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Microwave Engineering notes for undergraduate students
Typology: Lecture notes
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v.1.3.
When the lossless line is terminated by a load
Reflected waves occur.
Reflection coefficient at the load,
where and consists of a superposition of an incident and reflected waves called Standing Waves ( ).
This shows is constant at anywhere on the line.
is constant.
Matched load (No reflected power, maximum power is delivered). Total reflection, (All power reflected).
is not constant.
The maximum value occurs when
. The minimum value occurs when. When increases, increases as a measure of mismatch. Then Standing Wave Ratio ( ) is
when means matched line. In that case at , the reflection coefficient and input impedance
Using the definition of , more useful form known as Transmission Line Impedance Equation as
Insertion Loss :
The proper length of open or short circuited transmission line can provide any desired reactance or susceptance.
In practice, finite conductivity (or lossy dielectrics) lines can be evaluated as a Lossy Line.
Low-loss line ,. Then ignoring the last term of
Power lost in the line:
Power flow along lossy line:
Power loss per unit length:
Attenuation constant:
: Incremental inductance
: Loss per unit length
: Power at the.
Then, Attenuation constant:
where is skin depth and.
where.
2.2. Smith Chart
where is Resistance, is Reactance, is Conductance and is Susceptance. Whenever is normalized impedance
The apsis and ordinate of Smith chart are and.
Rearranging them
3. MICROWAVE NETWORKS
At low frequencies for electrically small circuits, lumped active and passive circuit elements are enough for analyzing the circuit leading a type of a Quasi-Static solution (assumption of negligible phase change in any where of the circuit) of Maxwell's equations and to Kirchoff Current and Voltage Laws with impedance concept. Moreover fields are considered as TEM type. But this way is not possible to analyze microwave circuits. The circuit concept should modify and apply to microwave network theory developed in MIT in 1940. The reasons of using it are as follow
3.1. Voltage, Current and Impedance
The measurements of and at microwave frequencies are difficult due to not easily defined terminals for non-TEM waves. Because of that the fields are measured and used as
than, the impedance can be defined as. Because the fields depend on the coordinates (like in waveguide), special attenuation should be given for extraction of and. The way is to do that
then, the impedance can be defined as
. The impedance concept first used by O. Heaviside, and then after application to transmission lines, to electromagnetics by Schelkunoff. In this manner, types of it
It can be shown that the real parts of , and are even in , but the imaginery parts of them are odd in. and are the even function in.
3.2. Impedance & Admittance Matrices
The and of a port microwave network having 'th terminal
The impedance matrix is in the form of
Similarly the admittance matrix is in the form of
Clear that. It can be shown that
Then and are known as Input Impedance and Transfer Impedance , respectively.
The form of the scattering matrix gives the complete description of the port networks with the incident and reflected waves as
Any element of the scattering matrix
and are Reflection and Transmission Coefficient (from port to port ), respectively. Network analyzer is used to measure parameters of a network. If network is reciprocal, is symmetric. If network is lossless, is unitary means that
. Using the relations as and with between and matrixes
In the , all the and are defined as a reference point at the end of every lines. If the reference point is shifted, then
where is the electrical length of the outward shift.
3.2.1.1. Generalized Scattering Matrix
In the previous chapter, is defined for networks with same characteristic impedance for all ports. Generally for not same impedance for all ports, a new set of wave amplitudes as
Then, generalized matrix
Many microwave networks consisting cascade connection and need building block fashion in practice. ABCD matrix is defined to satisfy this as
where port 1 and port 2 are completely isolated in the equation means that cascade multiplication is possible. The current direction is also specially designed for ABCD matrix. The relation between and matrix parameters are as
If the network is reciprocal ( ), then.
3.3. Equivalent Circuits for 2 Port Networks
3.4. Signal Flow Graphs
Signal flow graph is an additional technique to analyse microwave networks in terms of reflected and transmitted waves. Three different forms of it are given below with nodes and branches.
4. IMPEDANCE MATCHING
Impedance matching (or tuning) is an important issue for
Whenever has nonzero real part, impedance matching is possible with the factors such as
4.1. L Networks Matching
The normalized should be converted to with adding impedance, then adding – the impedance matching will be successful.
4.2. Quarter Wave Transformer
It is used for only real load impedance. Complex load impedance can always be transformed to real impedance by appropriate length of transmission line. But this generally alters the frequency dependency of the equivalent load reducing the bandwidth of the matching. The following relation has to be satisfied as
where
For multiple reflection, total is
The condition of is also enough to make total reflection (multiple) is zero.
If and , then and.
For each frequency, has to be equal to. Thus fixed line length is possible only for one frequency.
Bandwidth performance of QWT for wide band matching: If one set the maximum value of reflection coefficient, that can be tolerated, then the fractional bandwidth is
This shows that as becomes closer to , the bandwidth increases. This result is valid only for TEM lines. The step changes of reactance effect can be compensated by making a small adjustment in the length of the matching section. Approximate behavior of reflection coefficients is shown at below.
The QWT can be extended as a multisection form for matching of broader bandwidth.
4.3. Single Stub Tuning
A single open-circuited (or short-circuited) transmission line is connected either in parallel or series with the feed line at a certain distance from the load. Single Sub Tuning can match any load impedance to line, but suffer from disadvantage of requiring a variable length between the load and stub.
Since lumped elements are not required, single stub is convenient and easy to fabricate in microstrip form. Two adjustable parameters and susceptance or reactance provided by stub. Although for microstrip lines open circuit is easy to fabricate since a via-hole is enough, for coax or waveguide short circuit is preferred since open circuit line may be large for radiation. If the impedance has the form of at the
distance. Then stub reactance can be chosen as – , resulting for matching condition. For a given susceptance or reactance, the difference in lengths of open and short-circuited stub is.
4.4. Double Stub Tuning
The double stub tuner can not match all load impedances, but load may be arbitrary distance from the first stub.
Distance between the stubs should be generally chosen as or to reduce the frequency sensitivity.
Single Section Transformer : The reflection coefficient of single section transformer can be written when discontinuities between the impedances are small as
This shows that is dominated by and.
Multisection Transformer : If the applications require more bandwidth, multisection transformers consists of equal length sections of the lines can be used. The reflection coefficient can be written as
The importance of this result is that the desired reflection coefficients response as a function of frequency can be synthesized by proper choosing of. To obtain passband responses, binominal (maximally flat) and Chebysev (equal
ripple) multisection matching transformers can be used. In first one: the derivatives of is settled to zero, in second one: is equated to Chebyshev polynomial.
4.5. Tapered Lines
The line can be continuously tapered for decreasing the effect of the step changes in characteristic impedance between the discrete sections. The incremental reflection coefficient
Since , by using theory of small reflections
where from can be found. Chancing type of taper (Exponential, , Triangular, Klopfenstein), different band pass characteristic may be applied. Klopfenstein yields the shortest matching section. The Bode-Fano criterion for certain type of canonical load impedances will help us to define theoretical limit on the minimum reflection with the upper limit of matching performance and provide a benchmark against which a practical design can be compared.
It is a network with the useful property of being lossless when the output ports are matched, that is, only reflected power is dissipated. It is known that a lossy three port network can be made having all ports are matched with isolation between the output ports. Wilkinson Power Divider can be made in microstrip or stripline form with arbitrary power division of way Divider or Combiner. The even-odd mode technique is used for analysis.
5.2.3.1. Quadrature Hybrid ( Hybrid)
This is a directional coupler (knows as Branch Line
Hybrid ) with a phase difference in outputs (2 3). Even-
odd mode technique can be applied for analysis. matrix has a high degree of symmetry means any port can be used for input as given below
It is a four port network with a phase shift (2 3) between two outputs (also may be in phase). It can be used as a combiner and has unitary symmetric scattering matrix as
It may be produces as the form of ring hybrid (rate race), tapered matching lines and hybrid waveguide junction (Magic T, (Rate Race)) in which symmetrically (or antisymmetrical) placed tuning ports (or irises) can be used for matching.
Coupled lines of two (or more) transmission lines are closed together, power can be coupled between the lines. Generally TEM mode is assumed rigorously valid for striplines, but approximately valid for microstrips. Coupled Line Theory is based on types of excitations as even mode (strip currents are equal in amplitude with same directions) and odd mode (strip currents are equal in amplitude with opposite directions). Arbitrary excitation can be treated as a superposition of appropriate even and odd modes amplitudes. Moreover design graphs are present for coupled lines. Design Considerations:
To increase coupling factor, Lange Coupler (several lines) with phase difference between outputs is used as a 3 dB coupling ratio in an octave or more bandwidth can be achieved. The main disadvantage of it (a type of quadrature hybrid) is difficult to fabricate due to very narrow lines. Folded Lange coupler is also used for more easily analysis to model equivalent circuit.
5.3. Other Couplers
Moreno Crossed Guide Coupler Schwinger Reversed Phase Coupler Riblet Short Slot Coupler Symmetric Tapered Coupled Line Coupler Coupler with Apertures in Planar Lines
As an example of a device uses a directional coupler is Reflectometer isolate and sample the incident and reflected powers from a mismatch load as a heart of a scalar (or vectorial) network analyzer.
6. NOISE & ACTIVE COMPONENTS
Noise is usually generated by random motions of charges (or charge carriers in devices and materials). Such motions can be caused by the mechanism of
Thermal Noise: Thermal vibrations of bound charges. Shot Noise: Random fluctuations of charge carriers. Flicker Noise: noise. Plasma Noise: Random motions of charges. Quantum Noise: Quantized nature of charge carriers.
Noise is a random process and can be passed into a system from external sources or generated within the system itself. Noise level defining the system performance determines for minimum
signal reliability detected by a receiver.
Dynamic Range and Compression Point : The linearity and deterministic features of all components can be satisfied in a range called Dynamic Range. The floor level of noise dominates the output power at very low frequencies. Compression Point is defined as the input power for which the output is below that of an ideal amplifier.
Noise Power and Equivalent Noise Temperature : Rayleigh- Jeans approximation results Voltage Fluctuations as
where is Boltzmann’s constant, is temperature, is bandwidth and is resistance. Because of frequency independency, this is known as White Noise Source can be treated as Gaussian distributed variables. A noisy resistor can be replaced with a noiseless resistor and a voltage source of RMS. Then connecting a load resistor results in maximum power transfer called Noise Power as
If is not strong function of frequency ( White Noise ), an Equivalent Noise Temperature is defined as
where is Noise Power delivered to load. A noisy amplifier with a source of resistor at a temperature of can be replaced with a noiseless amplifier and a resistor having Equivalent Noise Temperature as
where is output noise power and is amplifier gain. Excess Noise Ratio (ENR) is also used to characterize Noise Power of active noise generator consisting of a diode or a tube as
where & are Noise Power & Equivalent Temperature of generator.
Measurement of Noise Power: factor method is applied as
where should be determined via power measurement. Then
where & are temperature of hot & cold load, respectively.
6.1. Noise Figure,
It is a measure of the degration in ratio between the input and output as
is defined for a matched input source and for a noise source consist of a resistor temperature.
Noise Figure of a Noisy Network : Having the parameters of with input noise and signal power , the output noise power and the output noise signal. Then Noise Figure is
If the network is noiseless ,.
Noise Figure of a Two-Port Passive and Lossy Network : Having such as attenuator (or lossy line) with a matched source resistor at , overall system temperature also at , noise factor
where is lossy factor. The equivalent noise temperature
6.3. RF Diode Characteristics
Shottky Barrier Diode Detectors : This is a nonlinear device consisting of semiconductor-metal junction resulting lower junction capacitance can be used frequency conversion (rectification, detection, mixing). It has a
with a Small Signal Model
These diodes are used as rectifiers, detectors and demodulation of an AM modulated RF carrier.
PIN Diode: This is used to construct an electronic switching for control circuits such as phase shifters and attenuators. These are preferable because of small size, high speed and inerrability with planar circuits. Especially single-pole PIN diode switches can be used in either a series or a shunt configuration to form a single pole RF switch. Insertion Loss of switches
where is diode impedance as
Varactor Diode : Junction capacitance varies with bias voltage used for electronically frequency tuning.
Impatt Diode : Similar to PIN diode, but based on avalanche effects exhibiting negative resistance over a broad frequency range, therefore used to directly convert DC to RF power.
Gunn Diode : It exhibits a negative differential resistance based on Gunn effect and used to generate RF power to DC.
Baritt Diode : Similar to junction transistor without a base contact and useful for detector and mixer applications with advantages of lower AM noise.
7. MICROWAVE AMPLIFIER DESIGN
7.1. Two-Port Power Gains
The gain and stability of a general two-port amplifier in terms of parameters of transistor will be investigated for amplifier and oscillator design. Three types of power gain can be derived as
where is independent of. is defined with an assumption that conjugate matching of both source and load depend on but not. depends on both and. Whenever input and output are both conjugately matched, gain is maximized and .
The average power delivered to network
The power delivered to load
Then, the power gain
where
If , then.
If , then , Unilateral Transducer Power Gain ,
More generally, most useful power definition is Transducer Power Gain account for both source and load mismatch
where
If transistor is unilateral, ; , , then
Similar relation can be obtained by Equivalent circuit parameter.
7.2. Stability
There are necessary conditions for a transistor amplifier to be stable based on the possible oscillation for input and output impedance has a negative real part as a two-sub group:
The stability condition is usually frequency dependent since matchings generally depend on frequency (stability may be possible for a frequency but not possible for others). Rigorous treatment of stability requires parameters of network have no poles in the right-half complex plane in addition to and. If device is unilateral , more simply results and are enough for stability.
Stability Circles : Applying the above requirement for unconditional stability, following conditions have to be satisfied
These conditions define a range for and where amplifier will be stable. Finding this range by using Smith chart, plotting the input and output Stability Circles are defined as loci in the (or ) plane for which (or ), then define boundaries between stable and unstable regions. The equations for input and output stability conditions can be extracted as
