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Preface
The chapter focuses on the design of the matching networks and considers the effects of these networks on transducer gain, noise, and stability. The chapter includes a discussion of crystal oscillators, along with an overview of the major types of frequency synthesizers that use phase-locked loops.
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RF Transmission Lines
Introduction
Voltage, Current, and Impedance Relationships on a Transmission Line
Furthermore, if a transmission line is terminated at its characteristic impedance, the impedance at the input of the line will be equal to the characteristic impedance; The voltage,V, and current,I, at any distance,z, along the transmission line can now be found in terms of the voltage and current at the transmitting side by substituting V1 and V2 from Eq.
Propagation Constant
- Dispersion
- Amplitude Distortion
The group velocity is the speed at which this envelope propagates along the transmission line. If 𝛽 is a linear function of frequency, then the 𝛽–𝜔response will be a straight line and the group delay will be constant and independent of frequency.
Lossless Transmission Lines
If this speed is independent of frequency, the line will be dispersionless and the phase relationships between the frequency components of the signal will be preserved. 𝛾=j𝛽 also changes the expression for the input impedance of a transmission line, and Eq.
Waves on a Transmission Line
- Derivation of the Smith Chart
- Properties of the Smith Chart
Impedance points are plotted on the graph by locating the intersection of the appropriate resistance and reactance lines. A radial line is then drawn from the center of the plot to pass through the required angle (60∘ in this example) on the reflection coefficient scale, which is printed around the periphery of the plot area.
Stubs
Distributed Matching Circuits
Any complex load impedance can be matched to a real impedance by attaching a stub of the correct length to a suitable point on a transmission line connecting the source to the load. In this example, the stub provides a matching susceptance across the line, and we have assumed that the characteristic impedance of the line and the stub are the same.
Manipulation of Lumped Impedances Using the Smith Chart
This gives the position of y1. Note that this movement represents the addition of the 6.93 pF capacitor; we move clockwise because a capacitor has positive susceptance. This gives the position ofy3. Note that this movement represents the addition of the 4.87 pF capacitor; we move clockwise because a capacitor has positive susceptance.
Lumped Impedance Matching
- Matching a Complex Load Impedance to a Real Source Impedance
- Matching a Complex Load Impedance to a Complex Source Impedance
The network must be composed of two lossless reactances, with the configuration shown in Figure 1.26. The network must be composed of two lossless reactances, with the configuration shown in Figure 1.34.
Equivalent Lumped Circuit of a Lossless Transmission Line
Supplementary Problems
Show that there are two possible solutions and find the values of the component in both cases. ii) Assume that the frequency decreases by 15%. Show that a network consisting of two inductors will provide a suitable match and find the values of the inductors.
Coaxial Cable
The speed of propagation along a coaxial cable depends only on the permittivity of the dielectric filling the cable, and since the propagation is TEM, the speed is given simply by The cutoff wavelength is related to the physical parameters of the cable with [2].
Coplanar Waveguide
At high microwave frequencies, the propagation becomes non-TEM as there are longitudinal components in the magnetic field. 2 The value of the complete elliptic integral with a given modulus k can be found from tables or as an approximation [1] by summing the first three terms in the following series:.
Metal Waveguide
The line LN in figure 1.56 represents a plane wave traveling at the speed of light, c, at an angle 𝜃to the axis of the waveguide. In a waveguide, the free space impedance is modified by the presence of the waveguide walls. Hz=HOJn(kcr)cos(n𝜃)ej(𝜔t−𝛽z). 1.95) Note that a function of the form Jn(x) is called a Bessel function of the first kind, of orderm and argumentx; the values of the function are usually found from tables [6].
Microstrip
The wavelength along the microstrip line (also known as the substrate wavelength) at a given frequency (f) will be 𝜆s=vp. Hammerstad and Bekkadal [9] give the following widely used expression for the conductor loss, 𝛼c, in microstrip 𝛼c = 0.072√. The dielectric loss in a microstrip line can be calculated from the expression given by Gupta [4] as 𝛼d=27.3𝜀r(𝜀MSTRIPr,eff −1)tan𝛿.
Equivalent Lumped Circuit Representation of a Transmission Line
A length of lossless transmission line, with a characteristic impedance ZO and a propagation constant𝛽, and shown in Figure 1.65, can also be represented by an ABCD matrix [10], and it is given in Eq. If the electrical performance of the π-network shown in Figure 1.64 is to be equivalent to that of a short transmission line, the ABCD matrix in Eq.
Planar Circuit Design I
Introduction
Electromagnetic Field Distribution Across a Microstrip Line
Effective Relative Permittivity, 𝜺 MSTRIP r,eff
Microstrip Design Graphs and CAD Software
Operating Frequency Limitations
The alternating current produces a changing magnetic field (magnetic flux) in the conductor, causing an induced voltage in the conductor. It is defined as the depth in the conductor at which the magnitude of the current has decreased to 1/e of the surface value (e=2.718), and is indicated in Figure 2.3, where the surface current is denoted by Is. The value of𝛿can be calculated. where 𝜇 is the permeability of the conductor, f is the operating frequency and 𝜎 is the conductivity of the conductor.
Examples of Microstrip Components
- Branch-Line Coupler
- Quarter-Wave Transformer
- Wilkinson Power Divider
For this reason, the physical appearance of the clutch is in the shape of a rectangle rather than a square. A broadband, maximally flat response (Butterworth response) will be obtained [3] when the input reflection coefficient, Γ, is of the form Example 2.6 Find the bandwidth of the transformer designed in Example 2.5 if the maximum VSWR that can be tolerated at the input is 1.5.
Microstrip Coupled-Line Structures
- Analysis of Microstrip Coupled Lines
- Microstrip Directional Couplers
- Design of Microstrip Directional Couplers
- Directivity of Microstrip Directional Couplers
- Improvements to Microstrip Directional Couplers
- Examples of Other Common Microstrip Coupled-Line Structures .1 Microstrip DC Break
- Edge-Coupled Microstrip Band-Pass Filter
- Lange Coupler
Equations relating odd and even mode impedances to the geometry of paired lines are given in [1]. The value of the coupling coefficient, k. ii) Values of odd and even mode impedances. Two techniques can be used to equalize the transmission phases of the odd and even modes over the length of the coupled region.
Summary
Supplementary Problems
P2.7 A microstrip circuit is to be fabricated on a substrate having a relative permittivity of 9.8 and a thickness of 0.635 mm. The transformer must be fabricated on a substrate that has a relative permittivity of 9.8 and a thickness of 0.5 mm. The connector must be fabricated on a substrate that has a relative permittivity of 9.8 and a thickness of 0.4 mm.
Microstrip Design Graphs
Question 2.17 Suppose a thick layer of dielectric, also with a relative permittivity of 9.8, is placed on top of the coupler designed in question 2.16. Question 2.18 A 50Ωmicrostrip edge-coupled directional coupler is to be designed to provide 14 dB of coupling at 15 GHz. Q2.19 Determine the frequency range over which the coupler designed in Q2.18 will maintain coupling to within 0.5 dB of the designed value.
Fabrication Processes for RF and Microwave Circuits
Introduction
Review of Essential Material Parameters
- Dielectrics
- Conductors
It can be shown [1] that the loss tangent and the components of the complex permittivity are related. Conductor losses can be divided into losses due to the bulk resistance of the material and losses due to surface roughness. The skin depth, 𝛿s, is found using Eq. where we have assumed that the relative permeability of the conductor is unity.
Requirements for RF Circuit Materials
It is essentially the impedance between two electrodes placed along opposite edges of a square on the surface of the conductor, and so are the units of Ohms per square, normally written in abbreviated form as Z◽. This is particularly important for the substrates used in thick film fabrication (which is covered later in the chapter) where the substrate can be subjected to temperatures of up to 1000∘C during the firing of the metal conductors. This is especially important for a microstrip where most of the lane current flows in the bottom of the metal.
Fabrication of Planar High-Frequency Circuits
- Etched Circuits
- Thick-Film Circuits (Direct Screen Printed)
- Thick Film Circuits (Using Photoimageable Materials)
- Low-Temperature Co-Fired Ceramic Circuits
8] showed that by printing a layer of low permittivity dielectric under the conductor on an aluminum oxide substrate (as shown in Fig. 3.6), the characteristic impedance of the line can be significantly increased without reducing its width. In some cases, the required spacing between two microstrip lines in the same plane can be on the order of 10 μm, which is difficult to manufacture. An enlarged view of the finger coupling is given in Figure 3.8, showing the details of the conductor overlap.
Use of Ink Jet Technology
Although one of the main advantages of inkjet technology is the ability to process high-frequency circuits on thin, flexible substrates at moderate temperatures, the process is also very attractive for the production of multilayer circuits on thick, rigid substrates. 19] demonstrated the capability of the inkjet process for printing BST materials by fabricating a tunable S-band phase shifter. Another example of the versatility of inkjet technology for high frequency work was provided by Chen and Wu [20] who showed that carbon nanotube solutions can be successfully printed as part of an all-inkjet printed phased array antenna operating at 5 GHz.
Characterization of Materials for RF and Microwave Circuits
- Measurement of Dielectric Loss and Dielectric Constant
- Cavity Resonators
- Dielectric Characterization by Cavity Perturbation
- Use of the Split Post Dielectric Resonator (SPDR)
- Open Resonator
- Free-Space Transmission Measurements
- Measurement of Planar Line Properties
- The Microstrip Resonant Ring
- Non-resonant Lines
- Physical Properties of Microstrip Lines
We can find an expression for the resonance frequency of the cavity by substituting from Eq. In Figure 3.18, a small circular coupling iris is shown in the center of the cross plate. The electric field lines in the TE01𝛿 state form closed loops in the plane of the sample as shown in Figure 3.25.
Supplementary Problems
Dielectric characterization by microwave cavity perturbation corrected for non-uniform fields. IEEE Transactions on Microwave Theory and Techniques. Accurate measurement of permittivity by means of an open resonator. Proceedings of the Royal Society of LondonA. Transmission line analysis of a capacitively coupled microstrip ring resonator. IEEE Transactions on Microwave Theory and Techniques.
Planar Circuit Design II
Introduction
Discontinuities in Microstrip
- Open-End Effect
- Step-Width
- Corners
- Gaps
- T-Junctions
Determine the required length of stub, apply the necessary compensation for the open effect. Normal practice is then to shorten the length of the narrow line by 𝛿lL to compensate for the series inductance. The value of L can be determined from the properties of the narrow microstrip lines using Eq.
Microstrip Enclosures
Dydyk [8] proposed a relatively simple compensation arrangement for microstrip T-junctions using short matching sections of transmission line to achieve a minimum mismatch at the junction, resulting in the general form of junction shown in Figure 4.12a be shown. Chadha and Gupta [9] proposed a simple compensation technique, shown in Figure 4.12b, which involved removing a triangular portion of microstrip trace at the junction. No general design equations were given for this technique, but published data showed that it gave good results up to frequencies around 8 GHz.
Packaged Lumped-Element Passive Components
- Typical Packages for RF Passive Components
- Lumped-Element Resistors
- Lumped-Element Capacitors
- Lumped-Element Inductors
Resistors using carbon composites as the resistive element can be formed in any of the package configurations shown in Figure 4.14. The value of ESR will depend on the resistance of the dielectric and the resistance of the capacitor cables. The resistance of the conductor will also increase with frequency due to the skin effect.
Miniature Planar Components
- Spiral Inductors
- Loop Inductors
- Interdigitated Capacitors
- Metal–Insulator–Metal Capacitor
Example 4.10 Determine the inductance provided by each of the following structures, and comment on the results. Each of the structures is formed from a flat (ribbon) conductor with a width of 0.8 mm of negligible thickness. i). Example 4.11 Calculate the capacitance of each of the following structures, and comment on the results:
Insertion Loss Due to a Microstrip Gap
- Parameters
- Introduction
- S-Parameter Definitions
- Signal Flow Graphs
- Mason’s Non-touching Loop Rule
- Reflection Coefficient of a Two-Port Network
- Power Gains of Two-Port Networks
- Stability
- Supplementary Problems
The loop gain is equal to the product of the S-parameters around the loop. The second-order loop gain is equal to the product of the two first-order loop gains. The gain of a third-order loop is equal to the product of the gains of three first-order loops.
Relationships Between Network Parameters
Q5.9 Assume that the FET specified in Q5.8 was connected between a 50Ω source and a 50Ω load and that source and load matching networks were used to provide conjugate impedance matching at the input and output of the FET. Conversions between S, Z, Y, h, ABCD and T parameters valid for complex source and load impedances. IEEE Transactions on Microwave Theory and Techniques. Comments on "conversions between S, Z, Y, h, ABCD and T parameters valid for complex source and load impedances". IEEE Transactions on Microwave Theory and Techniques.
Microwave Ferrites
Introduction
Basic Properties of Ferrite Materials
- Ferrite Materials
- Precession in Ferrite Materials
- Permeability Tensor
- Faraday Rotation
When the frequency of the disturbing AC field is equal to the natural precession frequency, then. Magnetization refers to the density of the magnetic field and is expressed in units of Weber per square meter (Wb/m2). We know that the spinning electrons inside the ferrite will precess around the direction of the DC magnetic field.
Ferrites in Metallic Waveguide
- Resonance Isolator
- Field Displacement Isolator
- Waveguide Circulator
Thus, the free precessional motion of electrons in ferrite will be counterclockwise. For clarity, the details of rotations are shown in figure 6.12, where time t=0, corresponds to the magnetic field distribution shown in figure 6.11. Comment: The presence of the isolator has significantly reduced the mismatch at the output of the signal source.
