Periodically Loaded Line Phase Shifters Report1

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1. Introduction

A periodically loaded line, as its name implies, is an infinite transmission line periodically

loaded with reactive elements and is refered to as a periodic structure [1]. Periodic structures

can take various forms, depending on the guiding structure being used. These structures find

application in filters, travelling-wave tubes, masers, phase shifters.

Consider a transmission line loaded periodically with variable capacitors illustrated in Fig 1.

Figure 1 Schematic of a periodically loaded line phase shifter

The loaded transmission line shown in Fig 1 behaves like a synthetic transmission line and

exhibits slow-wave behaviour. The phenomenon is similar to acoustic wave propagation in

crystal structures. In this structure the phase velocity of a propagating wave can be changed

by varying the capacitance value.

C Therefore it can be used as a continuously tunable time-delay/phase shift device. This characteristics is very attractive when a fixed phase resolution

is not enough or impractical to use.

As a variable capacitor, a varactor diode can be used. The variable capacitance is obtained

through a voltage-tuned capacitance of the varactor diode under a reverse-bias condition.

2. Theory

Consider a cascade of short sections of lossless transmission line shunted by a variable

capacitance ,C so as to form the periodic structure illustrated in Fig. 1. For this periodic

structure, we can define a unit cell which consists of a section of transmission line of length s and a shunt variable capacitance to ground. Each distributed transmission line section can

be approximated by a lumped LC circuit [2], as shown in Fig 2.



In

Figure 2 Equivalent circuit for the varactor diode loaded line

To find the basic difference equations, Kirchhoff’s current and voltage laws can be applied tothe th

n and st n 1+ nodes of the loaded line to obtain

11 ++=− n snn I L jV V ω

(2.1)

n snn V C C j I I )(1 +=−+

ω

(2.2)

which can be combined, leading to the discrete voltage wave equation

n s snnn V C C LV V V )(2

2

11+−=+−

−+ω

(2.3)

The relation between any adjacent nodes can be expressed as

nn V V /1+ = a complex constant (independent of n ) = jkse±

(2.4)

since the wave cannot distinguish between indiviual nodes and it travels a distance s

between them. Therefore, the voltage nV at the thn node is a superposition of forward and

backward waves

θ θ jn jnn eV eV V

−

−

++=

(2.5)

where the dimensionless parameter ks=θ may be complex even for a lossless loaded line. If

we substitude the forward–propagating voltage wave ( (2.5) with 0=−

V ) into the discrete

wave equation (2.3), we find

)(2 2C C Lee s s

j j+−=+−

−ω

θ θ (2.6)



After some manipulations, we obtain the dispersion relation for the loaded line as

20

222

4

)()2/(sinω

ω ω θ =+= C C L s s

(2.7)

where)(

20

C C L s s +

=ω is the Bragg frequency for this periodically loaded line, for if

0ω ω > , must be imaginary and the waves evanescent. In the case of 0ω ω << , the

dispersion relation (2.7) becomes [3]

4

)(

4

)

2

(sin22

2 C C L s s +=≈ω θ θ

(2.8)

,which means that

)/))((/( sC C s Lk s s +±≈ ω

(2.9)

(2.9) is the dispersion relation of a continuous, lossles TEM line. For frequencies well below

the Bragg frequency, the periodically loaded line may be treated as a synthetic transmission

line with a series inductance per unit length s L s / and a shunt capacitance per unit length

sC C s /)( + . In other words, we distribute the lumped variable capacitance over the length of

the unit cell. In the vicinity of the Bragg frequency, this approximation does not work.

Figure 3 Synthetic transmission line with voltage-tunable characteristic impedance)(V Z loaded and )(V vloaded . n is the number of sections.

Since the loading capacitors are voltage-tunable, the characteristic impedance and the phase

velocity of the loaded line are also tunable.



3. Design Equations

These equations are valid when the frequency of interest is well below the Bragg frequency.

There are two important design parameters [4]: loading factor and capacitance ratio defined asfollows

loading factor:unloaded C

sC x

/max= (3.1)

capacitance ratio:max

min

C

C y =

(3.2)

where sC C sunloaded /= is the capacitance per unit length of the unloaded line. minC andmaxC are the minimum and maximum loading capacitance values, respectively.

According to (2.9) the phase shift obtained from a single section at a given frequency well

below the Bragg frequency is s

C C

s

L sks s s )( +

== ω θ , which is maximum when maxC C =

and minimum when minC C = . Therefore for a given frequency the maximum possible

differential phase shift obtainable from a single section is given by

)11(minmaxmax xy x

v

s

s

C C

s

C C

s

L s

unloaded

s s s +−+=

+−

+= ω ω θ (3.3)

What (3.3) tells us is increasing loading factor x and/or the capacitance ratio y results in

more differential phase shift. Thus in order to realize a 0360 phase shifter at frequency f

the number of sections n required is given by

max

2

θ

π =n (3.4)

The Bragg frequency is depends on C and it is minimum when maxC C = . Using the

equations previously developed, the relationship among the diode spacing s , the minimum

Bragg frequency min,0 f and the loading factor can be obtained as

x f

v s unloaded

+

=

1min,0π

(3.5)



Once we decide what minimum Bragg frequency to use, s becomes a function of the loading

factor only.

4. Comparison of literature

Periodically loaded line phase shifters are capable of producing continuously variable00 3600 − phase shift, which is a desirable feature for adaptive antenna systems [5]. In [6]

diode-loaded lines have been used as true time-delay devices in prototype arrays. In this work

the loaded line phase shifter is realized using hybrid integration techniques. A 1.1 ns true time

delay with <4 dB insertion loss is measured. A monolithic implementation of a coplanar

waveguide periodically loaded with varactor diodes is discussed in [4]. The distributed phase

shifter described in this paper is capable of producing 00 3600 − phase shift at 20 GHz with a

maximum insertion loss of 4.2 dB and return loss better than 12 dB over all phase states. In

[7], BST capacitors, a potential low cost voltage tunable component for microwave circuits,

are investigated in the design of distributed phase shifters and coplanar waveguides

periodically loaded with tunable BST thin film capacitors are designed providing 0240 phase

shift with an insertion loss of 3 dB at 10 GHz at room temperature.



REFERENCES

[1] D. M. Pozar, Microwave Engineering, Second Edition, John Wiley & Sons, 1998.

[2] J. L. Volakis, Antenna Engineering Handbook, Fourth Edition, Chapter 21, Mc-Graw Hill,

2007[3] D. H. Staelin, A. W. Morgenthaler, J. A. Kong, Electromagnetic Waves, Prentice Hall,

1994.

[4] A. S. Nagra and R. A. York, Distributed Analog Phase Shifters with Low Insertion Loss,

IEEE Trans. on Microwave Theory and Tech. Vol. 47, No. 9, September 1999.

[5] F. Ellinger, H. Jäckel, W. Bächtold, Varactor-Loaded Transmission Line Phase Shifter at

C-Band Using Lumped Elements, IEEE Trans. on Microwave Theory and Tech. Vol. 51,

No. 4, April 2003

[6] W.M. Zhang, R.P. Hsia, C. Liang, G.Song, C.W. Domier, N. C. Nuhmann, Novel Low-

Loss Delay Line for Broadband Phased Antenna Array Applications, IEEE Microwave and

Guided Wave Letters Vol. 6, No. 11, November 1996.

[7] B. Acikel, T.R. Taylor, P. J. Hansen, J.S. Speck, and R.A.York, A New High PerformancePhase Shifter Using 31 TiOSr Ba x x − Thin Films, IEEE Microwave and Wireless

Components

Letters Vol. 12, No. 7, July 2002.

Periodically Loaded Line Phase Shifters Report1

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