Lumped Elements for RF and Microwave Circuits phần 10 potx

442 Lumped Elements for RF and Microwave Circuits

Figure 14.5 Total Q for a quarter-wave resonator on RT/Duroid (e r = 2.32), quartz (e r = 3.8),

and alumina (e r = 10.0) versus substrate thickness.

qc = tanh S1.043 + 0.121

h′

h − 1.164 h

h′D (14.25c)

Here F(W /h) is given by (14.3). Using the preceding equations, the

characteristic impedance of the shielded microstrip can be calculated from

Z0 = Z a

0 /√e re .

For the range of parameters, 1 ≤ e r ≤ 30, 0.05 ≤ W /h ≤ 20, t /h ≤ 0.1,

and 1 < h′/h < ∞, the maximum error in Z0 and e re is found to be less than

±1%. When h′/h ≥ 5, the effect of the top cover on the microstrip characteristics

becomes negligible.

The effect of sidewalls on the characteristics of microstrip must also be

included. It is found that the sidewall effect is negligible when S /h ≥ 5, where

Microstrip Overview 443

Figure 14.6 Enclosed microstrip configuration.

S is the separation between the microstrip conductor edge and the sidewall of

the enclosure.

14.2.5 Frequency Range of Operation

The maximum frequency of operation of a microstrip is limited due to several

factors such as excitation of spurious modes, higher losses, pronounced disconti￾nuity effects, low Q due to radiation from discontinuities, effect of dispersion

on pulse distortion, tight fabrication tolerances, handling fragility, and, of course,

technological processes. The frequency at which significant coupling occurs

between the quasi-TEM mode and the lowest order surface wave spurious mode

is given here [1, 3]:

fT = 150

ph √ 2

e r − 1 tan−1 (e r ) (14.26)

where fT is in gigahertz and h is in millimeters. Thus the maximum thickness

of the quartz substrate (e r ≅ 3.8) for microstrip circuits designed at 100 GHz

is less than 0.3 mm.

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

operating below the cutoff frequency of the first higher order mode, which is

given approximately by

fc ≅

300

√e r (2W + 0.8h) (14.27)

where fc is in gigahertz, and W and h are in millimeters. This limitation is

mostly applied to low impedance lines that have wide microstrip conductors.

444 Lumped Elements for RF and Microwave Circuits

14.2.6 Power-Handling Capability

The power-handling capacity of a microstrip, like that of any other dielectric

filled transmission line, is limited by heating as a result of ohmic and dielectric

losses and by dielectric breakdown. An increase in temperature due to conductor

and dielectric losses limits the average power of the microstrip line, whereas the

breakdown between the strip conductor and ground plane limits the peak power.

14.2.6.1 Average Power

Microstrip lines are well suited for medium power (about 100 to 200W) applica￾tions and have been extensively used in power MMIC amplifiers. Average power￾handling capability (APHC) of microstrip lines has been discussed in [1, 13–15].

Recent advancements in multilayer microstrip line technologies have made it

possible to realize compact MMICs [16], compact modules [17], low-loss micro￾strip lines [18], and high-Q inductors [19]. In multilayered components, along

with substrate materials, low dielectric constant materials such as polyimide or

BCB are used as a multilayer dielectric. The thermal resistance of polyimide or

BCB is about 200 times the thermal resistance of GaAs or alumina. To ensure

reliable operation of multilayered components such as inductors, capacitors,

crossovers, and inductor transformers for high-power applications, thermal mod￾els are needed for these structures. Bahl [20] discussed the average power￾handling capability of multilayer microstrip lines used in MICs and MMICs.

The APHC of a multilayer microstrip is determined by the temperature

rise of the strip conductor and the supporting dielectric layers and the substrate.

The parameters that play major roles in the calculation of average power capabil￾ity are (1) transmission-line losses, (2) the thermal conductivity of dielectric

layers and the substrate material, (3) the surface area of the strip conductor;

(4) the maximum allowed operating temperature of the microstrip structure,

and (5) ambient temperature; that is, the temperature of the medium surrounding

the microstrip. Therefore, dielectric layers and substrates with low-loss tangents

and large thermal conductivities will increase the average power-handling capabil￾ity of microstrip lines.

Typically a procedure for APHC calculation consists of the calculation of

conductor and dielectric losses, heat flow due to power dissipation, and the

temperature rise. The temperature rise of the strip conductor can be calculated

from the heat flow field in the microstrip cross section. An analogy between

the heat flow field and the electric field is provided in Table 14.7. The heat

generated by the conductor loss and the dielectric loss is discussed separately

in the following sections. It has been assumed that there are no nonuniformities

in the line and that the line is perfectly matched at two ends.

14.2.6.2 Density of Heat Flow Due to Conductor Loss

A loss of electromagnetic power in the strip conductor generates heat in the

strip. Because of the good heat conductivity of the strip metal, heat generation

Microstrip Overview 445

Table 14.7

Analogy Between Heat Flow and Electric Field

Heat Flow Field Electric Field

1. Temperature, T (°C) Potential, V (V)

2. Temperature gradient, Tg (°C/m) Electric field, E (V/m)

3. Heat flow rate, Q (W) Flux, f (coulomb)

4. Density of heat flow, q (W/m2

) Flux density, D (coulomb/m2

)

5. Thermal conductivity, K (W/m-°C) Permittivity, e (coulomb/m/V)

6. Density of heat generated, r h (W/m3

) Charge density, r (coulomb/m3

)

7. q = −K=T D = −e=V

8. = ? q = r h = ? D = r

is uniform along the width of the conductor. Because the ground plane of the

microstrip configuration is held at ambient temperature (i.e., acts as a heat

sink), this heat flows from the strip conductor to the ground plane through

the polyimide layer/layers and the GaAs/alumina substrate. The heat flow can

be calculated by considering the analogous electric field distribution. The heat

flow field in the microstrip structure corresponds to the electrostatic field (with￾out any dispersion) of the microstrip. The electric field lines (and the thermal

field lines in the case of heat flow) spread as they approach the ground plane.

As a first-order approximation, the heat flow from the microstrip conductor

can be considered to follow the rule of 45° thermal spread angle [21] as shown

in Figure 14.7 for a two-layered microstrip configuration. This means that the

heat generated in the microstrip conductor (assuming there are no other heat

sources and heat flow is mainly by conduction) flows down through the dielectric

materials through areas larger than the strip conductor as it approaches the

ground plane, where the ground plane acts as a heat sink. However, to account

accurately for the increase in area normal to heat flow lines, the parallel plate

Figure 14.7 Schematic of microstrip line heat flow based on 45° thermal spread angle rule.