Impedance Matching and Smith circle diagrams: Fundamentals

When dealing with the practical application of RF systems, there are always some very difficult work, and the matching of the different impedance of each part of the hierarchical circuit is one of them. In general, the circuits that need to be matched include the matching between antenna and low noise amplifier (LNA), the matching between power amplifier output (RFOUT) and antenna, and the matching between LNA/VCO output and mixer input. The purpose of matching is to ensure that the signal or energy is transmitted efficiently from the "source" to the "load".

At the high frequency end, parasitic elements (such as inductance on the wire, capacitance between the plates, and resistance of the conductor) have a noticeable and unpredictable effect on the matching network. At frequencies above tens of megahertz, theoretical calculations and simulations are far from sufficient, and RF testing in the laboratory and appropriate tuning must be taken into account in order to achieve proper finality. It is necessary to determine the circuit structure type and corresponding target element value with the calculated value.

There are many methods of impedance matching, including

• Computer simulation: This type of software is complicated to use because it is designed for different functions rather than just impedance matching. Designers must be familiar with entering large amounts of data in the correct format. Designers also need the skill to find useful data from a large number of outputs. In addition, circuit simulation software cannot be pre-installed on a computer unless the computer is built specifically for this purpose.

• Manual calculation: This is an extremely tedious method because of the long (" kilometers ") formulas and the complex numbers being processed.

• Experience: Only people who have worked in RF field for many years can use this method. In short, it is only suitable for experienced experts.

• Smith's circle diagram: the focus of this article.

The main purpose of this paper is to review the structure and background knowledge of Smith's circle diagram, and summarize its application in practice. Topics discussed include practical examples of parameters, such as finding values that match network components. Of course, Smith circle diagram can not only find the matching network of maximum power transmission, but also help designers to optimize the noise coefficient, determine the influence of quality factors and carry out stability analysis.

 

Figure 1. Impedance and basis of Smith circle diagram

Basic knowledge of

Before introducing the use of Smith circle diagram, it is best to review the electromagnetic wave propagation phenomenon of IC connection in RF environment (> 100MHz). This is valid for applications such as RS-485 transmission lines, connections between PA and antenna, and connections between LNA and down converter/mixers.

It is well known that the signal source impedance must be equal to the conjugate impedance of the load for maximum power transmission from the signal source to the load, i.e. :

RS + jXS = RL - jXL

 

FIG. 2. Equivalent diagram of the expression RS + jXS = rl-jxl

Under these conditions, the transfer of energy from the signal source to the load is maximum. In addition, for efficient transmission of power, meeting this condition can avoid energy being reflected from the load to the signal source, especially in high frequency applications such as video transmission, RF or microwave networks.

Smith circle diagram

Smith's circle is a graph with many circles interwoven together. Properly used, it is possible to obtain the matching impedance of a superficially complex system without doing any calculations, all that is required is reading and tracking the data along the circumference.

The Smith circle diagram is a polar coordinate diagram of the reflection coefficient (gamma, expressed by the symbol γ). The reflection coefficient can also be mathematically defined as the single-port scattering parameter, s11.

Smith circle diagrams are generated by verifying impedance matching loads. Instead of considering impedance directly, we use the reflection coefficient γ L, which reflects load characteristics (admittance, gain, transconductance), and γ L is more useful when dealing with RF frequencies.

We know that the reflection coefficient is defined as the ratio of reflected wave voltage to incident wave voltage:

 

Figure 3. Load impedance

The strength of the reflected signal depends on the mismatch between the source impedance and the load impedance. The expression of reflection coefficient is defined as:

 

Since the impedance is complex, so is the reflection coefficient.

To reduce the number of unknown parameters, you can solidify a parameter that occurs frequently and is often used in your application. Here Z0 (characteristic impedance) is usually a constant and a real number, and is commonly used as a normalized standard value, such as 50, 75, 100, and 600. Thus we can define the normalized load impedance:

 

Accordingly, the formula of reflection coefficient is rewritten as:

 

From the above equation we can see the direct relationship between the load impedance and its reflection coefficient. But this relationship is a complex number, so it's not practical. We can consider Smith's circle diagram as a graphical representation of the above equation.

To build a circle diagram, the equations must be rearranged to conform to standard geometry (such as circles or rays).

Firstly, it is solved by equation 2.3.

 

and

 

If the real and imaginary parts of equation 2.5 are equal, two independent relations can be obtained:

 

After rearranging equation 2.6, the final equation 2.14 was obtained through equation 2.8 to 2.13. This equation is in the complex plane (Γ Γ r, I), the parametric equation of circular on squared (x - a) + (b) y squared = r squared, it [r/(r + 1), 0] as the center, radius of 1 / (1 + r).

 

See Figure 4A for more details.

 

Figure 4a. Points on a circle represent impedances with the same real part. For example, a circle with r = 1 has a center of (0.5, 0) and a radius of 0.5. It contains the origin (0, 0) representing the reflection zero (the load matches the characteristic impedance). A circle with (0, 0) as the center and radius 1 represents a load short circuit. When the load is open, the circle degenerates into a point (center 1, 0, radius zero). Corresponding to this is the maximum reflection coefficient 1, that is, all incident waves are reflected back.

There are some problems that need to be noticed when making Smith circle diagrams. Here are some of the most important:

• All circles have the same, unique intersection (1, 0).

• The circle representing 0 ω, that is, no resistance (r = 0), is the largest circle.

• The circle corresponding to the infinite resistance degenerates into a point (1, 0)

• There is no negative resistance in practice. If there is a negative resistance value, oscillation may occur.

• Selecting a circle corresponding to the new resistance value is equivalent to selecting a new resistance.

drawing

By transforming equations 2.15 to 2.18, Equation 2.7 can be derived from another parametric equation, equation 2.19.

 

γ r, γ I = (x - a) ^ 2 + (y - b) ^ 2 = r ^ 2, with a center of (1, 1/x) and a radius of 1/x.

See Figure 4b for more details.

 

Figure 4b. Points on a circle represent impedances with the same imaginary part X. For example, a circle with × = 1 is centered on (1, 1) and has radius 1. All circles (x is a constant) include points (1, 0). Unlike a real part of a circle, x can be either positive or negative. This means that the lower half of the complex plane is a mirror image of the upper half. The center of all circles lies on a vertical line passing 1 point on the horizontal axis.

Complete the chart

To complete the Smith circle diagram, we put two clusters of circles together. You can see that all the circles in one cluster intersect all the circles in another cluster. If the impedance is given to be R + jx, it is only necessary to find the intersection of two circles corresponding to r and x to obtain the corresponding reflection coefficient.

interchangeability

The above process is reversible, and if the reflection coefficient is known, the intersection of the two circles can be found to read the corresponding values of r and cross. The process is as follows:

• Determine the corresponding point of impedance on the Smith circle diagram

• Find the reflection coefficient corresponding to this impedance (γ)

• Given the characteristic impedance and γ, find the impedance

• Convert impedance to admittance

• Find the equivalent impedance

• Find the component values that correspond to the reflection coefficient (especially the component that matches the network, see Figure 7)

inference

Because Smith's circle graph is a graphic-based solution, the accuracy of the results directly depends on the accuracy of the graph. Here is an example of RF application represented by Smith circle graph:

Example: The known characteristic impedance is 50 ω, the load impedance is as follows:

Z1 = 100 + J50 ω Z2 = 75-J100 ω Z3 = J200 ω Z4 = 150 ω

Z6 = 0 (a short circuit) Z7 = 50 ω Z8 = 184-J900 ω

Normalize the values above and label them in the circle (see Figure 5) :

Z1 = 2 + j z2 = 1.5 - j2 z3 = j4 z4 = 3

Z5 = 8 z6 = 0 z7 = 1 z8 = 3.68-j18

 

Figure 5. Points on Smith's circle

The reflection coefficient γ can now be solved directly from the circular diagram in Figure 5. Drawing the impedance points (the intersection of the iso-impedance circle and the iso-reactance circle) and reading their projections on the horizontal and vertical axes of the cartesian coordinates gives the real and imaginary parts of the reflection coefficients γ R and γ I (see FIG. 6).

There are eight possible cases in this example, and the corresponding reflection coefficient γ can be obtained directly on the Smith circle shown in FIG. 6:

1 = 0.4 + 0.2 j Γ Γ 2 = 0.51-0.4 j Γ 3 = 0.875 + 0.48 j Γ 4 = 0.5

γ 6 = 0 γ 6 = 0 γ 6 = 0 γ 6 = 0

 

Figure 6. Direct reading of the real and imaginary parts of the reflection coefficient γ from the X-y axis

In terms of admittance

The Smith circle diagram is constructed using impedance (resistance and reactance). Once the Smith circle diagram is worked out, it can be used to analyze parameters in series and parallel cases. New series elements can be added, and the effect of the new elements can be determined by moving around the circle to their corresponding values. However, when adding parallel components, the analysis process is not so simple, and other parameters need to be considered. In general, it is easier to handle parallel components with admittance.