Impedance matching and the basic principles of the Smith chart[Copy link]
When dealing with practical application problems of RF systems, there are always some very difficult tasks, and matching the different impedances of the cascaded circuits is one of them. In general, the circuits that need to be matched include the matching between the antenna and the low noise amplifier (LNA), the matching between the power amplifier output (RFOUT) and the antenna, and the matching between the LNA/VCO output and the mixer input. The purpose of matching is to ensure that the signal or energy is effectively transmitted from the "signal source" to the "load". At high frequencies, parasitic elements (such as inductance on the connection, capacitance between board layers, and resistance of conductors) have a significant and unpredictable effect on the matching network. When the frequency is above tens of megahertz, theoretical calculations and simulations are far from meeting the requirements. In order to obtain appropriate final results, RF tests in the laboratory must also be considered and properly tuned. Calculated values are needed to determine the structural type of the circuit and the corresponding target component values. There are many methods of impedance matching, including: Computer simulation: Because this type of software is designed for different functions than just impedance matching, it is more complicated to use. Designers must be familiar with entering a lot of data in the correct format. Designers also need to have the skills to find useful data from the large amount of output results. In addition, circuit simulation software may not be pre-installed on the computer unless the computer is specially built for this purpose. Manual calculation: This is an extremely tedious method because it requires long ("several kilometers") calculation formulas and the data being processed are mostly complex numbers. Experience: Only those who have worked in the RF field for many years can use this method. In short, it is only suitable for experienced experts. Smith chart: This is the focus of this article. The main purpose of this article is to review the structure and background of the Smith chart and summarize how it is used in practice. Topics discussed include practical examples of parameters, such as finding the values of matching network components. Of course, the Smith chart can not only help us find matching networks for maximum power transfer, but also help designers optimize noise figure, determine the impact of quality factors, and perform stability analysis.
Figure 1. Impedance and Smith Chart Basics
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Basics Before introducing the use of the Smith chart, it is helpful to review the phenomenon of electromagnetic wave propagation in IC wiring in an RF environment (greater than 100MHz). This is valid for applications such as RS-485 transmission lines, connections between PAs and antennas, and connections between LNAs and downconverters/mixers. As we all know, to maximize the power delivered from the source to the load, the source impedance must be equal to the conjugate impedance of the load, that is: Rs + jXs = RL - jXL Under this condition, the energy transferred from the source to the load is maximized. In addition, for efficient power transfer, meeting this condition can avoid energy reflection from the load to the source, especially in high-frequency application environments such as video transmission, RF or microwave networks. The Smith chart is a graph consisting of many circles interwoven together. Used correctly, it is possible to obtain the matched impedance of a seemingly complex system without any calculations. The only thing that needs to be done is to read and track the data along the circular line. The Smith chart is a polar coordinate plot of the reflection coefficient (gamma, expressed in symbols). The reflection coefficient can also be mathematically defined as a single-port scattering parameter, namely s11. The Smith chart is generated by verifying the impedance matching of the load. Here we do not consider the impedance directly, but the reflection coefficient L, which can reflect the characteristics of the load (such as admittance, gain, transconductance). When dealing with RF frequency problems, L is more useful. We know that the reflection coefficient is defined as the ratio of the reflected wave voltage to the incident wave voltage:
The strength of the load reflected signal depends on the mismatch between the source impedance and the load impedance. The expression of the reflection coefficient is defined as:
Since impedance is a complex number, the reflection coefficient is also a complex number. In order to reduce the number of unknown parameters, a parameter that often appears and is often used in applications can be solidified. Here Zo (characteristic impedance) is usually a constant and a real number, which is a commonly used normalized standard value, such as 50, 75, 100 and 600. We can then define the normalized load impedance:
From the above equation, we can see the direct relationship between the load impedance and its reflection coefficient. However, this relationship is a complex number, so it is not practical. We can think of the Smith chart as a graphical representation of the above equation. To create the circle chart, the equation must be rearranged to conform to the form of a standard geometric figure (such as a circle or ray).
Figure 4a. Points on a circle represent impedances with the same real part. For example, the circle with R = 1 has a center at (0.5,0) and a radius of 0.5. It contains the origin (0,0) (the load matches the characteristic impedance). The circle with (0,0) as the center and a radius of 1 represents a short-circuited load. When the load is open, the circle degenerates to a point (with 1,0 as the center and a radius of zero). Corresponding to this is the maximum reflection coefficient of 1, that is, all incident waves are reflected back.
There are some issues that need to be paid attention to when making a Smith chart. The following are the most important aspects: All circles have only one identical, unique intersection point (1,0). The circle representing 0, that is, no resistance (r=0) is the largest circle. The circle corresponding to infinite resistance degenerates to a point (1,0) In practice, there is no negative resistance. If negative resistance appears, oscillation may occur. Selecting a circle corresponding to a new resistance value is equivalent to selecting a new resistance.
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