Design of Reduced Size Power Divider for Lower RF Band using

Periodically Loaded Slow Wave Structure

Karun Rawat, F.M Ghannouchi

iRadio Lab., Department of Electrical and Computer Engineering, Schulich School of Engineering,

University of Calgary, Calgary, AB, T2N1N4, Canada

Abstract — In conventional Wilkinson Power divider the

quarter wave transformers are used to match the split ports to

the common port. These are realized by 90° transmission lines, which are quite large especially at lower RF range of frequencies. This paper reports the design and development of a reduced size

two-way Wilkinson power divider at 850 MHz and 620 MHz. Approximately, 25%-35% length reduction is achieved by loading transmission line by capacitance at fixed intervals, which

reduces the phase velocity and hence physical length at the desired frequency of operation. The required dimensions of the periodically loaded line are calculated using basic transmission

line theory. The paper describes the method of theoretical calculation of the factor by which phase velocity is reduced at the desired frequency of operation. The EM simulation and

experimental results validates the design approach. The circuit is realized with microstrip technology and hence can easily be fabricated using conventional printed circuit processes.

Index Terms — Slow wave structure, phase velocity, Wilkinson power divider, k0/β ratio.

I. INTRODUCTION

Power dividers find many uses in microwave and Radio

frequency application [1], [2], [3]. Several circuits are

proposed for power division depending upon the performance

required. The matched symmetrical n-way power divider,

proposed by E.J Wilkinson [4], is one of the topology used for

equal amplitude equal phase power division. It is well known

that a reciprocal three port T- junction cannot be matched at

all the three port simultaneously [5]. Wilkinson proposed the

basic configuration of such matched power divider with all the

three ports simultaneously matched by introducing a resistor

between the two output ports, improving the isolation between

them. A two-way matched power divider based on Wilkinson

topology utilizes quarter wave transformer to match the split

lines with the input port. These quarter wavelength lines are

quite large and hence the circuit occupies large space when the

frequency of operation is less as in radio frequency ranges.

There are several miniaturization techniques reported [6]-[10]

for such kind of circuits. Meandering of the transmission lines

is one of the methods often used to realize compact structures.

The number of meandered section and how much tight

meandering is done determines the level of miniaturization, in

such cases. The meandering of lines (by hybrid- meandering)

requires proper adjustment of coupling between the adjacent

lines and avoid coupling between non-adjacent lines [6]. Very

tight meandering introduces coupling between the non-

adjacent lines and a very weak coupling makes the meander

section far apart and it loses our purpose of miniaturization.

Moreover each meander section adds approximately four

discontinuities (bends) and thus, many such sections result in

many discontinuities. Several further improvements in

miniaturizing by meandering are reported [7]-[8] which use

narrow width lines with less coupling (almost negligible)

between the non- adjacent lines. Such narrow width lines can

be achieved by using thin film microstrip lines [7]. However,

these thin film lines have reported high insertion loss, which

can be reduced by valley microstrip line structures [8].

Lumped circuit realization is also reported [9]-[10] but

requires precise inductor models for design accuracy. These

circuits are in general narrow band and for broadband

realization require large number of sections. However,

incorporation of lattice structure reduces number of sections

but it is not suited for planar configuration [10].

Slow wave structure is one of the concepts, which can be used

for reducing the size of the circuits, especially the length of

the transmission lines. For the planar circuit technology the

periodic shunt loading of the transmission lines can behave as

the slow wave structure. This periodic shunt loading reduces

the phase velocity and hence increasing the effective electric

length of the line. In this paper, we present a design of a

reduced size Two-Way Wilkinson Power divider using quarter

wave transformer realized by periodically shunt loaded

microstrip, which can be fabricated with the standard printed

circuit technology.

II. SLOW WAVE STRUCTURES FOR PLANER CIRCUITS

A simple transmission line section of characteristic

impedance Zc and phase velocity vp is given as under [11]

C

L

cZ = (1)

LC

1

pv = (2)

where, L and C are the inductance and capacitance per unit

length of the line respectively. In a physically smooth line if

we increase the capacitance per unit length (by changing its

configuration) it automatically decreases its inductance per

unit length. However, if capacitances in shunt are added at

periodic intervals along the length and the spacing between

the added capacitors is small enough compared to the

978-1-4244-2804-5/09/$25.00 © 2009 IEEE IMS 2009613

wavelength, it may be anticipated that the line will appear to

be electrically smooth with the effective characteristic

impedance and the phase velocity given as under [11].

d

pCC

L

c_loadedZ

+

= (3)

+

=

d

pCCL

1

p_loadedv (4)

where, Cp is the periodically loaded capacitance at ‘d’ distance

apart over the line. Zc_loaded and Zc are the characteristic

impedance of loaded and unloaded line respectively. Clearly,

the effect of periodic loading is the lowered effective

characteristic impedance and the phase velocity. A lowered

phase velocity means that an effectively long electric length

can be realized with a shorter physical length. Moreover, to

realize a line with certain characteristic impedance, the line

used for loading should have higher characteristic impedance

so that after loading its value reduced to the desired value. The

electric length thus achieved after periodic loading is related

to the physical length l and is given as under

×=

p_loadedv

0ωlΦ (5)

The phase velocity of the loaded line vp_loaded and that of

unloaded line vp is related through their respective propagation

constants. If k0 and β0 is the phase constant of the unloaded

and the loaded line respectively then the two phase velocities

can be related as under.

=

β

0k

pvp_loadedv (6)

Following Figure 1 shows the circuit diagram and schematic

of periodically loaded line.

(a) (b)

Fig. 1. (a) Circuit diagram of single periodic section of periodically loaded transmission line. (b) Schematic of periodically loaded line with open stub used as loaded capacitance.

The above (1) to (6) are general equations for the periodically

loaded transmission lines. The formulae used for the physical

realization are derived from these as under [12].

From, (1) to (6) we can write

cZ

c_loadedZ

β

0k= (7)

Thus, in other words, for a periodic loaded line to behave as

slow wave structure, it is necessary that the ratio given by (7)

is less than one. This implies that we use a high impedance

line for loading and more is the impedance of this line more

will be the reduction factor. Using (5) and (6) we can express

the distance d of the periodic loading as under [12]

0ω

pv

cZ

c_loadedZ

N

Φd

= (8)

Similarly, using equations (1) to (6) we can express the loaded

capacitance Cp, in terms of known parameters in the design, as

in [12]

2 2Z ZcΦ c_loaded

CP 2Nω Z Z0 c c_loaded

−=

(9)

If the capacitance calculated by equation (10) is realized using

open circuit stubs, the following formula can be used [11]

lstubC

P Z v0_stub p_stub= for

ω0lstub

vp_stub

<< 1 (10)

Where Z0_stub and vp_stub are the characteristic impedance and

phase velocity of transmission line used as stub of length lstub.

III. METHOD AND DESIGN CONSIDERATIONS

Thus for realizing a line with characteristic impedance of say

Zeff and an electric length of Φeff , with the periodic shunt

loading of the transmission line with given characteristic

impedance Zc and phase velocity vp we can calculate k0/β from

(7) by replacing the parameters as under

effZc_loadedZ = (11)

effΦΦ = (12)

Zc is selected according to the degree of miniaturization

required and it should be at its highest value possible for a

maximum size reduction.

614

For a given microstrip line technology the highest

characteristic impedance line width is determined by the

thinnest line that can be manufactured using the fabrication

process of the facility. Phase velocity vp however depends

more on the substrate parameters. The number of periodic

sections N is independent of the realized length but it is

chosen according to the capacitance value to be realized (by

the open stub) and the value of d (chosen to avoid coupling

between stubs). The value of d is chosen such that N has the

value that gives the capacitance value that can be realized by

optimum (in terms of less interference and size occupancy)

length of the stub. For high value of N we can have lower

value of capacitance and hence the length of the open stub

(realizing the capacitance) is less. However, to avoid the

interference among the stubs their distance should be greater

than 3h where, h is the height of the substrate used. Moreover,

reducing the characteristic impedance of the stub may also

reduce the stub length. This will increase its width again

making more prone for interference among each other. The

limit d > 3h can be crossed (without producing interference

among stubs) by placing alternate stubs on the other sides of

the microstrip lines. Thus once selecting velocity reduction

factor k0/β we can get the reduced value of length for the given

electric length from (5). This reduced value of length is then

realized by loading the line with ‘N’ open stubs, at the interval

of d distance. The value of N, d, CP can directly be calculated

using (8) to (10).

III. EXPERIMENT AND DISCUSSION

The reduced size 2-way Wilkinson power divider is designed

at 850 MHz and 620 MHz. In the case of 850 MHz design out

of 90° electric length of the transformers, only 58° is realized

by the periodically loaded transmission line. Whereas, in

620 MHz design the whole 90° electric length of the

transformer is miniaturized. The substrate used has dielectric

constant of 9.9. Figure 2 is the photograph of the fabricated

Wilkinson power divider with periodically loaded

transmission lines. The circuits are realized in substrate size of

1"×1". The input feeding line is bent as per the orientation of

the system requirement.

(a) (b)

Fig. 2. (a) Photograph of 850 MHz Power Divider. (b) Photograph of 620 MHz Power Divider.

To analyze the combined effects of the various discontinuities

we have performed EM simulation for planar circuits using

momentum engine of Agilent ADS. Table 1 compares the

values of different design parameters calculated theoretically

and compensated for momentum co-simulation results. A

result in terms of miniaturization achieved is listed in table 2.

For both 850 MHz and 620 MHz design, the achieved

insertion loss over the band of ±150 MHz is less than 0.7 dB

with return loss greater than 15 dB at all the ports over the

band. The isolation achieved between the two output ports is

greater than 15 dB. The gain and phase error is less than 0.8

dB and 3.5 deg respectively within the entire band. Some

important measured and EM simulated results are illustrated

Figure 3 and Figure 4 for design at 850 MHz and 620 MHz

respectively.

TABLE I

SUMMARY OF DESIGN PARAMETERS

Design Parameters

850 MHz Design 620 MHz Design

Theoretically

Calculated Values

Compensated Values

after EM simulation

Theoretically

Calculated Values

Compensated Values

after EM simulation

Distance between

periodic loading (d)

1.95 mm 1.9 mm 2.1 mm 2.0 mm

Width of Unloaded Line 0.1 mm 0.1 mm 0.06 mm 0.06 mm

Width of open Stubs 0.1 mm 0.1 mm 0.2 mm 0.2 mm

Length of open Stubs 1.5 mm 2 mm 1.84 mm 2.56 mm

TABLE II

SUMMARY OF RESULTS IN TERMS OF MINIATURIZATION

Design Frequency Actual Length of Line

lactual

Realized Reduced

Length lreduced

1-k0

β×100

Theoretical Percentage

Reduction

1-lreduced

lactual

×100

Practically Realized

Percentage Reduction

850 MHz Design 23.46 mm 17.3 mm 24.78% (Zc =94 Ω) 26.26%

620 MHz Design 50.28 mm 33.61 mm 34.54% (Zc= 108 Ω) 33.15

615

(a) Return Losses at different ports

(b) Isolation between two output ports

(c) Gain and phase error between two paths

Fig.3. Results of 850 MHz design

(a) Return Losses at different ports

(b) Isolation between two output ports

(c) Gain and phase error between two paths

Fig.4. Results of 620 MHz design

IV. CONCLUSION

In this paper, a theoretical method is presented for the design

of miniaturized two-way Wilkinson power divider using

periodically loaded transmission line to realize 90˚

transformers. The design approach is validated for the

realization of power divider at 850 MHz and 620 MHz. The

theoretically calculated response is validated through

Electromagnetic Simulation and the various design variables

are chosen accordingly.

[5] Bharathi Bhatt, Shiban K. Koul , Stripline like Ttransmission Lines for MIC’s, New Delhi: Willey Eastern Limited, 1989.

[6] E.G. Crystal, “Meander-line and hybrid-meander line transformers,” IEEE Trans. Microwave Theory & Tech., vol. 21, no.2 pp. 69-76, Feb 1973.

[7] T.Hiraoka, T.Tokomitsu, M.Aikawa, “Very small wide band MMIC magic T’s using microstrip on a thin dielectric film,” IEEE Trans. Microwave Theory & Tech., vol. 37, no.10, pp. 1569-1575, Oct 1989.

[8] T.Hasegawa, S.Banba, H. Ogawa, “A branch line using valley microstrip line,” IEEE Microwave Guided Lett., vol. 2, no.2, pp. 76-78, Feb 1992.

[9] S. J. Parisi, “180° lumped element hybrid,” 1989 IEEE MTT-S Int. Microwave Symp. Dig., vol. 3, pp. 1243-1246, June 1989.

[10] T.Kawai, Y. Kokubo, I. Ohta, “Broadband lumped element 180-degree hybrids utilizing lattice circuits,” 2001 IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, pp. 47-50, May 2001.

[11] R.E. Collin, Foundation for Microwave Engineering,New York: IEEE Press Series on Electromagnetic Wave Theory, 1998.

[12] K.W. Eccleston, S.H.M Ong, “Compact planar microstrip line branch-line and rat-race couplers,” IEEE Trans. Microwave Theory & Tech., vol. 51, no.10, pp. 2119-2124, Oct 2003.

-45

-40

-35

-30

-25

-20

-15

-10

700 750 800 850 900 950 1000

Magnitude (dB)

Frequency (MHz)

RETURN LOSSES AT PORTS

S11 S11(EM) S22 S22(EM) S33 S33(EM)

-30

-25

-20

-15

-10

700 750 800 850 900 950 1000

Magnitude (dB)

Frequency (MHz)

ISOLATION

S(2,3) S(2,3)(EM)

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

700 750 800 850 900 950 1000

Phase Error (degree)

Magnitude Error(dB)

Frequency (MHz)

Gain & Phase Error

Mag_Error Phase_Error

-45.00

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

470 520 570 620 670 720 770

Magnitude (dB)

Frequency(MHz)

RETURN LOSSES AT PORTS

S11(dB) S11(EM) S22 S22(EM) S33 S33(EM)

-45.00

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

470 520 570 620 670 720 770

Magnitude (dB)

Frequency (MHz)

ISOLATION

S(2,3) S23(EM)

-0.80

-0.75

-0.70

-0.65

-0.60

-0.55

-0.50

-0.45

-0.40

-0.17

-0.14

-0.12

-0.09

-0.07

-0.04

-0.02

0.01

470 520 570 620 670 720 770

Magnitude (Degree)

Magnitude (dB)

Frequency(MHz)

GAIN & PHASE ERROR

Mag_Error Phase_Error

REFERENCES

[1] I. Bahl, P. Bhartia, Microwave Solid-State Circuit Design, New York: Wiley, 1998.

[2] S.A. Mass, Nonlinear Microwave Circuits, Norwood: Artech House, 1988.

[3] D.M. Pozar, Microwave Engineering, New York: Willey, 1998. [4] E.J. Wilkinson, “A N-way hybrid power divider,” IRE Trans.

Microwave Theory & Tech., vol. 8, pp. 116-118, Jan 1960.

616