Microstrip bộ lọc cho các ứng dụng lò vi sóng RF (P4)

However,with the presence of the two guided-wave media the dielectric substrate and theair, the waves in a microstrip line will have no vanished longitudinal components be-of electric an

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 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 TEMwave has only transverse components, and its propagation velocity depends only onthe material properties, namely the permittivity
and the permeability  However,with the presence of the two guided-wave media (the dielectric substrate and theair), the waves in a microstrip line will have no vanished longitudinal components

be-of electric and magnetic fields, and their propagation velocities will depend notonly on the material properties, but also on the physical dimensions of the mi-crostrip

77

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)

4.1.3Quasi-TEM Approximation

When the longitudinal components of the fields for the dominant mode of a crostrip line remain very much smaller than the transverse components, they may beneglected In this case, the dominant mode then behaves like a TEM mode, and theTEM transmission line theory is applicable for the microstrip line as well This iscalled the quasi-TEM approximation and it is valid over most of the operating fre-quency ranges of microstrip

mi-4.1.4 Effective Dielectric Constant and Characteristic Impedance

In the quasi-TEM approximation, a homogeneous dielectric material with an 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 Z c,which may then be obtained by quasistatic analysis [1] In quasi-static analysis, thefundamental 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 oftwo capacitances as follows

effec-
re= C

C d a



(4.1)

Z c=

in which C dis the capacitance per unit length with the dielectric substrate present,

C ais 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 × 108m/s)

1



c 兹C苶a苶C d苶

Ground planeConducting strip

For very thin conductors (i.e., t씮 0), the closed-form expressions that provide

an accuracy better than one percent are given [2] as follows

The more accurate expression for the characteristic impedance is

1

18.7

u4+ 冢5

r+ 1

2

r+ 1

2

r+ 1

2

4.1 MICROSTRIP LINES 79

4.1.5 Guided Wavelength, Propagation Constant, Phase Velocity, and Electrical Length

Once the effective dielectric constant of a microstrip is determined, the guidedwavelength of the quasi-TEM mode of microstrip is given by

where 0 is the free space wavelength at operation frequency f More conveniently,

where the frequency is given in gigahertz (GHz), the guided wavelength can beevaluated directly in millimeters as follows:

The associated propagation constant and phase velocity v pcan be determined by

where c is the velocity of light (c⬇ 3.0 × 108m/s) in free space

The electrical length for a given physical length l of the microstrip is defined by

Therefore, = /2 when l = g/4, and = when l = g/2 These so-called wavelength and half-wavelength microstrip lines are important for design of mi-crostrip filters

Z c

60

and for W/h 2

= 冦(B – 1) – ln(2B – 1) + 冤ln(B – 1) + 0.39 – 冥冧 (4.11)with

B =

These expressions also provide accuracy better than one percent If more accuratevalues are needed, an iterative or optimization process based on the more accurateanalysis 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]

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

4.1.8 Dispersion in Microstrip

Generally speaking, there is dispersion in microstrips; namely, its phase velocity isnot a constant but depends on frequency It follows that its effective dielectric con-stant
reis a function of frequency and can in general be defined as the frequency-dependent effective dielectric constant
re ( f ) The previous expressions for
reareobtained based on the quasi-TEM or quasistatic approximation, and therefore arerigorous only with DC At low microwave frequencies, these expressions provide agood approximation To take into account the effect of dispersion, the formula of

re ( f ) reported in [6] may be used, and is given as follows:

40

5f

冣冧 for W/h 0.7

(4.16c)

where c is the velocity of light in free space, and whenever the product m0m cis

greater than 2.32 the parameter m is chosen equal to 2.32 The dispersion model

shows that the
re ( f ) increases with frequency, and
re ( f ) 씮
r as f

racy is estimated to be within 0.6% for 0.1  W/h 10, 1 
r 128 and for any

4.1.9 Microstrip Losses

The loss components of a single microstrip line include conductor loss, dielectricloss and radiation loss, while the magnetic loss plays a role only for magnetic sub-strates such as ferrites [8–9] The propagation constant on a lossy transmission line

is complex; namely, = + j , where the real part in nepers per unit length is theattenuation constant, which is the sum of the attenuation constants arising fromeach effect In practice, one may prefer to express in decibels (dB) per unit length,which can be related by

(dB/unit length) = (20 log10e) (nepers/unit length)

⬇ 8.686(nepers/unit length)

A simple expression for the estimation of the attenuation produced by the conductorloss is given by [9]

in which Z c is the characteristic impedance of the microstrip of the width W, and R s

represents the surface resistance in ohms per square for the strip conductor andground plane For a conductor

R s= 冪莦where is the conductivity, 0is the permeability of free space, and is the angu-lar frequency The surface resistance of superconductors is expressed differently;this will be addressed in Chapter 7 Strictly speaking, the simple expression of(4.18) is only valid for large strip widths because it assumes that the current distrib-ution across the microstrip is uniform, and therefore it would overestimate the con-ductor loss for narrower microstrip lines Nevertheless, it may be found to be accu-rate enough in many practical situations, due to extraneous sources of loss, such asconductor surface roughness

The attenuation due to the dielectric loss in microstrip can be determined by[8–9]

where tan denotes the loss tangent of the dielectric substrate

Since the microstrip is a semiopen structure, any radiation is either free to gate away or to induce currents on the metallic enclosure, causing the radiation loss

propa-or the so-called housing loss

4.1.10 Effect of Enclosure

A metallic enclosure is normally required for most microstrip circuit applications,such as filters The presence of conducting top and side walls will affect both thecharacteristic impedance and the effective dielectric constant Closed formulae areavailable in [1] for a microstrip shielded with a conducting top cover (without sidewalls), which show how both the parameters are modified in comparison with theunshielded ones given previously In practice, a rule of thumb may be applied in thefilter design to reduce the effect of enclosure: the height up to the cover should bemore than eight times and the distance to walls more than five times the substratethickness For more accurate design, the effect of enclosure, including the housingloss, can be taken into account by using full-wave EM simulation

4.1.11 Surface Waves and Higher-Order Modes

A surface wave is a propagating mode guided by the air–dielectric surface for a electric substrate on the conductor ground plane, even without the upper conductorstrip Although the lowest surface wave mode can propagate at any frequency (it has

di-no cutoff), its coupling to the quasi-TEM mode of the microstrip only becomes nificant at the frequency

at which the phase velocities of the two modes are close [10]

The excitation of higher-order modes in a microstrip can be avoided by operating

it below the cutoff frequency of the first higher-order mode, which is given by [10]

In practice, the lowest value (the worst case) of the two frequencies given by(4.20) and (4.21) is taken as the upper limit of operating frequency of a microstripline

structure supports two quasi-TEM modes, i.e., the even mode and the odd mode,

as shown in Figure 4.3 For an even-mode excitation, both microstrip lines havethe same voltage potentials or carry the same sign charges, say the positive ones,

resulting in a magnetic wall at the symmetry plane, as Figure 4.3(a) shows In the

case where an odd mode is excited, both microstrip lines have the opposite voltagepotentials or carry the opposite sign charges, so that the symmetric plane is an

electric wall, as indicated in Figure 4.3(b) In general, these two modes will be

ex-cited at the same time However, they propagate with different phase velocities cause they are not pure TEM modes, which means that they experience differentpermittivities Therefore, the coupled microstrip lines are characterized by thecharacteristic impedances as well as the effective dielectric constants for the twomodes

be-4.2.1 Even- and Odd-Mode Capacitances

In a static approach similar to the single microstrip, the even- and odd-mode teristic impedances and effective dielectric constants of the coupled microstrip lines

charac-may be obtained in terms of the even- and odd-mode capacitances, denoted by C e and C o As shown in Figure 4.3, the even- and odd-mode capacitances C e and C o

FIGURE 4.2 Cross section of coupled microstrip lines.

FIGURE 4.3 Quasi-TEM modes of a pair of coupled microstrip lines: (a) even mode; (b) odd mode.

C fis the fringe capacitance as if for an uncoupled single microstrip line, and is uated by

eval-2C f= 兹
苶re 苶/(cZ c ) – C p (4.25)

The term Cf accounts for the modification of fringe capacitance C f of a single line

due the presence of another line An empirical expression for Cf is given below

where

A = exp[–0.1 exp(2.33 – 2.53W/h)]

For the odd-mode, C ga and C gdrepresent, respectively, the fringe capacitances for

the air and dielectric regions across the coupling gap The capacitance C gdmay befound from the corresponding coupled stripline geometry, with the spacing between

the ground planes given by 2h A closed-form expression for C gdis



(4.28b)

k = 兹1苶 –苶 k苶2苶and the ratio of the elliptic functions is given by





1

 ln冢211

+–兹

兹

k

k

苶

苶

冣 for 0  k2 0.5

K

K

((

k k

)

+–兹

4.2.2 Even- and Odd-Mode Characteristic Impedances and Effective Dielectric Constants

The even- and odd-mode characteristic impedances Z ce and Z co can be obtainedfrom the capacitances This yields

refor even and odd modes, respectively,

can be obtained from C e and C oby using the relations

4.2.3More Accurate Design Equations

More accurate closed-form expressions for the effective dielectric constants and thecharacteristic impedances of coupled microstrip are available [12] For a static ap-proximation, namely, without considering dispersion, these are given as follows:

e

(4.33)With

冥

b e= 0.564冢 冣0.053

where u = W/h and g = s/h The error in
e

reis within 0.7% over the ranges of 0.1 

1

18.7

r+ 1

2

4.2 COUPLED LINES 87

where
re is the static effective dielectric constant of single microstrip of width W as

discussed previously The error in
o

reis stated to be on the order of 0.5%

The even- and odd-mode characteristic impedances given by the followingclosed-form expressions are accurate to within 0.6% over the ranges 0.1  u  10,

Q9= ln(Q7)·(Q8+ 1/16.5)

Closed-form expressions for characteristic impedance and effective dielectricconstants, as given above, may also be used to obtain accurate values of capaci-tances for the even and odd modes from the relationships defined in (4.29) to(4.32) The formulations that include the effect of dispersion can be found in [12]

4.3DISCONTINUITIES AND COMPONENTS

In this section, some typical microstrip discontinuities and components that are ten encountered in microstrip filter designs are described

of-4.3.1 Microstrip Discontinuities

Microstrip discontinuities commonly encountered in the layout of practical filtersinclude steps, open-ends, bends, gaps, and junctions Figure 4.4 illustrates sometypical structures and their equivalent circuits Generally speaking, the effects ofdiscontinuities can be more accurately modeled and taken into account in the filterdesigns with full-wave electromagnetic (EM) simulations, which will be addressed

in due course later on Nevertheless, closed-form expressions for equivalent circuitmodels of these discontinuities are still useful whenever they are appropriate Theseexpressions are used in many circuit analysis programs There are numerous closed-form expressions for microstrip discontinuities available in open literature [1,13–16], for convenience some typical ones are given as follows

4.3.1.1 Steps in Width

For a symmetrical step, the capacitance and inductances of the equivalent circuit

in-dicated in Figure 4.4(a) may be approximated by the following formulation [1]

where L wi for i = 1, 2 are the inductances per unit length of the appropriate crostrips, having widths W1and W2, respectively While Z ciand
reidenote the char-

mi-acteristic impedance and effective dielectric constant corresponding to width W i , c

is the light velocity in free space, and h is the substrate thickness in micrometers.

4.3.1.2 Open Ends

At the open end of a microstrip line with a width of W, the fields do not stop

abrupt-ly but extend slightabrupt-ly further due to the effect of the fringing field This effect can

be modeled either with an equivalent shunt capacitance C por with an equivalentlength of transmission line l, as shown in Figure 4.4(b) The equivalent length isusually more convenient for filter design The relation between the two equivalentparameters may be found by [13]

l = (4.39)

where c is the light velocity in free space A closed-form expression for l/h is

giv-en by [14]

= (4.40)where

A microstrip gap can be represented by an equivalent circuit, as shown in Figure

4.4(c) The shunt and series capacitances C p and C gmay be determined by [1]