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Microstrip Filters for RF/Microwave Applications. Jia-Sheng Hong, M. J. Lancaster Copyright © 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic)
CHAPTER 4
Transmission Lines and Components
In this chapter, basic concepts and design equations for microstrip lines, coupled microstrip lines, discontinuities, and components useful for design of filters are briefly described. Though comprehensive treatments of these topics can be found in the open literature, they are summarized here for easy reference.
4.1 MICROSTRIP LINES
4.1.1 Microstrip Structure
The general structure of a microstrip is illustrated in Figure 4.1. A conducting strip (microstrip line) with a width W and a thickness t is on the top of a dielectric sub-strate that has a relative dielectric constant «r and a thickness h, and the bottom of the substrate is a ground (conducting) plane.
4.1.2 Waves in Microstrips
The fields in the microstrip extend within two media—air above and dielectric be-low—so that the structure is inhomogeneous. Due to this inhomogeneous nature, the microstrip does not support a pure TEM wave. This is because that a pure TEM wave has only transverse components, and its propagation velocity depends only on the material properties, namely the permittivity « and the permeability m. However, with the presence of the two guided-wave media (the dielectric substrate and the air), the waves in a microstrip line will have no vanished longitudinal components of electric and magnetic fields, and their propagation velocities will depend not only on the material properties, but also on the physical dimensions of the mi-crostrip.
77
78 TRANSMISSION LINES AND COMPONENTS
Conducting strip W t
εr Ground plane
h Dielectric substrate
FIGURE 4.1 General microstrip structure.
4.1.3Quasi-TEM Approximation
When the longitudinal components of the fields for the dominant mode of a mi-crostrip line remain very much smaller than the transverse components, they may be neglected. In this case, the dominant mode then behaves like a TEM mode, and the TEM transmission line theory is applicable for the microstrip line as well. This is called the quasi-TEM approximation and it is valid over most of the operating fre-quency ranges of microstrip.
4.1.4 Effective Dielectric Constant and Characteristic Impedance
In the quasi-TEM approximation, a homogeneous dielectric material with an effec-tive dielectric permittivity replaces the inhomogeneous dielectric–air media of mi-crostrip. Transmission characteristics of microstrips are described by two parame-ters, namely, the effective dielectric constant «re and characteristic impedance Zc, which may then be obtained by quasistatic analysis [1]. In quasi-static analysis, the fundamental mode of wave propagation in a microstrip is assumed to be pure TEM. The above two parameters of microstrips are then determined from the values of two capacitances as follows
«re = }}
a (4.1)
Zc = cÏCwwCw
in which Cd is the capacitance per unit length with the dielectric substrate present, Ca is the capacitance per unit length with the dielectric substrate replaced by air, and c is the velocity of electromagnetic waves in free space (c < 3.0 × 108 m/s).
4.1 MICROSTRIP LINES 79
For very thin conductors (i.e., t R 0), the closed-form expressions that provide an accuracy better than one percent are given [2] as follows.
For W/h # 1:
«re = «r + 1 + «r – 1 511 + 12 W 2–0.5 + 0.0411 – W 226 (4.2a)
Zc = 2pÏw«w ln1} + 0.25}2 (4.2b)
where h = 120p ohms is the wave impedance in free space. For W/h $ 1:
«re = «r + 1 + «r – 1 11 + 12 h 2–0.5 (4.3a)
Zc = Ïw«w5} + 1.393 + 0.677 ln1} + 1.44426–1 (4.3b)
Hammerstad and Jensen [3] report more accurate expressions for the effective dielectric constant and characteristic impedance:
«re = «r + 1 + «r – 1 11 + }2–ab (4.4)
where u = W/h, and
a = 1 + 49 ln1u4 + 0.432 2+ 18.7 ln31 + 118.1 234
b = 0.5641«r – 0.9 20.053 r
The accuracy of this model is better than 0.2% for «r # 128 and 0.01 # u # 100. The more accurate expression for the characteristic impedance is
Zc = 2pÏw«we ln3} + !1 +§§§4 (4.5) where u = W/h, h = 120p ohms, and
F = 6 + (2p – 6)exp3– 130.666 20.75284
The accuracy for Zcϫww is better than 0.01% for u # 1 and 0.03% for u # 1000.
80 TRANSMISSION LINES AND COMPONENTS
4.1.5 Guided Wavelength, Propagation Constant, Phase Velocity, and Electrical Length
Once the effective dielectric constant of a microstrip is determined, the guided wavelength of the quasi-TEM mode of microstrip is given by
lg = Ïww (4.6a)
where l0 is the free space wavelength at operation frequency f. More conveniently, where the frequency is given in gigahertz (GHz), the guided wavelength can be evaluated directly in millimeters as follows:
lg = f(GHz)Ïwre mm (4.6b)
The associated propagation constant b and phase velocity vp can be determined by
b = 2p (4.7) g
vp = } = Ïww (4.8)
where c is the velocity of light (c < 3.0 × 108 m/s) in free space.
The electrical length ufor a given physical length l of the microstrip is defined by
u = bl (4.9)
Therefore, u = p/2 when l = lg/4, and u = p when l = lg/2. These so-called quarter-wavelength and half-wavelength microstrip lines are important for design of mi-crostrip filters.
4.1.6 Synthesis of W/h
Approximate expressions for W/h in terms of Zc and «r, derived by Wheeler [4] and Hammerstad [2], are available.
For W/h # 2
W 8 exp(A) h exp(2A) – 2
with
A = 60 5«r + 1 60.5 + «r – 1 50.23 + 0.11 6
(4.10)
4.1 MICROSTRIP LINES 81
and for W/h $ 2
} = }5(B – 1) – ln(2B – 1) + «2–r1 3ln(B – 1) + 0.39 – 0.61 46 (4.11) with
60p2 ZcÏww
These expressions also provide accuracy better than one percent. If more accurate values are needed, an iterative or optimization process based on the more accurate analysis models described previously can be employed.
4.1.7 Effect of Strip Thickness
So far we have not considered the effect of conducting strip thickness t (as referring to Figure 4.1). The thickness t is usually very small when the microstrip line is real-ized by conducting thin films; therefore, its effect may quite often be neglected. Nevertheless, its effect on the characteristic impedance and effective dielectric con-stant may be included [5].
For W/h # 1:
Zc(t) = 2pϫww ln5 We(t)/h + 0.25 W(t) 6 (4.12a) For W/h $ 1:
Zc(t) = Ïw«w 5 e(t) + 1.393 + 0.667 ln1 e(t) + 1.44426–1 (4.12b) where
e(t) }} + } 5 }11 + ln}}2(W/h # 0.5p) h }} + } 5 }}11 + ln}}2(W/h $ 0.5p)
«re(t) = «re – «4– 1 ÏwW/wh
(4.13a)
(4.13b)
In the above expressions, «re is the effective dielectric constant for t = 0. It can be observed that the effect of strip thickness on both the characteristic impedance and effective dielectric constant is insignificant for small values of t/h. However, the ef-fect of strip thickness is significant for conductor loss of the microstrip line.
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