Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 699
The Characteristics of Parallel-connected Transmission
Lines
Jan-Dong Tseng
National Chin Yi Institute of Technology, Taiwan
Abstract
The characteristics of two uncoupled parallel-connected trans mission lines in are investigated by transmission line method analysis and circuit layout experiments. By adequate setting the characteristic impedances values and the electrical length of two transmission lines, the band pass or band rejection phenomenon will achieve at the designed frequency point. The validity of the derived results from transmission line method is verified by the experimental results. The numerical and the experimental results show a good agreement in the testing frequency range.
Introduction
Parallel-connected transmission lines networks are widely used in acoustic, microwave and optical circuits for their frequency dependent characteristics and, therefore, can be used as resonators, phase shifters, baluns, filters, and oscillators [1-8].
In this research, the electrical characteristics of uncoupled parallel-connected lines are analyzed by using symmetrical circuit analysis. By setting the scattering parameters values we could obtain the required band pass and band reject properties. In addition, based on the derived results from transmission line method, two circuit designs, band pass and band reject, are performed and simulated by microwave circuit simulator, IE3D.
The band pass and band reject circuits are also fabricated to verify the validity of the derived results.
Circuit Analysis
The schematic diagram of two uncoupled parallel connected transmission lines is shown in Fig.1(a). It consists of two transmission lines, with the characteristic impedance, Z
1 and Z
2
, and the electric length, θ
1 and θ
2
, respectively. Two transmission lines are connected together at each end as input and output ports and these two ports have the characteristic impedance as Zo . The parallel-connected transmission lines show a symmetrical structure and the analysis therefore performs by using the even mode and odd mode analysis to get the eigenvalues of the even and odd mode symmetrical circuit.
Figure 1: Schematic diagram of (a) Parallel-connected transmission line, (b) Odd mode circuit, (c) Even mode circuit.
For even mode condition, the total load admittance is the sum of input admittances of two transmission lines.
Y O.C
L
= jY
2 tan
θ
2
2
+ jY
1 tan
θ
1
2
(1)

700 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
The eignvalue of even mode has the relation with Y o , and Y O.C
L and is shown as follows
S
1
=
Z o.c
L
Z o.c
L
− Z o
+ Z o
=
Y
1 o.c
L
Y
1 o.c
L
−
+
1
Y o
1
Y o
=
Y o
− Y o.c
L
Y o
+ Y o.c
L
=
Y o
− jY
2 tan
Y o
+ jY
2 tan
θ
2
2
θ
2
2
N = tan
θ
1
2 then the expression of S
1 has the simple form
− jY
1 tan
+ jY
1 tan
θ
1
2
θ
1
2
(2) let M = tan
θ
2
2
S
1
=
Y o
− jY
2
M − jY
1
N
Y o
+ jY
2
M + jY
1
N
(3)
In the similar way, the total load admittance of the odd mode is the sum of input admittances of two transmission lines and has the form as
Y s.c
L
= − jY
2 cot
θ
2
2
− jY
1 cot
θ
1
2
The eignvalue of odd mode has the relation with Y o , and Y S.C
L and is shown as follows
(4)
S
2
=
Y o
− Y s.c
L
Y o
+ Y s.c
L
=
Y o
Y o
+ jY
2 cot
− jY
2 cot
θ
2
2
θ
2
2
+ jY
1 cot
− jY
1 cot
θ
1
2
θ
1
2
=
Y o
Y o
+ jY
2
− jY
2
1
M
1
M
+ jY
1
− jY
1
1
N
1
N
(5)
The scattering parameter S
11
, S
21 is the half value of the sum and the difference of S
1 and their expressions are shown as follows and S
2
, respectively,
S
11
=
1
2
( S
1
+ S
2
) =
Y
0
2 + jY o
Y
2
( M
Y
0
2
−
1
M
− ( Y
2
2
) +
+ jY o
Y
1
Y
2
M
N
Y
1
( N −
+ Y
1
Y
2
1
N
N
M
) + Y
2
2
+ Y
+ Y
1
2
1
2 )
+ Y
1
Y
2
(
N
M
+
M
N
)
S
21
=
1
2
( S
1
− S
2
) =
Y o
2 + Y
1
2 + Y
2
2
− j { Y o
Y
2
( M +
+ Y
1
Y
2
( M
N
+ N
M
1
M
) +
) + Y o
Y
1
( N + j { Y o
Y
2
( M −
1
N
1
M
) }
) + Y o
Y
1
( N − 1
N
) }
(6)
(7)
The Conditions of Band Pass and Band Rejection
The band pass and band reject conditions can obtain by setting the scattering parameters S
11 and S
21 to be zero value. From the scattering parameters we have just obtained, the condition of band pass is letting the numerator of eq. (6) be zero.
Y o
2
− Y
1
2
− Y
2
2
− Y
1
Y
2
(
M
N
+
N
M
) = 0 (8)
In eq. (8), except the admittances Y o , Y
1
, and Y
2 we have two values M and N which are the tangent value of the electrical length of two transmission lines. In order to get more specific band band conditions, some special cases of M, N are considered.
(1). Let M = N = 1, then M
N
The electrical lengths satisfy the above expression are
The numerator now becomes
+
Y o
2 tan tan
−
θ
2
2
θ 1
1
Y
1
2
= 1
− Y
2
2
θ
1
2
=
π
4
+
π
2 n and
θ
2
2
=
π
4
+
π
2 n
− 2 Y
1
Y
2
, the condition for band pass is Y
.
o
2 − ( Y
1
+ Y
2
) 2 = 0 or
Y o
= Y
1
+ Y
2
.
(2) Let M = 1 and N = − 1, then M
N
+ tan tan
θ 2
2
θ
1
1
The electrical lengths satisfy this condition are
=
θ
2
2
− 1
+
θ
1
2
= nπ . The numerator now becomes
2 Y
1
Y
2 and we have either Y o
= Y
1
− Y
2 or Y o
= Y
2
− Y
1 for the band pass conditions.
Y o
2 = Y
1
2 − Y
2
2 −
(3) General case. When M and N are not given, the numerator of S
11 is equal to zero for the band pass condition, then Y o
2 = Y
1
2 + Y
2
2 + Y
1
Y
2
(
M
N
+
N
M
) or Y o
2 = ( Y
1
+
M
N
Y
2
)( Y
1
+
N
M
Y
2
) or Y o
2 = ( Y
The band reject condition is S
21
= 0 and the numerator of S
21 set to zero will lead to
2
+
M
N
Y
1
)( Y
2
+
N
M
Y
1
).
Y o
Y
2
( M +
1
M
) + Y o
Y
1
( N +
1
N
) = 0
Some special cases of M and N are considered for the band reject condition.
(1). M=1 and N=-1. The electrical lengths of two transmission lines are θ
In this condition the admittances of two transmission lines should be equal, Y
1
2
= 2 nπ −
− Y
1
= 0.
π
2
(9) and θ
2
= 2 mπ + π
2
.

Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 701
(2). General case. M and N are not specified then equation (9) can further manipulate and has the form as
−
Y
2
Y
1
=
N +
M +
1
N
1
M
= tan tan
θ
1
2
θ
2
2
+ cot
+ cot
θ
1
2
θ
2
2
=
2 sin θ
1
2 sin θ
2
= sin θ
2 sin θ
1
(10)
The range selection of 2 θ
1 i.e. [0 , π ], [ π, 2 π ], [2 π, 3 π ], [3 π, 4 π ], [4 π, 5 π ]. Furthermore, the magnitude of Y
2 relation and 2 θ
2 should be in different regions to ensure their values having opposite sign, and Y
1 has to have the following
Y
2
= | sin θ
2 sin θ
1
| Y
1
(11)
Numerical and Experimental Results
Based on the above derived results, two circuits, band pass and band reject, are designed at center frequency of 2.45GHz. The band pass circuit is designed by given Z
1
= 50Ω , θ
1
= 270 ◦ @2 .
45 GHz ; Z
2
= 25Ω , θ
2
=
90
90
◦ @2 .
45 GHz . The frequency response from 0 to 5 GHz of the designed circuit and the layout are shown in
Fig. 3(a) and 3(b). The band reject circuit designed by given Z
1
= 100Ω , θ
1
= 270
◦
@2 .
45 GHz ; Z
2
= 100 W, θ
2
=
◦
@2 .
45 GHz . The frequency response from 1 to 5 GHz of the designed circuit and the layout are shown in
Fig.4(a) and Fig.4(b).The circuits are fabricated on FR4 substrate with 1.6mm thickness. The simulation of both band pass and band reject circuits are performed by microwave circuit simulator IE3D.The simulated and experimental results show a good agreement within the testing frequency range.
Figure 3: Band pass characteristics (a)simulation and experimental frequency response from 0 to 5GHz, (b)
The circuit layout ( Z
1
= 50Ω, θ
1
= 270 ◦ @2 .
45 GHz ; Z
2
= 25Ω , q 2 = 90 ◦ @2 .
45 GHz )
Figure 4: Band stop characteristics (a)simulation and experimental frequency response from 1 to 5 GHz, (b)
The circuit layout ( Z
1
= 100Ω, θ
1
= 270 ◦ @2 .
45 GHz ; Z
2
= 100Ω , θ
2
= 90 ◦ @2 .
45 GHz )
Conclusion
The characteristics of parallel-connected transmission lines are obtained by symmetrical circuit analysis.
By properly adjusting the electrical lengths and the characteristic impedances of the transmission lines, the

702 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 frequency characteristic of band pass or band stop have obtained. Two different circuits, band pass and band reject, designed at the center frequency of 2.45GHz are verified by experiment.
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