Design considerations for the traditional low frequency circuits and the RF circuits are quite different. In low frequency design, the maximum signal transfer occurs when the source presents low impedance while the load presents high impedance. A typical example is a buffer, where the input impedance is high and the output impedance is low. As long as that requirement is fulfilled, the designer is capable of choosing arbitrary levels of impedance that best suits the circuit requirements or applications.
Therefore this chapter aims to provide background on impedance matching between source and load, with or without a transmission line. The analysis can be conducted by using Smith Charts and S-Parameters, which are also presented in this chapter. The analysis in this chapter is oriented to RFID applications whereas other books provide general analysis.
During RF design, the impedances should be matched for maximum signal transfer. Additionally, when the circuits are connected using transmission lines, they should match also the standard values of the transmission lines. At very low frequencies, transmission lines can be thought as just a wire. Nevertheless, at high frequencies, the signal wavelength is comparable to or smaller than the length of the transmissions line and power can be seen as traveling waves. As a matter of fact, even a conductor can be thought as a transmission line in a high frequency circuit.
Most RF equipments and coaxial cables use the standard impedances of 50 or 75 Ω. The value of 75 Ω is used, as an example, in cable TV equipment, since this value provides the minimum losses, as it is desired in transmitting the signal over long distances. In fact, the value of impedance for minimum loss should be 77 Ω, but it was rounded to 75 Ω by convenience.
The value of 50 Ω corresponds to a reasonable compromise, the average, between the minimum loss of a 77 Ω and the maximum power handling capability given of 30 Ω.
2. Transmission line
Fig. 1 shows the lumped component model of a real (lossy) transmission line. The segment indicated corresponds to an infinitesimal segment of the transmission line. The characteristic impedance
As can be observed, the characteristic impedance
If the value of
2.1. Reflection coefficient
If a transmission line is terminated by an impedance
The incident signal is given by:
At the load end, the mismatch in impedances gives rise to a reflected signal. Since the system is still linear, the total voltage at any point in the system is the sum of incident and reflected voltages. The net current is superposition of incident and reflected currents. However, since the currents are traveling in opposite directions, the net current is the difference between them. Therefore, the load impedance is given by:
Expression (4) can be rewritten to express
The ratio of reflected to incident quantities at the load end of the line is called Reflection Coefficient
As can be observed from expression (7), if the impedances of the load and the line are equal, there will be no reflection. If the line is terminated in either a short or an open circuit, the reflection will be maximum, with a magnitude of 1 .
Therefore, if a transmission line is terminated by its characteristic impedance there will be no reflection since all the transmitted power is absorbed by the load and the energy flows in just one direction.
When the line is terminated by a short circuit a reflected wave is sent back to the source since the short can not sustain any voltage, and therefore dissipates zero power. The incident and the reflected voltage waves are of the same magnitude. They are 180 out of phase at the load and they travel in opposite directions.
If the line is terminated by an open circuit a reflected wave is sent back to the source since the open can not sustain any current, and therefore dissipates zero power. The incident and the reflected current waves are of the same magnitude and travel in opposite directions. The current waves are 180 out of phase at the load, but the incident and reflected voltage waves are in phase .
If the line is terminated by an impedance different of the short, open and characteristic impedance, part of the signal will be absorbed by the load and part will be reflected back. The amount of reflected signal is given by expression (7).
3. Smith chart
The reflection coefficient
The reflection coefficient can become even more convenient by normalizing it to
On the same way, normalizing (6) results in:
Considering the normalized real and imaginary parts of both
After some algebraic manipulation (using conjugate), the real and imaginary parts are of
Expression (11) can be manipulated as:
Similarly, expression (12) into:
When the two parametric equations (13) and (14) are drawn on a complex coordinate, they build the Smithchart. Equation (13) forms resistance circles, and equation (14) generates reactance circles, as shown in Fig. 2 and Fig. 3, respectively. The resulting Smithchart is illustrated in Fig. 4.
As can be verified from expression (13), the imaginary axis in the
As can be verified from expression (14), lines of constant reactance are perpendicular to lines of constant resistance in the
The Smith, as shown in Fig. 4, is just the plotting of both constant resistance and constant reactance, but without the presence of the
The upper half of the Smith chart corresponds to the upper half part of the
Although the Smith chart presents many interesting and useful properties, they will no be presented here due to the focus of this material.
2.1. Admittance chart
The Smithchart is built by considering impedance (resistance and reactance). Once the Smithchart is developed, a similar approach can be used for admittance analysis. The concept that admittance is the inverse of impedance is very important for parallel circuit synthesis. Adding new elements in series can be resolved easily by adding the impedance values. However, summing elements in parallel can be cumbersome in terms of impedance. Thus, admittance is often considered for parallel elements.
By definition, admittance is expressed as:
where G is conductance and B is susceptance of the element. The reflection coefficient
The admittance reflection coefficient
As can be observed, the value of the admittance reflection coefficient
The admittance Smithchart can be obtained using the same procedure used to construct the impedance Smithchart. The normalized real and imaginary parts of
After some algebraic manipulation (using conjugate), the real and imaginary parts are of
When the two parametric equations (21) and (22) are drawn on a complex coordinate, they build the Admittance Smithchart. Equation (21) forms resistance circles, and equation (22) generates reactance circles, as shown in Fig. 5.
3. S Parameters
At low frequencies, linear systems can be analyzed by means of voltages and currents applied to its ports. The two port circuit shown in Fig. 5 could be analyzed from its impedance (Z-parameters), admittance (Y-parameters), or a mixture of them, which could be hybrid (H-parameters) and inverse-hybrid (G-parameters).
As an example, the circuit of Fig. 5 could be analyzed using H parameters whose equations are:
As can be observed from the equations, the value of
By the same way,
Shorting or opening terminals is feasible at low frequencies but virtually impossible at high frequencies, particularly over a broad range of frequencies. Additionally, RF circuits are very sensitive to impedances, and they may oscillate or just quit working when terminated with open or short circuits. Therefore, Z-parameters, Y-parameters, H-parameters and G-parameters are not suitable to high frequency operations.
For high frequency operations, the
The S-parameters takes advantage of the fact that there is no reflection in a line terminated in its characteristic impedance. Therefore, it is necessary a circuit representation for S-parameters, where source and the load terminations are
The S-parameters equations are:
The normalization by is very convenient since the square of
The magnitudes of
The magnitudes of
3.1. Measurements of S-Parameters
The circuit topology used to measure S-Parameters is given in Fig. 7.
The input reflection coefficient
This expression, using the concepts of voltage division, corresponds to:
The value of
3.1. S Parameters in the Smith chart
The center point of the Smith chart corresponds to the point of zero reflection, where
As the frequency increases, the S-Parameters plots in the Smith chart move clockwise.
Given the value of
3.2. Application example
Fig. 10 shows its
The same parameters could be plotted in a standard dB format, as shown in Fig. 11. The graph of Fig. 10 provides more information and insight than the graph of Fig. 11. The last one provides only the magnitude, whereas the first one provides both the imaginary and real part, so that it is possible to infer a capacitive and/or inductive behavior of the circuit, among other information.
Unfortunately, it is not always possible to analyze S-parameters using Smith chart. One such case is
4. Noise figure/factor
In analog circuits at low frequency, the signal-to-noise figure (SNR), defined as the ratio of signal power to the noise power, is an important and very used parameter. As an example, in a radio receiver, it indicates the quality of the demodulated signal [2-5].
Nevertheless, as the signal passes through the RF circuits, the SNR changes. This signal-to-noise degradation along the system is described by the noise factor (F), as:
where the index
If a system has no noise, then
which can be seen as the total output noise power over the output noise due to the input source.
The total output noise is the sum of the original noise at the input (which was amplified) and the noise added by the circuit. This can be denoted as:
Therefore, expression (18) can be expressed also as:
Again, if the circuit adds no noise, F becomes 1.
Another important figure of merit is the noise figure,
While the noise factor of a noiseless circuit is 1, the noise figure is 0dB.
4.1. Noise figure of a cascade system
Fig. 12 shows a cascade amplifying system, whose gain of each stage is
The output noise due to the source is:
While the total output noise is:
which is the input noise multiplied by the gain of the three stages, plus the noise generated by the first stage and amplified by the stages 2 and 3, plus the noise generated by the second stage and amplified by the last stage, and plus the noise generated by the last stage.
This expression, with the aid of expression (27), can be re-written as:
As can be observed from expression (32), the noise factor of the first stage is the most relevant to the total noise factor. That is the reason for putting most effort in the first stage in terms of noise minimization, thus requiring low noise amplifiers at the front of the system.
The basic knowledge of impedance matching between source and load, either with or without a transmission line is essential to the design of RF circuits. The analysis presented can be conducted by using Smith Charts and S-Parameters. The analysis in this chapter was oriented to RFID applications whereas other books provide general analysis.