3.9 z −dependent parameters
3.9.1 Solution based on propagators
We again consider an inhomogeneous transmission line on the section 0 < z < `. At z = 0 our line is connected to a homogeneous line z < 0 with characteristic impedance Z0. At z= ` we have connected a load impedance Zb. The inhomogeneous line is excited by an incident voltage wave V+(0). For 0 < z < ` we define V±(z) by
V+(z) V−(z)
= 1 2
1 Z(z) 1 −Z(z)
V(z) I(z)
= [A]
V(z) I(z)
We introduce the propagators g±(z) such that
V+(z) V−(z)
=
g+(z) g−(z)
V+(0) (3.48)
This relation is differentiated w.r.t. z d
dz
V+(z) V−(z)
= d dz
g+(z) g−(z)
V+(0) and then we utilize (3.45) and (3.48) and obtain
−[P (z)]
g+(z) g−(z)
V+(0) = d dz
g+(z) g−(z)
V+(0)
where [P (z)] = [A(z)][D(z)][A(z)]−1− d[A(z)]
dz [A(z)]−1. Thus the propagators satisfy the same equations as V±(z)
d dz
g+(z) g−(z)
=−[P (z)]
g+(z) g−(z)
(3.49) The boundary conditions for g±(z) are g+(0) = 1 and g−(`) = Γ(`)g+(`) = ZZb−Z(`)
b+Z(`)g+(`).
Problem 61
Problems in Chapter 3
3.1 a) The time-harmonic voltage is v(t) = V0cos(ωt + π/4). Determine the correspond-ing complex voltage V .
b)The complex voltage is V = V0(1− j)/√
2 where V0 is real. Determine v(t).
3.2 Draw the equivalent circuit for a two-port based on the a) impedance matrix
b) admittance matrix c) hybrid matrix
b) the inverse hybrid matrix
3.3 Derive the relation between [Z] and [H] for a general two-port.
3.4 Show that Z and Y are symmetric matrices for a reciprocal N -port.
3.5 Show that the impedance matrix for a lossless N -port is purely imaginary.
3.6 A lossy transmission line has the following data at 100 MHz:
Z0 = 50 Ω (real) α= 10−3m−1 β= 0,95π m−1 Determine L, C, R and G at 100 MHz.
3.7
t = 0
-+
-+
V0
0 ℓ x
Z0 R
R =
4
v2(t)
A lossless transmission line has the length `. The line has the characteristic impedance R and the phase speed vp. In the left end the line is connected to an ideal voltage source in series with a resistance 4R, see figure. The other end of the line is open.
That means that i(`, t) = 0. The voltage source gives the voltage v(t) = 0 t <0
v(t) = V0 t≥ 0
where V0 is constant voltage. We are interested in the voltage v2(t) in the right end of the line. The source and the resistor are connected to the line via short wires.
a) Determine v2(t) when t→ +∞.
b) Determine v2(`/2vp).
c) Determine v2(3`/2vp).
d) Determine v2(5`/2vp).
e) Determine v2(7`/2vp).
3.8 A lossy transmission line is terminated by a matched load. The voltage along the line is measured at two positions 20 m apart.The result was 2,8 V and 2,1 V.
a) Determine the attenuation constant α.
b) Determine the attenuation in dB/km.
3.9 A lossless transmission line has the length 30 m. The input impedance of the line was measured when the other end was shortened, and when it was open. When it was open the input impedance was j360 V/A and when it was shortened the impedance was j10 V/A. The wavelength along the line was larger than 1 km. Determine the characteristic impedance Z0 and the phase coefficient β.
3.10 An antenna with the purely resistive impedance 300 Ω is to be matched to a coaxial cable with the characteristic impedance 60 Ω. For this purpose a quarter wave transmission line is used. The quarter wave line consists of a coaxial cable with the relative permittivity = 2 between the conductors. Determine the length ` and the characteristic impedance Z0of the quarter wave line when the frequency is 200 MHz.
3.11 A lossless transmission line has the characteristic impedance Z0 = 60 Ω. One end is connected to a load resistance Rb= 180 Ω. Determine the reflection coefficient at the load Γ and the standing wave ratio SWR.
3.12 A lossless line with the characteristic impedance Z0 = 50 Ω has a resistive load Rb. The standing wave ratio is SWR= 3. Determine Rb.
3.13 A lossless transmission line with characteristic impedance Z0 = 60 Ω is terminated by a load impedance Zb = (60 + j60) Ω. The length of the line is λ/8, where λ is the wavelength along the line. Determine the input impedance of the line.
3.14
L
L R
R
t = 0 V0
Zb= R
Z=
Zb=R
+
-v(t)
a) A voltage source with output resistance R has the open circuit voltage V0. At time t = 0 the source is connected to a circuit that consists of three lossless transmission lines, see figure. The three transmission lines are identical and has the length L, the characteristic impedance Z = R and the phase speed vp. Determine the voltage v(t) over the load Zb1 for all times.
b) Assume that the three transmission lines are connected to a time harmonic volt-age source. The three lines can then be replaced by an equivalent impedance Zin. Determine Zin if the frequency is chosen such that each of the lines is a quarter of a wave length long.
Summary 63
3.15
Zb
z +
-+
-0
Z0
Z0
= j
`= λ (z)
V0 V
The figure depicts a lossless transmission line with an ideal voltage source and a purely reactive load. The line is one wavelength long. The reactance of the load equals the characteristic impedance of the line. Determine the z−values in the interval [0, 2π/β] for which the amplitude of v(z, t) has its maximum.
3.16 ` z
+
-0
+- Z(0) ,
Z0
V0 V0
= R
Zb= R(1 + j )
The figure shows a lossless line with the load impedance Zb = R(1 + j). The characteristic impedance of the line is R. It is possible to chose the length ` such that the input impedance Z(0) is real.
a) Determine the values of β` in the interval 0 < β` < π for which Z(0) is real.
b) Determine for each of the β`-value in a) the corresponding value of Z(0).
Summary of chapter 3
Transmission lines
Time domain line equations
−∂v(z, t)
∂z = Ri(z, t) + L∂i(z, t)
∂t
−∂i(z, t)
∂z = Gv(z, t) + C∂v(z, t)
∂t
Frequency domain line equations
−dV(z)
dz = (R + jωL)I(z)
−dI(z)
dz = (G + jωC)V (z)
Characteristic impedance
Z0 =
rL
C lossless line rR+ jωL
G+ jωC lossy line
Reflection coefficient at the load
Γ = vr(`, t)
vi(`, t) = RL− Z0
RL+ Z0
Input impedance
Z(0) =
Z0ZLcos(β`) + jZ0sin(β`)
Z0cos(β`) + jZLsin(β`), lossless line
Z0ZLcosh γ` + Z0sinh γ`
ZLsinh γ` + Z0cosh γ` lossy line
Chapter 4
Electromagnetic fields with a preferred direction
In this chapter we decompose an arbitrary vector field in the longitudinal component along the z−direction and the transverse vector in the x-y-plane. We apply this decomposition to the Maxwell equations and analyze the solutions to these equations for a geometry that is constant in the z−direction. The equations that are derived in this chapter form the basis for the following chapters.
From now on we use the time dependence e−iωt in contrast to the transmission line chapter where we used ejωt. The reason is that most literature on waveguides uses this convention.
4.1 Decomposition of vector fields
An arbitrary vector field F (r) can always be decomposed in two perpendicular vectors1. We let one component be along the z−axis and the other in the xy-plane. A similar decomposition is used for the ∇-operator. We introduce the notations:
∇ = ∇T + ˆz ∂
∂z
F (r) = FT(r) + ˆzFz(r)
where FT denotes the vector in the xy-plane. The two components of the vector F are uniquely determined and are obtained as
(Fz(r) = ˆz· F (r)
FT(r) = F (r)− ˆz (ˆz · F (r)) = F (r) − ˆzFz(r) = ˆz× (F (r) × ˆz)
In the last equality we used the BAC-CAB-rule, A× (B × C) = B(A · C) − C(A · B).
The z−component of a vector is called the longitudinal component and the xy-component the transverse component. We use a related decomposition of the position vector r
r = ˆxx + ˆyy + ˆzz = ρ + ˆzz
1The dependence of ω or t is not written explicitly in the argument of the fields in this chapter
65
Since∇T and ˆz are perpendicular we decompose the rotation of a vector as zˆ· (∇ × F (r)) = ˆz ·
∇T + ˆz ∂
∂z
× (FT(r) + ˆzFz(r))
= ˆz· (∇T × FT(r)) =−∇T · (ˆz × FT(r))
(4.1)
In the last equality we applied a cyclic permutation. The transverse component of the rotation is
∇ × F (r) − ˆz (ˆz · (∇ × F (r)))
=
∇T + ˆz ∂
∂z
× (FT(r) + ˆzFz(r))
− ˆz (ˆz · (∇T × FT(r)))
=∇T × ˆzFz(r) + ˆz ∂
∂z × FT(r) = ˆz× ∂
∂zFT(r)− ˆz × ∇TFz(r)
(4.2)
since∇T × FT(r) = ˆz (ˆz· (∇T × FT(r))).
The decompositions we have described so far are valid for all vector fields. We now apply them to the fields in the Maxwell equations.