Lecture 13
Multi-Junction Transmission
Lines, Duality Principle
13.1 Multi-Junction Transmission Lines
By concatenating sections of transmission lines of different characteristic impedances, a large
variety of devices such as resonators, filters, radiators, and matching networks can be formed.
We will start with a single junction transmission line first. A good reference for such problem
is the book by Collin [83], but much of the treatment here is not found in any textbooks.
13.1.1 Single-Junction Transmission Lines
Consider two transmission line connected at a single junction as shown in Figure 13.1. For
simplicity, we assume that the transmission line to the right is infinitely long so that there is no
reflected wave. And that the two transmission lines have different characteristic impedances,
Z01 and Z02.
121
122 Electromagnetic Field Theory
Figure 13.1: A single junction transmission line can be modeled by a equivalent transmission
line terminated in a load Zin2.
The impedance of the transmission line at junction 1 looking to the right,using the formula
from previously derived,1 is
Zin2 = Z02
1 + ΓL,∞e−2jβ2l2
1 − ΓL,∞e−2jβ2l2 = Z02 (13.1.1)
since no reflected wave exists, ΓL,∞ = 0, the above is just Z02. Transmission line 1 sees a
load of ZL = Zin2 = Z02 hooked to its end. The equivalent circuit is shown in Figure 13.1 as
well. Hence, we deduce that the reflection coefficient at junction 1 between line 1 and line 2,
using the knowledge from the previous lecture, is Γ12, and is given by
Γ12 = ZL − Z01
ZL + Z01
= Zin2 − Z01
Zin2 + Z01
= Z02 − Z01
Z02 + Z01
(13.1.2)
13.1.2 Two-Junction Transmission Lines
Now, we look at the two-junction case. To this end, we first look at when line 2 is terminated
by a load ZL at its end as shown in Figure 13.2
1We should always remember that the relations between the reflection coefficient Γ and the normalized
impedance Zn are Γ = Zn−1
Zn+1 and Zn = 1+Γ
1−Γ .
Multi-Junction Transmission Lines, Duality Principle 123
Figure 13.2: A single-junction transmission line with a load ZL at the far end of the second
line.
Then, using the formula derived in the previous lecture,
Zin2 = Z02
1 + Γ(−l2)
1 − Γ(−l2) = Z02
1 + ΓL2e−2jβ2l2
1 − ΓL2e−2jβ2l2 (13.1.3)
where we have used the fact that Γ(−l2) = ΓL2e−2jβ2l2 . It is to be noted that here, using
knowledge from the previous lecture, that
ΓL2 = ZL − Z02
ZL + Z02
(13.1.4)
Now, line 1 sees a load of Zin2 hooked at its end. The equivalent circuit is the same as
that shown in Figure 13.1. The generalized reflection coefficient at junction 1, which includes
all the reflection of waves from its right, is now
˜Γ12 = Zin2 − Z01
Zin2 + Z01
(13.1.5)
Substituting (13.1.3) into (13.1.5), we have
˜Γ12 = Z02( 1+Γ
1−Γ ) − Z01
Z02( 1+Γ
1−Γ ) + Z01
(13.1.6)
where Γ = ΓL2e−2jβ2l2 . The above can be rearranged to give
˜Γ12 = Z02(1 + Γ) − Z01(1 − Γ)
Z02(1 + Γ) + Z01(1 − Γ) (13.1.7)
Finally, by further rearranging terms, it can be shown that the above becomes
˜Γ12 = Γ12 + Γ
1 + Γ12Γ = Γ12 + ΓL2e−2jβ2l2
1 + Γ12ΓL2e−2jβ2l2 (13.1.8)
124 Electromagnetic Field Theory
where Γ12, the local reflection coefficient, is given by (13.1.2), and Γ = ΓL2e−2jβ2l2 is the
general reflection coefficient2 at z = −l2 due to the load ZL. In other words,
ΓL2 = ZL − Z02
ZL + Z02
(13.1.9)
Figure 13.3: A two-junction transmission line with a load ZL at the far end. The input
impedance looking in from the far left can be found recursively.
Figure 13.4: Different kinds of waveguides operating in different frequencies in power lines,
RF, microwave, and optics. (courtesy of Owen Casha.)
2We will use the term “general reflection coefficient” to mean the ratio between the amplitudes of the
left-traveling wave and the right-traveling wave on a transmission line.
Multi-Junction Transmission Lines, Duality Principle 125
Equation (13.1.8) is a powerful formula for multi-junction transmission lines. Imagine
now that we add another section of transmission line as shown in Figure 13.3. We can use
the aforementioned method to first find ˜Γ23, the generalized reflection coefficient at junction
2. Using formula (13.1.8), it is given by
˜Γ23 = Γ23 + ΓL3e−2jβ3l3
1 + Γ23ΓL3e−2jβ3l3 (13.1.10)
where ΓL3 is the load reflection coefficient due to the load ZL hooked to the end of transmission
line 3 as shown in Figure 13.3. Here, it is given as
ΓL3 = ZL − Z03
ZL + Z03
(13.1.11)
Given the knowledge of ˜Γ23, we can use (13.1.8) again to find the new ˜Γ12 at junction 1.
It is now
˜Γ12 = Γ12 + ˜Γ23e−2jβ2l2
1 + Γ12 ˜Γ23e−2jβ2l2
(13.1.12)
The equivalent circuit is again that shown in Figure 13.1. Therefore, we can use (13.1.8)
recursively to find the generalized reflection coefficient for a multi-junction transmission line.
Once the reflection coefficient is known, the impedance at that location can also be found.
For instance, at junction 1, the impedance is now given by
Zin2 = Z01
1 + ˜Γ12
1 − ˜Γ12
(13.1.13)
instead of (13.1.3). In the above, Z01 is used because the generalized reflection coefficient
˜Γ12 is the total reflection coefficient for an incident wave from transmission line 1 that is sent
toward the junction 1. Previously, Z02 was used in (13.1.3) because the reflection coefficients
in that equation was for an incident wave sent from transmission line 2.
If the incident wave were to have come from line 2, then one can write Zin2 as
Zin2 = Z02
1 + ˜Γ23e−2jβ2l2
1 − ˜Γ23e−2jβ2l2
(13.1.14)
With some algebraic manipulation, it can be shown that (13.1.13) are (13.1.14) identical. But
(13.1.13) is closer to an experimental scenario where one measures the reflection coefficient
by sending a wave from line 1 with no knowledge of what is to the right of junction 1.
Transmission lines can be made easily in microwave integrated circuit (MIC) by etching
or milling. A picture of a microstrip line waveguide or transmission line is shown in Figure
13.5.
126 Electromagnetic Field Theory
Figure 13.5: Schematic of a microstrip line with the signal line above, and a ground plane
below (left). A strip line with each strip carrying currents of opposite polarity (right). A
ground plane is not needed in this case.
13.1.3 Stray Capacitance and Inductance
Figure 13.6: A general microwave integrated circuit with different kinds of elements.
Multi-Junction Transmission Lines, Duality Principle 127
Figure 13.7: A generic microwave integrated circuit.
The junction between two transmission lines is not as simple as we have assumed. In the real
world, or in MIC, the waveguide junction has discontinuities in line width, or shape. This
can give rise to excess charge cumulation. Excess charge gives rise to excess electric field
which corresponds to excess electric stored energy. This can be modeled by stray or parasitic
capacitances.
Alternatively, there could be excess current flow that give rise to excess magnetic field.
Excess magnetic field gives rise to excess magnetic stored energy. This can be modeled by
stray or parasitic inductances. Hence, a junction can be approximated by a circuit model as
shown in Figure 13.8 to account for these effects. The Smith chart or the method we have
outlined above can still be used to solve for the input impedances of a transmission circuit
when these parasitic circuit elements are added.
Notice that when the frequency is zero or low, these stray capacitances and inductances
are negligible. But they are instrumental in modeling high frequency circuits.
128 Electromagnetic Field Theory
Figure 13.8: A junction between two microstrip lines can be modeled with a stray junction
capacitance and stray inductances. The capacitance is used to account for excess charges at
the junction, while the inductances model the excess current at the junction.
13.2 Duality Principle
Duality principle exploits the inherent symmetry of Maxwell’s equations. Once a set of E
and H has been found to solve Maxwell’s equations for a certain geometry, another set for a
similar geometry can be found by invoking this principle. Maxwell’s equations in the frequency
domain, including the fictitious magnetic sources, are
∇ × E(r, ω) = −jωB(r, ω) − M(r, ω) (13.2.1)
∇ × H(r, ω) = jωD(r, ω) + J(r, ω) (13.2.2)
∇ · B(r, ω) = %m(r, ω) (13.2.3)
∇ · D(r, ω) = %(r, ω) (13.2.4)
One way to make Maxwell’s equations invariant is to do the following substitution.
E → H, H → −E, D → B, B → −D (13.2.5)
M → −J, J → M, %m → %, % → %m (13.2.6)
The above swaps retain the right-hand rule for plane waves. When material media is included,
such that D = ε · E, B = μ · H, for anisotropic media, Maxwell’s equations become
∇ × E = −jωμ · H − M (13.2.7)
∇ × H = jωε · E + J (13.2.8)
∇ · μ · H = %m (13.2.9)
∇ · ε · E = % (13.2.10)
In addition to the above swaps, one need further to swap for material parameters, namely,
μ → ε, ε → μ (13.2.11)
Multi-Junction Transmission Lines, Duality Principle 129
13.2.1 Unusual Swaps
If one adopts swaps where seemingly the right-hand rule is not preserved, e.g.,
E → H, H → E, M → −J, J → −M, (13.2.12)
%m → −%, % → −%m, μ → −ε, ε → −μ (13.2.13)
The above swaps will leave Maxwell’s equations invariant, but when applied to a plane wave,
the right-hand rule seems violated.
The deeper reason is that solutions to Maxwell’s equations are not unique, since there is
a time-forward as well as a time-reverse solution. In the frequency domain, this shows up in
the choice of the sign of the k vector where in a plane wave k = ±ω√με. When one does
a swap of μ → −ε and ε → −μ, k is still indeterminate, and one can always choose a root
where the right-hand rule is retained.
13.2.2 Fictitious Magnetic Currents
Even though magnetic charges or monopoles do not exist, magnetic dipoles do. For instance,
a magnet can be regarded as a magnetic dipole. Also, it is believed that electrons have spins,
and these spins make electrons behave like tiny magnetic dipoles in the presence of a magnetic
field.
Also if we form electric current into a loop, it produces a magnetic field that looks like the
electric field of an electric dipole. This resembles a magnetic dipole field. Hence, a magnetic
dipole can be made using a small electric current loop (see Figure 13.9).
130 Electromagnetic Field Theory
Figure 13.9: Sketches of the electric field due to an electric dipole and the magnetic field due
to a electric current loop. The E and H fields have the same pattern, and can be described
by the same formula.
Because of these similarities, it is common to introduce fictitious magnetic charges and
magnetic currents into Maxwell’s equations. One can think that these magnetic charges
always occur in pair and together. Thus, they do not contradict the absence of magnetic
monopole.
The electric current loops can be connected in series to make a toroidal antenna as shown
in Figure 13.10. The toroidal antenna is used to drive a current in an electric dipole. Notice
that the toroidal antenna acts as the primary winding of a transformer circuit.
Multi-Junction Transmission Lines, Duality Principle 131
Figure 13.10: A toroidal antenna used to drive an electric current through a conducting cylin-
der of a dipole. One can think of them as the primary and secondary turns of a transformer
(courtesy of Q.S. Liu).
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