Comparative Study of Passive Intermodulation Distortion in Wilkinson Power Dividers/Combiners and Branch Line Couplers
Eslam N. Mohamed Ayman G. Sobih Ayman M. ElTager Electronic Eng. Dpt., Electronic Eng. Dpt., Electronic Eng. Dpt.,
MTC, Cairo, Egypt.
MTC, Cairo, Egypt
.
MTC, Cairo, Egypt.
Eslamnasr388@gmail.com
Ayman_sobih@yahoo.co.uk
prof.ayman.eltager@ieee.org
Abstract
—
Passive intermodulation distortion manifests itself as a nonlinear mixing product in passive devices. The nonlinearity in microstrip lines (MSL) has a distributed nature and can be associated with the dielectric substrate and/or the printed conductor. In this paper, the nonlinearity modeling of uniform microstrip line is discussed. It is partitioned into short segments; each described by its equivalent RLCG circuit, and has been analysed using Harmonic Balance nonlinear simulation to evaluate PIM (passive intermodulation) effect of the microstrip line for different lengths and widths. The proposed PIM MSL model is compared to Shitvov's model and recent reported measurements, and verified using Xparameter simulation. The generalized PIM MSL model is applied in device level such as Wilkinson power divider (WPD) and branch line coupler to predict the PIM effect of each. Finally, a comparison between conventional WPD, inductive loaded WPD and branch line coupler, is conducted based on the introduced MSL model.
Keywords nonlinear distortion, passive intermodulation distrotion, nonlinear model of microstrip line, power divider, coupler, PIM3, Xparameter simulation.
I.
I
NTRODUCTION
PIM is known for its drawback effect on the performance of base stations used in the space, military and civil telecommunications [12]. The major disadvantages of PIM are: raising the noise floor, increasing the bit error rate, reducing the coverage area and blocking the receiver. PIM products resulting from nonlinear frequency mixing in passive devices usually occur in the reception band of the system. PIM is initially discovered as a product of nonlinear mixing on rusty metallic contacts [34]. A PIM phenomenon is further observed in
ferrite circulators [5], waveguide
and cable joints [6], duplexers [7], attenuators [8] and antennas [9]. PIM is usually confined in contact phenomena such as tunneling, thermionic emission and fritting, and noncontact phenomena such as ferromagnetic, thermal ionization and field emission. Recently, the electrothermal theory of PIM in antennas is studied in [10]. On the other hand, the PIM effect in printed MSL is studied in [11]. Furthermore, the telegraph equation is solved to build a PIM theoretical model in [12]. In this paper, a unit cell of nonlinear microstrip line (NLMSL) model of 50
Ω
MSL has been verified with the
theoretical model in [12]. Then a generalized model for different widths and lengths has been produced and applied in different devices such as Wilkinson power dividers/combiners and branch line couplers to predict their PIM performance. Moreover, Xparameter simulation is used to verify the proposed NLMSL model which is utilized in Wilkinson power dividers and branch line couplers. II.
T
HEORTICAL
B
ACKGROUND
The srcin of PIM nonlinearity could be from one of two sources; namely, microstrip conductor line and substrate dielectric material [13]. Therefore, one or more of the model elements shown in Fig. 1 should be nonlinear.
Figure 1. Nonlinear model of an infinitesimal length of a MSL [14].
Moreover, the nonlinearity in microstrip line is modelled by adding nonlinear parameter R
2
to the linear resistance, and the nonlinear resistance can be expressed as:
20 2
( )
R I R R I
(1) Where R
0
is the linear resistance and R
2
is the nonlinear coefficient. The linear resistance in telegraph equation is replaced by R (I) in equation (1). Assuming complete matching
at input and output ports, the telegraph equations can be solved to obtain the third order intermodulation distortion current as follows [12]:
2, 1 1,0 0, 12, 1
( X) ( (2 )X)2, 1,1( (2 x) 2 )
I X (1 )e ee
l l
(2)
57
2018 International Conference on High Performance Computing & Simulation
9781538678794/18/$31.00 ©2018 IEEEDOI 10.1109/HPCS.2018.00024
Where:
2 21,0 0, 120 1,0 0 0, 1 0 2, 1 2, 1 2, 1
3 and 32 ( ) ( ) ( )( 2 )
V V Z Z Z j
I
2,1, 1
Third order intermodulation current Z
0
(
ω
2,1
) The characteristics impedance of third IMD Z
0
(
ω
1,0
) The characteristics impedance at 1
st
Fund. tone Z
0
(
ω
0,1
) The characteristics impedance at 2
n
Fund. tone
α
The attenuation constant
β
2,1
The phase constant at third IMD
1,0
The propagation constant at 1
st
Fund. tone
0, 1
The propagation constant at 2
n
Fund. tone
2, 1
The propagation constant of third IMD V
q,p
The input voltage
l
The physical length of MSL The forward PIM current at the output port and the reverse PIM current at the input port can be obtained by substituting X= 0 and X =
l
in equation (2) for the forward and reverse PIM respectively.
2, 1
2 4
(0) (1 )
j l l
I
e
(3)
2, 1
2
( )
(1 )
( 1 )
l l
I l
e e
(4) The voltage U(X) and impedance expressed as:
( )1 0 and ( )
R j LdI U Z j C G dx G j C
1 2 22, 1[ (1 ) (1 )2, 1 2, 1,22, 12 ]
l l l U e e e j C Gq pl l e e
(5) Where R
0
, L, C and G are perunitlength parameters of the transmission line. Finally, forward PIM3 power P
forw
can be expressed as:
22, 1220 2, 1
( )( )2 ( )
forw
U x P x R Z
(6) This analytical PIM model (Shitvov
’s
model) has been verified using 50
Ω
microstrip line of 914 mm length realized
on TLG30 substrate with thickness of 0.76 mm, dielectric constant of 3, and tangent loss of 0.0026. The nonlinear parameter R
2
is
assumed to be 2.6x10
4
Ω
/A
2
.m, where this value has been obtained from curve fitting as reported in [12]. The PIM characteristics of the line can be checked by terminating it with lowPIM matched load and feeding it by two 44dBm carriers at frequencies 935 MHz and 960 MHz. The simulation results of Shitvov
’s
theoretical NLMSL model are compared to measurements in Fig. 5 of [15]. III.
N
ONLINEAR
M
ODELING OF
M
ICROSTRIP
L
INES
In fact, building of lossy MSL
can be achieved by cascading unit cells of RLCG equivalent circuit as shown in Fig. 2. This unit cell, as the smallest unit in the line, has an electrical length about 2° to verify the assumption of infinitesimal TL section which followed in the telegraph equations derived by Shitvov. The RLCG parameters of the unit cell equivalent circuit can be defined as follows:
1 0 0
R R x
Ω
/m
(7)
0 0
R x L
H/m
(8)
0 02 20 0
R xG R x
S/m
(9)
0 02 20 0
( )
R xC R x
F/m
(10) Where;
R
0
and
x
0
are the real and imaginary parts of the impedance. After that, adding a nonlinear resistor RM1 that represents a weak nonlinear conductor, different lengths of MSL can be represented by connecting cascaded unit cells. This modeling technique can be generalized for different substrates after retrieving their nonlinear resistors from the experimental data reported if PIM measurement facilities are available. A length of uniform NLMSL has been analyzed using the Harmonic Balance solver. The nonlinear resistance parameter R
2
, reported from the experimental data, is assumed to be current dependent, but the nonlinear parameter RM1in ADS (Keysight Advanced Design System) is voltage dependent. Therefore, it is necessary to map the nonlinear parameter from current dependent to be voltage dependent. The nonlinear resistor RM1 can be expressed by a nonlinear polynomial function as:
1 2
12
1(1 )
R RM C V C V
(11) Where, C
1
and C
2
are the coefficients of the nonlinear resistor RM1 for the 2
nd
and 3
rd
order of the intermodulation voltage respectively. Thus, the third order nonlinear coefficient can be expressed
as:
22 31
32
R l C R
(12) Where: R
2
The nonlinear resistor parameter in (
Ω
/A
.m) R
1
The linear resistor
for unit cell in (
Ω
)
A 50
Ω
NMSL of electrical length 2° is designed on TLG
30 substrate with nonlinear parameter R
2
=2.6x10
4
Ω
/A
2
.m, the RLCG equivalent circuit parameters and nonlinear coefficient C
2
are calculated to be; R
1
=0.0047
Ω
C=0.115 pF L=0.29 nH G=1.9
S
C
2
=3.756 V

Figure 2. An equivalent circuit of a unit cell of NTL.
The desired length of NLMSL can be obtained by cascading a series of 2° unit cells as illustrated in Fig. 3.
58
Figure 3. Cascaded unit cells for different lengths of nonlinear MSL.
Fig. 4 illustrates a comparison between the analytical model based on Shitvov
’s
model and this proposed NLMSL simulation model designed on TLG30 substrate, and a very good agreement is obtained between them. Therefore, the proposed NLMSL model is verified for different lengths.
Figure 4. Analytical versus NLMSL simulation models with different lengths
of 50
Ω
line.
The microstrip line is a basic structure in building passive RF devices/circuits. These devices have microstrip lines with different lengths and widths. Therefore, the effect of microstrip line with variable width should be discussed using the proposed passive intermodulation distortion model. The variation of line width changes the surface current density and hence, the nonlinear resistor parameter is changed. This change can be expressed as [16]:
223
eff
RW
(13)
4 2 2ln(2) (1 ln(4 ))
eff r
h h wW wh
(14) Where
ρ
2
,
W
eff
,
w
,
h
and
ε
r
are nonlinear surface resistivity, effective width of the MSL, the strip width, height and the dielectric constant of the substrate respectively. The nonlinear polynomial coefficient, C
2
, also changes with the strip width depending on R
2
as illustrated in equation (12). Working on different substrate such as TLX9 with dielectric constant of 2.5, tangent loss of 0.0019, and copper thickness of
35
μ
m, its nonlinear surface resistivity
ρ
2
has been found to be 10
11
Ω
.A
2
.m as reported in [16]. On this substrate, different strip lines with different impedances (strip widths) have been designed and their RLCG equivalent circuits have been obtained after calculating the corresponding nonlinear parameter R
2
using equation (13).The PIM performance of the designed lines has been checked using the Harmonic Balance solver with two 43 dBm tones of frequencies 935 MHz and 960 MHz for 1m physical length consisting of cascaded 2° unit cells. This 1m length is chosen to be high enough to obtain considerable nonlinearity. Then it could be easily measured with practical dynamic range setup. Moreover, the PIM3 level of these lines has been calculated using the analytical model and all results are listed in Table I. From these results it can be deduced that the PIM level increases with larger impedances (smaller line width) and vice versa.
TABLE I. PIM
L
EVEL
D
UE TO
V
ARIATIONS IN
I
MPEDANCE
W
IDTH
w
(mm)
W
eff
(mm) R
2
(
Ω
/A
2
.m) C
2 (V2)
Z
(
Ω
)
Shitvov’s
model (dBm) Proposed model (dBm)
2.22 5.04 1.05×10
5
0.17 75.3 115.1 115.1 4.43 7.44 3.3×10
6
0.18 50 118.1 118.2 6.65 9.8 1.4×10
6
0.17 37.9 120.5 120.7 8.86 12.1 7.5×10
7
0.18 30.6 122.3 122.3 11.1 14.4 4.5×10
7
0.2 25.7 123.8 123.5 13.3 16.7 2.9×10

0.16 22.2 125.1 124.4
A very good agreement has been obtained between the results of the proposed NLMSL model and the analytical model for different impedance lines as shown in Fig.5. As a result this verifies the proposed NLMSL simulation model.
Figure 5. Analytical versus NLMSL simulation model with different width relative to that of
50
Ω
.
IV.
P
IM IN
D
EVICE
L
EVEL
The microstrip line is a nonlinear structure in high power applications, and it is the basic building block of many passive devices such as dividers/combiners, couplers, and filters. The PIM distortion effect in filters is studied in [14]. On the other hand, in this Section, the generalized NLMSL model is used to analyze and check the PIM performance of the most commonly used devices; the Wilkinson power divider (WPD) and branch line coupler (BLC).
A.
PIM performance of conventional WPD
A conventional WPD has been designed at 900 MHz with 200 MHz bandwidth on TLG30 substrate using the proposed NLMSL model. Its main parts, the 50
Ω
MSL and 70.7

Ω
MSL
of electrical lengths 90°, have been built by cascading unit cells of 2° electrical length as shown in Fig. 6. Fig.6 (b) shows 9 unit cells that are modeling each block in Fig.6 (a).Therefore, 2°x 9x 5 = 90° section. Fig.6 (c) is built after calculating the RLCG equivalent circuit and its nonlinear parameter R
2
.Assuming the 100
Ω
lumped resistor is linear
component at high power, the PIM performance of the device has been checked using two 44 dBm tones of frequencies 935
59
MHz and 960 MHz with PIM3 frequency of 910 MHz (2f
1
–
f
2
). The simulation has been performed using the harmonic balance and the results are illustrated in Fig. 7 from which it can be seen that about 100.7 dBm of PIM power is predicted for the conventional WPD.
Figure 6(a).The WPD with whole NLMSL model. Figure 6(b).The model of each block in fig. 6(a). Figure 6(c).The model of unit cell in fig. 6(b). Figure 7Simulated PIM performance of conventional WPD.
B.
Approximate Analysis of PIM in WPD
The total passive intermodulation distortion at output port is usually a combination of all nonlinearities distributed in the device from input to output as shown in Fig.8.
Figure 8. Two cascaded nonlinear sources.
Therefore, the total forward PIM3 at each output port can be roughly estimated as:
1 21
N forwardPIM PIMn PIM PIM n
P P P P
(15)
22, 1 2
PIM
P I R
(16) Where
P
PIM
is intermodulation power generated at each nonlinearity source. In the conventional WPD as shown in Fig. 9, the main sources of PIM are its 50
Ω
and 70.7

Ω
nonlinear
quarter wavelength MSLs. The nonlinear resistor, R
2
, is calculated using, equation (13), for the different impedance sections to be 2.6x10
4
Ω
/A
2
.m and 7.21x10
4
Ω
/A
2
.m for 50
Ω
and 70.7
Ω
respectively.
The third intermodulation current, I
2,1
is calculated using equation (4) to be 3.8 mA
3
/
Ω
and 0.56 mA
3
/
Ω
for the 50
Ω
and 70.7
Ω
respectively.
The PIM at output ports of WPD are composed of the total PIM generated by 50
Ω
and PIM
generated by 70.7
Ω
. Therefore, one can calculate each PIM source individually based upon the previously obtained parameters from equation (4).
Figure 9. Conventional WPD structure.
The P
IM of 50
Ω
can be calculated as:
2 2 21 2, 1,50 50 50 2,50
. . .
PIM
P I Z l R
(
17
)
Where:
2, 1,50
I
Third intermodulation distortion for 50
Ω
50
Z
characteristic impedance at third IMD
50
l
Physical length of 50
Ω
impedance
2,50
R
Nonlinear parameter for 50
Ω
The PIM of
70.7
Ω
can be calculated as:
2 2 22 2, 1,70.7 70.7 70.7 2,70.7
. . .
PIM
P I Z l R
(
18
)
The P
PIM1
is suffering 3dB power division before going out to the output port. Thus, the total PIM at each output port can be calculated as:
1100.91 22
P P P dBm PIM PIM forwardPIM
(
19
)
Though this 100.9 dBm value is considered as rough estimate, it is very close to the simulation result which verifies the proposed generalized model.
60
C.
SParameter of WPD using NLMSL model
A Convectional WPD has been designed at 900 MHz operating frequency with the following specifications; 200 MHz bandwidth, return loss less than 10 dB, excess insertion loss less than 0.2 dB and isolation between output ports lower than 20 dB. The WPD was built based on series cascading RLCG equivalent circuit of 2° electrical length unit cells for 50
Ω
and 70.7
Ω
. Moreover, S
parameter simulation is conducted to check that the performance of WPD is not affected by the nonlinear model. The simulation results are depicted in Fig. 10 with excess insertion loss about 0.12 dB within the band of operation, from 0.8 to 1 GHz. Fig.11 shows perfect matching at all ports with return loss better than 20 dB over the whole band. Finally, Fig.12 shows excellent isolation between the output ports with almost 25 dB within the band. Therefore, the Sparameters verify the operation of the conventional WPD as well as the modeling methodology.
Figure 10. Insertion loss of WPD. Figure 11. Return loss of WPD.
D.
XParameter verification of the WPD NLMSLmodel
As it has been discussed in previous sections, the WPD has a nonlinearity considered in design. Therefore, Sparameter is not adequate to represent WPD characteristics due to nonlinearity. Hence, an Xparameter model representing the linear and nonlinear behavior of RF components is used. X parameter can be generated in one of two ways, either from a nonlinear simulation of a circuitlevel design or through actual measurements using Nonlinear Vector Network Analyzer (NVNA). To verify the proposed NLMSL model, X parameters are generated for the WPD nonlinear model as shown in Fig.13. Then nonlinear simulation is developed for the generated Xparameters using two tones harmonic balance simulation as illustrated in Fig.14.
Figure 12. Isolation of WPD. Figure 13. Xparameter simulation of WPD. Figure 14. Harmonic balance simulation of WPD Xparameter.
Finally, a comparison between harmonic balance simulations for nonlinear WPD generalized model and the X parameter generated model is made. Fig.15 illustrates that the third intermodulation distortion for generated Xparameter
61