Packaged Lumped-Element Passive Components

Planar Circuit Design II

4.4 Packaged Lumped-Element Passive Components

a a

h w

ε_r Metallic

enclosure

Figure 4.13 Cross-section of microstrip line surrounded by a metallic enclosure.

It is normally assumed that the enclosure will have little effect on the electromagnetic field surrounding the microstrip line ifa>5wandb>5h. If these conditions are not met, then the presence of the enclosure will affect both the character- istic impedance (Z_o) and the effective dielectric constant (𝜀^MSTRIP_r,eff ) of the microstrip line. March [11] has provided design equations to enable the modified values ofZ_oand𝜀_r,eff to be calculated for an enclosure of given dimensions.

In addition to modifying the transmission properties of a microstrip line, the presence of a metallic enclosure can lead to undesirable cavity modes. These modes refer to the electromagnetic field patterns within the enclosure, which have similar properties to those in a resonant cavity. The occurrence of these modes is due to spurious radiation from a microstrip circuit, primarily from circuit discontinuities. These modes can be suppressed by including some absorbing material within the metal package. Williams and Paananen [12] showed that this suppression can be achieved very effectively by including a dielectric substrate coated with a resistive film on the underside of the lid of the package. In general, including any lossy material within the enclosure will suppress unwanted modes, but care is needed to avoid the material causing excess loss in the microstrip circuit itself.

4.4 Packaged Lumped-Element Passive Components

4.4.1 Typical Packages for RF Passive Components

Component packaging is an important issue for components used at RF and microwave frequencies, as parasitic resistances and reactances associated with the package can have a significant effect on the electrical performance. Figure 4.14 shows the three most common types of package for RF passive components.

One of the main differences between the various packages is the configuration of the connecting leads. Connecting leads, even straight sections of wire, will have self-inductance. This inductance is small, and negligible at low frequencies, but at

Body of component Wire leads

Beam leads (a)

(b)

Metallization (c)

Figure 4.14 Typical packages for high-frequency passive components: (a) wire ended, (b) beam lead, and (c) Chip.

k k component leads should be made as small as possible as the frequency of operation increases.

Wire-ended components are convenient for manual assembly of circuits, and have been the traditional type of component used up to the UHF region. But, as will be shown later in this section, the leads on these components contribute unaccept- able amounts of inductance at high frequencies. Beam-lead components are an improvement, as the flat leads have less self-inductance, and if the component is correctly bonded onto the circuit this inductance can effectively be absorbed into the feed line to the component. Chip components provide the best configuration for high-frequency usage, because there is minimal lead inductance. But the small size of chip components usually precludes manual assembly, and accurate posi- tioning and bonding of chip passives requires the use of precise surface-mount equipment, although this type of equipment is ideal for automatic assembly of circuits.

It is useful to compare the high-frequency performance of different package configurations by comparing the self-inductance of circular wires with that of flat leads. Wadell [13] quotes the following expression for calculating the self-inductance of a conducting circular wire lead

L=0.002l [

ln (4l

d )

−1+ d

2l+𝜇rT(x) 4

]

μH, (4.26)

where

lis length of the wire in cm dis the wire diameter in cm

𝜇ris the relative permeability of the wire material (usually 1.0), and

T(x) = 0.873+0.00186x

1−0.278x+0.128x², (4.27)

where

x=2𝜋r

√2μf

𝜎 , (4.28)

and where

ris the radius of the wire in cm

𝜇is the permeability of the wire material (usually𝜇o) fis the frequency in Hz

𝜎is the conductivity of the wire material.

It should be noted that the self-inductance of a circular wire is influenced by the skin depth in the material, and con- sequently the self-inductance is frequency dependent as can be seen from Eq. (4.26). However, for wire diameters used for discrete components up to frequencies around 5 GHz, the effect of the frequency-dependent termT(x) in Eq. (4.26) is relatively small. Above frequencies around 5 GHz discrete components with circular wire leads are not normally used, so as to avoid unwanted parasitics.

At frequencies above 5 GHz, active and passive components are normally packaged with very thin beam leads, where the lead self-inductance is considered to be independent of frequency. Wadell [13] also quotes a corresponding expression for the self-inductance of a flat lead, such as would be found in a beam-lead component, as

L=0.002l [

ln ( 2l

w+t )

+0.5+0.2235 (w+t

)]μH, (4.29)

where

lis the length of the lead in cm wis the width of the lead in cm tis the thickness of the lead in cm.

In Example 4.7, which follows, the lead inductances of a circular wire and a flat lead are calculated, using lead dimensions typically found in practical components.

It is worth noting that whilst Eqs. (4.26) and (4.29) yield reasonably accurate values of self-inductance for leads with regular shapes, the leads bonded in a practical circuit have non-uniform shapes that may significantly affect their

k k

4.4 Packaged Lumped-Element Passive Components 141

inductance. Ndip et al. [14] addressed the issue of non-uniform bonding structures of various shapes, and developed analytical models that showed good agreement between theory and practical measurement.

Example 4.7 Compare the self-inductances at 1 GHz of the leads, assumed to be made of copper, on each of the following components:

(i) A wire-ended component, of the form shown in Figure 4.14a, which has 5 mm long circular leads, each of diameter 0.6 mm.

(ii) A beam-lead component, of the form shown in Figure 4.14b, in which each lead has a length of 210μm, a width of 115μm, and a thickness of 10μm.

Comment upon the results.

Take the resistivity of copper as 1.56×10⁻⁸Ωm Solution

(i) Using Eq. (4.28):x=2𝜋×0.03×

√

2×4𝜋×10⁻⁷×10⁹

(1.56×10⁻⁸)⁻¹ =1.18×10⁻³. Substituting in Eq. (4.27):

T(x) = 0.873+ (0.00186×1.18×10⁻³)

1− (0.278×1.18×10⁻³) + (0.128× [1.18×10⁻³]²)≃0.873.

Substituting in Eq. (4.26):

L(cir.wire) =0.002×0.5× [

(4×0.5 0.06

)

−1+ 0.06

2×0.5+1×0.873 4

]

μH=2.78 nH.

(ii) Using Eq. (4.29):

L(beam lead)

=0.002×0.021× [

( 2×0.021 0.0115+0.0010

)

+0.5+0.2235

(0.0115+0.0010 0.021

)]

μH

=0.078 nH.

Comment: A flat beam-lead has significantly less self-inductance than a circular wire. Also, Eq. (4.26)indicates that the self-inductance of a circular wire will increase with frequency, whereas Eq. (4.29)shows that the inductance of a flat beam-lead will not.

4.4.2 Lumped-Element Resistors

There are a number of methods available for implementing resistors in high-frequency circuits. Resistors using carbon compositions as the resistive element can be formed into any of the package configurations shown in Figure 4.14. However, the granular nature of carbon composition resistors tends to degrade the performance at high frequencies. This degradation in performance has been attributed to parasitic capacitance effects between the granules of the composition. Wire-wound resistors are not suitable for high frequency work, because the winding tends to generate excess inductance. Metal-film resistors are the best type of resistor for high-frequency work, although in situations where thick-film technology is used to fabricate a circuit printed thick-film resistors are another good option. Thick-film resistors can also be laser-trimmed after printing, where it is necessary to have precise values of resistance.

In general, all lumped-element RF resistors (apart from printed resistors) will suffer from packaging parasitics which become more significant as the frequency increases. Figure 4.15 shows the equivalent circuit which is normally used to represent a high-frequency resistor with a nominal value ofR.

The circuit shown in Figure 4.15 includes two series inductors, each of valueL_S, to represent the lead inductances, and a capacitor,C_pk, to represent the packaging capacitance between the ends of the structure. The presence of both inductance and capacitance in the equivalent circuit means that the structure will exhibit a resonant frequency. Although the

k k

C_pk

Figure 4.15 Equivalent circuit representing a packaged high-frequency resistor.

resonant frequency is normally quite high, it can become a significant issue for circuits working in the microwave frequency range. For these frequencies, careful consideration must be given to the type of resistor used so as to minimize the parasitics.

Example 4.8 A packaged 5 kΩresistor has the equivalent circuit shown in Figure 4.15. The self-inductance of each lead is 2.8 nH and the packaging capacitance is 0.14 pF. Determine the impedance of the resistor at the following frequencies: 5, 50, and 500 MHz. Comment upon the result.

Solution

Referring to Figure 4.15, the total impedance,Z_T, is given by 1

Z_T = 1

R+j2𝜔L_S +j𝜔C_pk, which can be rearranged to give

Z_T = R+j2𝜔L_S 1−2𝜔²C_pkL_S+j𝜔C_pkR.

At 5 MHz∶ 𝜔L_S =2𝜋×5×10⁶×2.8×10⁻⁹=0.088Ω,

𝜔²C_pkL_S= (2𝜋×5×10⁶)²×0.14×10⁻¹²×2.8×10⁻⁹≃0, 𝜔CpkR=0.022Ω,

Z_T =5000+j2×0.088

1+j0.022 Ω = (4997.51−j109.77) Ω.

At 50 MHz∶ 𝜔LS=0.088×10Ω =0.88Ω, 𝜔²C_pkL_S≃0,

𝜔CpkR=0.022×10Ω =0.22Ω, Z_T =5000+j2×0.88

1+j0.22 Ω = (4771.36−j1047.94) Ω. At 500 MHz∶ 𝜔LS=0.088×100Ω =8.8Ω,

𝜔²C_pkL_S=0.004,

𝜔C_pkR=0.022×100Ω =2.2Ω, Z_T =5000+j2×8.8

1+j2.2 Ω = (858.30−j1885.74) Ω.

k k

4.4 Packaged Lumped-Element Passive Components 143

Summary:

Frequency (MHz) Z_T(𝛀)

5 4997.51−j109.77

50 4771.36−j1047.94

500 858.30−j1885.74

Comment: At high frequencies within the RF range, relatively small capacitive and inductive parasitics begin to have a very significant effect on the impedance of the resistor. At 500 MHz the reactive component of the impedance is larger than the resistive component.

4.4.3 Lumped-Element Capacitors

RF lumped-element capacitors are straightforward developments of the parallel-plate capacitor, in which the capacitance is obtained from two oppositely charged parallel conducting plates separated by a dielectric. The capacitance of the structure is then given simply by

C=𝜀A

d , (4.30)

where𝜀is the permittivity of the dielectric,Ais the area of the plates, anddis the distance between the plates. RF and microwave capacitors come in several configurations, although some traditional methods of construction such as those using wound strips of foil and dielectric are not suitable because they have excess inductance. Generally, capacitors for high-frequency usage have a planar or chip format, and are selected for the RF and microwave properties of the dielectric.

Suitable dielectrics will have low-loss, a reasonably high dielectric constant, and good thermal properties, which means a low temperature coefficient. The temperature coefficient represents the change in capacitance value for a given change in temperature, and is specified in ppm/∘C. Two of the most popular dielectric materials are ceramic and mica. Ceramic is a good choice because the material is normally low loss, has good thermal characteristics, and the component can be made small in size due to the high dielectric constants available with ceramic materials. Moreover, it is possible to obtain ceramic materials that have either positive or negative temperature coefficients. For example, magnesium titanate is a ceramic material with a low dielectric constant and positive temperature coefficient, whereas calcium titanate has a negative temperature coefficient. By mixing these two materials a ceramic dielectric is obtained that has high temperature stability.

Mixed dielectric materials are often referred to as NPO (negative–positive–zero) dielectrics, and used in circuits such as resonators and filters, where stability is essential. Mica is also a popular choice for high-frequency capacitors, because it is low loss and has a very low temperature coefficient. It is usually employed in film capacitors in which a conductor such as silver is deposited onto a mica film. Like NPO dielectric capacitors, silvered mica film capacitors are used in situations where temperature stability is critical. However, one slight drawback to the use of mica capacitors is that mica has a relatively low dielectric constant, around 6, and so the capacitor tends to be physically quite large.

The equivalent circuit of a capacitor is shown in Figure 4.16a, whereCis the nominal capacitance value,R_sis the series resistance of each lead,R_dis the resistance of the dielectric,L_Sis the inductance of each lead, or the equivalent inductance

R_S R_S

R_d

L_S L_S

(a)

(b)

C 2R_S 2L_S

Figure 4.16 Equivalent circuits representing capacitors.

k k equivalent circuit can be redrawn as a simple series resonant circuit shown in Figure 4.16b.

The presence of lead inductance together with significant capacitance means that the capacitor will have a resonant frequency, and care must be taken to ensure this is not within the working RF range of the component.

Example 4.9 What is the resonant frequency of a 10 pF capacitor, in which each conducting lead has an effective inductance of 0.7 nH? Comment upon the result.

Solution

The capacitor forms a series resonant circuit in which the capacitance is 10 pF and the inductance is (2×0.7) nH. The resonant frequency is then:

f_o= 1 2𝜋√

= 1

2𝜋√

1.4×10⁻⁹×10×10⁻² Hz

=1.34 GHz.

Comment: Realistic values were given in the question for C and L_S and these led to a relatively low resonant frequency, indicating that some caution is needed in the selection and application of lumped-element capacitors in high-frequency circuit designs.

Another useful parameter to specify the performance of an RF capacitor is the quality factor, represented byQ. Using the normal definition of the quality factor of a series resonant circuit containing a capacitor we have

1∕_𝜔C

R = 1

𝜔CR, (4.31)

whereCis capacitance andRis the effective series resistance (ESR). The expression forQin Eq. (4.31) is often written in the form

1∕_𝜔C

ESR. (4.32)

The value of ESR will depend on the resistance of the dielectric and the resistance of the capacitor leads.

4.4.4 Lumped-Element Inductors

The traditional method of constructing an inductor is to wind a coil of wire on a cylindrical former. The inductance is then given by the well-known formula

L= 0.394r²N²

9r+10l μH, (4.33)

where

r=radius of coil (cm)

N=number of turns on the coil l=length of coil (cm).

The accuracy of the expression for inductance given in Eq. (4.33) is reportedly better than 1% whenl>0.7r. Eq. (4.33) indicates that there is a range of inductor geometries (i.e. combinations ofrandl) that will give a particular inductance, but an optimum value ofQ¹results whenl=2r.

At RF, the parasitics associated with a simple coil inductor can be very significant. In particular, there is significant capacitance between the turns of the coil, and significant series resistance due to the relatively long length of conductor forming the coil. The resistance of the conductor will also increase with frequency due to the skin effect. The equivalent circuit for an inductor is shown in Figure 4.17.

Clearly there will be a resonant frequency associated with the circuit shown in Figure 4.17. Below resonance the component will be primarily inductive, and above resonance it will look capacitive.

1 Since an inductor will have both reactive and resistive components in series we can refer to theQof an inductor simply asQ=^𝜔L∕R.

k k

Dalam dokumen RF and Microwave Circuit Design (Halaman 157-163)