= e + e
η1 η1
+
Ex1
= 0
e−jβ1 z + e+jβ1 z
η1
+
2Ex1 0
= cos(β1 z)
η1
= Re Hys1 ejωt
Hy1
+
2Ex1 0
= cos(β1 z) cos(ωt) (18)
η1
3.1 The standing wave
We notice that the equations for the total electric and magnetic fields in medium
1 are different than the regular equations of traveling waves, this is because these
waves are of a different nature than the traveling waves, we call them standing
waves.
A standing wave is a wave whose peak amplitude does not change with respect
to space, it only chnages with respect to time, as shown in figure 5.
Figure 5: Illustration of the oscillation of a standing wave with respect to time
This means that we can, for example, find where this standing wave is equal to 0.
E1 = 0 ∀t
+
2Ex10
sin(β1 z) sin(ωt) = 0 ∀t
sin(β1 z) = 0
β1 z = nπ n ∈ N
nπ
z= n∈N
β1
We can also rewrite this relation with respect to to the wavelength of the wave:
2π nλ1
λ1 = →z= n∈N (19)
β1 2
5
3.2 Power transfer
The incident average power density will be:
1 −→ −−→ 1 E 2
x10
Pi = Re E1+ × H1+ =
2 2 η1
which corresponds to the formula of the average power density in a perfect dielectric
+ +
Since Ex2 and Hy2 are both equal to 0 in this case, the transmitted power will
be 0 as well.
The reflected average power density should then be equal to the average incident
power, let’s verify that:
1 1 −Ex10 E2
Pr = − E1− H1− = − Ex10 = x10
2 2 η1 2η1
4 General case of normal incidence
After getting our feet wet with the simplest case of normal incidence (medium 1
perfect dielectric, medium 2 perfect conductor), we can now study the general case
of single boundary normal incidence for any 2 media. We reprise equations (11),
and (12) to get the total electric field in medium 1:
+ −
Exs1 = Exs1
+ Exs 1
+ −α1 z −jβ1 z + −α1 z +jβ1 z
= Ex1 0
e e + ΓEx10
e e
Which we can rearrange to get the following form:
+ −α1 z −jβ1 z +2α1 z +j2β1 z
Exs1 = Ex1 e e 1 + Γe e (20)
| 0 {z }| {z }
traveling wave standing wave
We can see then that the total electric field in medium will be comprised of 2 parts:
a travelling wave and a standing wave. The same can be said for the magnetic field:
Hys1 = Hy+1 + Hy−1
+
Ex1 E+
= 0 −α1 z −jβ1 z
e e + x10 e−α1 z e+jβ1 z
η1 η1
+
E
= x10 e−α1 z e−jβ1 z 1 − Γe+2α1 z e+j2β1 z
η
| 1 {z }| {z }
traveling wave standing wave
6
Figure 6: Electric and magnetic field intensities, and electric current density distributions
in a lossy earth [1: p200]
4.1 Average power densities
The equations for the average power (incident, transferred, and reflected) are a
little more
→ interesting in this general case. Recalling that the average power density
1 − → −
is 2 Re E × H we have these relations:
( !)
+
1
+ −α1 z −jβ1 z
Ex1 0 −α1 z −jβ1 z
Pi = Re Ex10 e e × e e
2 η1
2
Ex1
0 −2α1 z
1
= e Re (21)
2 η1∗
Γ2 Ex12
0 +2α1 z
1
Pr = e Re (22)
2 η1∗
T2 Ex12
0 −2α2 z
1
Pt = e Re (23)
2 η2∗
4.2 The standing wave ratio
In order to talk about the 2 types of waves coexisting at the same time in the
same medium, we talk about the standing wave ratio:
Exs1 , max
SWR = (24)
Exs1 , min
7
We can get the value of each of these components by looking closer at equation (20)
and try to determine its magnitude:
|Exs1 |2 = Exs1 × Exs∗
1
−jβz
=e 1 + Γej2βz Ex10 e+jβz 1 + Γ∗ e−j2βz Ex1
∗
0
h n oi
= 1 + |Γ|2 + 2Re (Γej2βz ) |Ex10 |2
Since the third term has a modulus of 1, Re (Γej2βz ) ∈ (−Γ, Γ). This means that
the the maximum value of Exs1 will be (1 + Γ)Ex10 and the minimum value will be
(1 − Γ)Ex10 , giving us a final expression for the standing wave ratio:
1 + |Γ|
SW R = (25)
1 − |Γ|
5 Transmission lines
In most transmission lines, the electric and magnetic fields are transverse to the
direction of propagation, much like the waves we were studying in general media in
previous sections, as such, we can establish a direct analogy between the uniform
plane wave and transmission lines. Establishing this analogy is useful because then
the equations and relations derived for one type of problems can be applied to
the other. We will however, talk about voltages and currents when it comes to
transmission lines.
To establish this analogy, we will take a differential length of a generalized
representation of a random transmission line:
Figure 7: General representation of a transmission line
Where R, G, C, and L are dependent on the choice of material for the trans-
mission line and the type of line.
Then, from applying Ohm’s laws in the circuit1 , we can get (Z here being the
impedance):
dV
= −ZI (26)
dz
and
dI
= −Y V (27)
dz
1 ∆V dV
Remember here that we took z to e a differential length so ∆Z → dz
8
with Y being the admittance of the line. If we differentiate the first equation and
plug it into the second we get
d2 V
= γ2V (28)
dz 2
where √
γ = ZY (29)
The solution to this differential equation would then be:
V = V + e−γz + V − eγz (30)
This can be verified by replacing back in the initial differential equation.
We then can use the relation between V and I to get
1
I= (V + e−γz + V − eγz ) (31)
Z0
Where Z0 is the characteristic impedance of the circuit
r
Z Z
Z0 = = (32)
γ Y
γ being
√ the propagation constant in this case as well, we can re-write γ =
α + jβ = ZY , so we can rewrite the equation of V in terms of α and β:
V = V + e−γz + V − eγz = V + e−αz e−jβz + V − e−αz ejβz (33)
5.1 Analogy with waves
As we saw previously, the equation of the voltage resembles very closely the
equation of the electric field that we had derived earlier. And as we said before,
there exists an analogy between transmission lines and plane waves, so let us clarify
it in this section.
p Transmission lines Uniform
p plane waves
Propagation constant (R + jωL)(G
s + jωC) (σ +
sjωε)(jωµ)
R + jωL σ + jωε
Characteristic impedance Z= η=
G + jωC jωµ
dVs dExs
Voltage/Electric field = −(R + jωL)I = −jωµHys
dz dz
dIs dHys
Current/Magnetic field = −(G + jωC)V = −(σ + jωε)Exs
dz dz
Figure 8: Table summarizing the analogy between transmission lines and uniform plane
waves
We still have of course
V− ZL − Z0
Γ = reflection coefficient =
+
=
V ZL + Z0
When we add a load ZL to the end of the transmission line depicted in figure 7.
The transmission coefficient becomes
2ZL
τ=
ZL + Z0
And the standing wave ratio:
1 + |Γ|
SW R =
1 − |Γ|
9
5.2 Input impedance
To find the input impedance of the circuit we set:
V
Zi =
I at z=−l
V + e−γz + V − eγz
= 1 + −γz + V − eγz )
(34)
Z0 (V e z=−l
+ −γz − γz
V e +V e
= Z0 (35)
V + e−γz + V − eγz z=−l
1 + Γe−2γl
= Z0
1 − Γe−2γl
V−
With Γ still being the reflection coefficient Γ = . In terms of load impedance
V+
we can write (by substituting for Γ):
ZL cosh γl + Z0 sinh γl
Zi = Z0 (36)
Z0 cosh γl + ZL sinh γl
6 Meaning of the standing wave ratio
The standing wave ratio is a measure of how close the pair of traveling wave +
standing wave is close to being a pure standing wave.
Figure 9: Different standing waves on a transmission line. SWR = 4, 2, 9.
10
6.1 In uniform plane waves
From equation (25), and section 3, we know that in a perfect standing wave the
minimum value of the reflected electric field is 0. We deduce that when SW R → ∞,
we have a perfect standing wave. This happens when we hit a boundary between
a perfect dielectric and a perfect conductor and we have Γ = −1.
When we have Γ = 0, we have no reflection, this is when in reality we have no
boundary as η1 = η2 , or the boundary is perfect.
Of course, most of the time reality is very far from theoretical cases, and we
will have lossy materials on both sides of the boundary, in this case the SWR will
be in between 1 and ∞.
6.2 In transmission lines
L −Z0
In transmission lines, we have a new equation for the SWR: SW R = Z ZL +Z0 .
This shows a new meaning for the SWR: how close the load impedance is to the
line impedance.
This measure proves to actually be very important: since we want to maxi-
mize power transfer from the generating source to the consuming load, we want to
maximize the transmitted part of the waves and minimize the reflected part. This
means we want to get as close as possible to a perfect traveling wave, and we wish
to minimize the SWR. Furthermore, if we have a perfect standing wave it means
that the load cannot absorb any power as all of it is being reflected.
This means that in practical application, we aim to match the impedances of
the load and of the line.
11
References
[1] Balanis, C. A. (2012). Chapter 5: Reflection and Transmission. In Advanced
engineering electromagnetics. Hoboken, NJ: John Wiley & Sons.
[2] David Staelin. 6.013 Electromagnetics and Applications. Spring
2009. Massachusetts Institute of Technology: MIT OpenCourseWare,
https://ocw.mit.edu. License: Creative Commons BY-NC-SA.
[3] Ramo, S., Duzer, T. V., & Whinnery, J. R. (1993). Fields and waves in
communication electronics (3rd ed.). Wiley.
[4] Anant Agarwal. 6.002 Circuits and Electronics. Spring 2007. Massachusetts
Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. Li-
cense: Creative Commons BY-NC-SA.
12
A Maxwell’s equations
Z →
−
→
− −∂B − →
→ − ∂B − →
I
∇× E = ⇐⇒ E · dl = − dS (1)
∂t S ∂t
→
− Z →
−
→
− −∂ D − →
→ − ∂D −
→
I
∇ × H = J⃗ + ⇐⇒ H · dl = I + − dS (2)
∂t ∂t
Z S
→
− − −
→ →
I
∇ · D = ρv ⇐⇒ D · dS = ρ dv (3)
S V
→
− − −
→ →
I
∇· B =0 ⇐⇒ B · dS = 0 (4)
S
13
