“Usually, the impedance matching at a certain frequency point can be carried out using the SMITH chart tool, and the two devices can definitely be done, that is, the impedance matching from any point on the chart to another point can be achieved through series + parallel inductors or capacitors, but this is single frequency. The mobile phone antenna is dual-frequency, matching one of the frequency points will inevitably affect the other frequency point, so the impedance matching can only be a compromise between the two frequency bands.

“

Usually, the impedance matching at a certain frequency point can be carried out using the SMITH chart tool, and the two devices can definitely be done, that is, the impedance matching from any point on the chart to another point can be achieved through series + parallel inductors or capacitors, but this is single frequency. The mobile phone antenna is dual-frequency, matching one of the frequency points will inevitably affect the other frequency point, so the impedance matching can only be a compromise between the two frequency bands.

It is easy to match at a certain frequency, but it is more complicated for more than two frequencies. Because it is completely matched at 900M, then the match will not be achieved at 1800, and a suitable matching circuit must be considered. It is best to use simulation software or a point to match, and adjust it under the S11 parameter on the network analyzer, because the matching point of the dual frequency is definitely not too far from here, only the two components are uniquely matched, but the pi type Network matching, there are countless solutions. At this time, you need to choose simulation, preferably with experience.

Simulation tools are of little use in practice. Because the simulation tool does not know the model of your component. You have to enter the model of the actual component, that is, the various distribution parameters, so that your results may match the actual. An actual Inductor is not simply measured by inductance, it should be simulated by an equivalent network. I usually only use simulation tools to do some theoretical research.

In the actual design, it is necessary to fully understand the principle of the Smith chart, and then use the circle chart tool of the network analyzer to debug more. Knowing the principles lets you know qualitatively what pieces to use, and polyphony lets you familiarize yourself with how the components you use will move on an actual circle. (Due to the different distribution parameters and the frequency response characteristics of the components, the movement of the actual piece on the circle diagram will be different from the movement you calculated theoretically).

Dual frequency matching is indeed a compromise process. You must add a piece on purpose. Taking GSM and DCS dual-band as an example, if you want to adjust GSM but don’t want to change DCS, you should choose series capacitors and parallel inductors. Similarly, if you want to adjust DCS, you should choose series inductor and parallel capacitor.

In theory, 2 pieces are needed to tune one frequency point, so the actual mobile phone or mobile terminal usually arranges the matching circuit according to the following rules: For simpler ones, the antenna space is relatively large, and the reflection is originally small, the Pai type (2 in one) is used. Serial), such as conventional candy bar phone, conventional flip phone; slightly more complex use double L type (2 series 2 parallel): for more complex, use L+Pai type (2 series 3 parallel), such as a mobile phone with a telescopic antenna.

Remember, matching circuits reduce reflections, but they also introduce losses. In some cases, although the standing wave ratio is good, the efficiency of the antenna system will be reduced. Therefore, the design of matching circuits is somewhat taboo; for example, in matching circuits in GSM and DCS mobile phones, the series inductance is generally not greater than 5.6nH. In addition, when the reflection of the antenna itself is relatively large and the bandwidth is not enough, it can be seen on the Smith diagram that the radius of the boundary point of each frequency band from the center of the circle is large. Generally, adding matching cannot improve the radiation.

The reflection index (VSWR, return loss) of the antenna is generally only used as a reference in the design process. The key parameters are transport parameters (such as efficiency, gain, etc.). Some people insist on return loss, a mouth should be -10dB, and the standing wave ratio should be less than 1.5, which is meaningless. When I met such a person, I joked that as long as you have a good reflection index, I will connect you with a 50 ohm matching resistor, and the standing wave will be less than 1.1. As for whether your mobile phone can work or not, I don’t care!

The SWR standing wave ratio only describes the matching degree of the ports, that is, the impedance matching degree. The matching is good, the SWR is small, and the power reflected back at the antenna input port is small. If the matching is not good, the power reflected back will be large. As for whether the part of the power entering the antenna is radiated, you don’t know at all. The efficiency of an antenna is the ratio of the total power radiated into space to the total power at the input port. So if the SWR is good, it cannot be judged that the antenna efficiency must be high (connect a 50ohm matching resistor, the SWR is very good, but is there radiation?). However, the SWR is not good, and the reflected power is large, so it is certain that the efficiency of the antenna will not be high. Good SWR is a necessary but not sufficient condition for good antenna efficiency. Good SWR and high radiation efficiency are sufficient and necessary conditions for high antenna efficiency. When the SWR is an ideal value (1), the ports are ideally matched, and the antenna efficiency is equal to the radiation efficiency at this time.

In today’s mobile phones, the space of the antenna is getting smaller and smaller, at the cost of sacrificing the performance of the antenna. For some multi-band antennas, even the VSWR reaches 6. In the past, everyone used external antennas more, and the average efficiency was low at 50%. Now the efficiency of more than 50% is very good! Take a look at the mobile phones on the market, even those from famous companies, such as Nokia, are less than 20% efficient. Some mobile phones (sliding, rotating) are only about 10% efficient at certain frequencies.

I have seen several test reports of built-in antennas in mobile phones. The antenna efficiency is basically around 30-40%. At the time, I thought it was really bad enough (compared to the microstrip antenna I designed). However, in actual engineering, it seems that the loss caused by S11 and the loss of the matching circuit are included in the efficiency. According to the antenna principle, there are only dielectric losses (including those caused by the substrate and the magnets in the mobile phone) and metal losses (although very small). ) are included in the antenna losses, while the return loss and matching circuit losses should not be included. But engineering is engineering, so it’s easy to test.

By the way, add another sentence, software simulation is helpful to engineering to a certain extent. Of course, the accuracy of the simulation results cannot be compared with the test, but the trend of the antenna performance with the parameters obtained by the parameter sweep simulation is still useful, which is much faster than the data obtained by the test, especially for some less commonly used parameter.

“Simulation tools are of no use in actual engineering”, which means that when designing matching circuits, more specifically, when designing dual-band GSM and DCS mobile phone antenna matching circuits. If you understand this sentence alone, it is undoubtedly wrong. In fact, I’ve been doing antenna simulations with HFSS and the results are based on the simulation results as well.

By the way, soldering components is really a laborious thing, and there are also methods. The so-called practice makes perfect. A big company might give you a dedicated welder, so you might just have to say what to weld. However, what we are discussing here is how to efficiently design a matching circuit. Pay attention to effectiveness! Effectiveness includes time spent and accuracy of component selection. If you don’t have practical experience, you can only get a matching design through software simulation but use it on the actual antenna input. Haha, I can say that in all likelihood, your design will not work, and it may even be very different from your imagination!

In actual design, there is another situation that you cannot consider in simulation (unless you measure it in advance). That is, the effect of distribution parameters on PIFA. Since antenna heights are getting smaller these days, and the matching circuit is either below (inside) or above (outside) the antenna, whichever is close, adding an actual element would actually introduce a change in the distribution parameters. Especially if the layout of the circuit board is not good, this effect will be more obvious. In actual welding, even if a piece is not well welded, re-soldering will bring about changes in impedance.

Therefore, in the design of PIFA, we usually do not use matching circuits (or 0ohm matching). This requires you to carefully adjust and optimize your antenna. Generally speaking, it is easier to do with the current flexible circuit board design (Flexfilm), because it is easier to modify the radiator. For another design scheme that is used more often, stamping metal is relatively more difficult. One is that the hardness is high, and due to the limitation of the process, all the space cannot be fully justified. Second, it is also difficult to modify the design of the radiant sheet many times once the mold is formed.

Is the simulation tool very useful in matching design? Not many people can use the simulation tool to calculate the matching. Besides, how to measure whether it has a great effect? Engineering is fast and accurate. Emulation for emulation’s sake is moot. In order to get a 2, 3, up to 5 piece match you have to model the inductance, capacitance, it’s not worth it. Also, how do you account for the change in the distribution parameters of the PIFA match I mentioned above? Earlier I also mentioned some taboos of matching circuits, not from theory, but entirely from practice. Because the design of the antenna is to improve its radiation efficiency (total efficiency)! I have not succeeded in finding an accurate matching circuit (say GSM, DCS) dual-band through the simulation tool within 1 hour, (in practice, it is possible to use the optical illusion method).

When dealing with practical application problems of RF systems, there are always some very difficult tasks, one of which is to match the different impedances of each part of the cascade circuit. 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 efficiently transferred from the “signal source” to the “load”.

On the high frequency side, parasitic elements (such as inductance on the traces, capacitance between board layers, and resistance of conductors) have significant and unpredictable effects on the matching network. At frequencies above tens of megahertz, theoretical calculations and simulations are far from sufficient, and RF testing in the laboratory must also be considered and properly tuned for proper final results. Calculated values are required to determine the type of construction of the circuit and the corresponding target component values.

**There are many methods of impedance matching, including**

• Computer simulation: This type of software is more complicated to use because it is designed for different functions and not just for impedance matching. Designers must be familiar with entering large amounts of data in the correct format. Designers also need to have the skills to find useful data from a large number of output results. Also, circuit simulation software cannot be pre-installed on a computer unless the computer is specifically built for this purpose.

• Manual calculation: This is an extremely cumbersome method, as long (“a few kilometers”) formulas are required and the data being processed is mostly complex.

• Experience: Only those who have worked in the RF field for many years should use this method. In short, it is only suitable for senior experts.

• Smith Chart: The focus of this article.

The main purpose of this paper is to review the structure and background knowledge of the Smith chart, and to summarize its application in practice. Topics discussed include practical examples of parameters, such as finding the values of matching network elements. Of course, the Smith chart can not only help us find the matching network for maximum power transfer, but also help designers optimize noise figure, determine the impact of quality factor, and perform stability analysis.

Figure 1. Impedance and Smith Chart Basics

**Basic knowledge**

Before introducing the use of the Smith chart, it is good to review the electromagnetic wave propagation phenomenon in IC wiring in 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, in order to maximize the power delivered by the signal source to the load, the signal source impedance must be equal to the conjugate impedance of the load, namely:

RS + jXS = RL – jXL

Figure 2. Equivalent diagram of the expression RS + jXS = RL – jXL

Under this condition, the maximum energy is transferred from the signal source to the load. In addition, to efficiently transmit power, meeting this condition can avoid energy reflection from the load to the signal source, especially in high frequency applications such as video transmission, RF or microwave networks.

**Smith Chart**

A Smith chart is a graph of many circles intertwined. Using it correctly, the matched impedance of an apparently complex system can be obtained without any calculations, the only thing that needs to be done is to read and trace the data along the circumference.

A Smith chart is a polar plot of reflection coefficients (gamma, denoted by the symbol Γ). The reflection coefficient can also be mathematically defined as the one-port scattering parameter, s11.

Smith charts are generated by verifying impedance matched loads. Here we do not directly consider the impedance, but use the reflection coefficient ΓL, which can reflect the characteristics of the load (such as admittance, gain, transconductance), and ΓL is more useful when dealing with RF frequencies.

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

Figure 3. Load Impedance

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

Since the impedance is a complex number, the reflection coefficient is also a complex number.

In order to reduce the number of unknown parameters, it is possible to fix a parameter that occurs frequently and is often used in the application. Here Z0 (characteristic impedance) is usually a constant and a real number, and is a normalized standard value that is commonly used, such as 50Ω, 75Ω, 100Ω, and 600Ω. Then we can define the normalized load impedance:

Accordingly, the formula for the 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 relation is a complex number, so it is not practical. We can think of the Smith chart as a graphical representation of the above equation.

In order to create a circle diagram, the equations must be rearranged to conform to the form of standard geometric figures (such as circles or rays).

First, it is solved by equation 2.3;

and

Equating the real and imaginary parts of Equation 2.5 yields two independent relations:

Rearranging Equation 2.6, going through Equations 2.8 to 2.13 yields the final Equation 2.14.This equation is the parametric equation of the circle (x – a)² + (y – b)² = R² in the complex plane (Γr, Γi), which is given by[r/(r + 1)，0]is the center of the circle and the radius is 1/(1 + r).

See Figure 4a for more details.

Figure 4a. Points on the circumference represent impedances with the same real part

For example, a circle with r = 1 has a center at (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 a radius of 1 represents the load short circuit. When the load is open, the circle degenerates to a point (centered at 1, 0, and the radius is zero). This corresponds to a maximum reflection coefficient of 1, ie all incident waves are reflected back.

There are a few things to keep in mind when making a Smith chart. Here are the most important aspects:

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

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

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

• In practice, there is no negative resistance. If there is a negative resistance, oscillation may occur.

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

**drawing**

After the transformation of Equations 2.15 to 2.18, Equation 2.7 can lead to another parametric equation, Equation 2.19.

Likewise, 2.19 is also the parametric equation (x – a)² + (y – b)² = R² of a circle on the complex plane (Γr, Γi) with its center at (1, 1/x) and radius 1/x .

See Figure 4b for more details.

Figure 4b. Points on the circumference represent impedances with the same imaginary part x

For example, a circle with × = 1 has a center at (1, 1) and a radius of 1. All circles (x constant) include the point (1, 0). Unlike the real circumference, x can be either positive or negative. This means that the lower half of the complex plane is a mirror image of its upper half. The centers of all circles are on a vertical line passing through 1 point on the horizontal axis.

**Complete the circle diagram**

To complete the Smith chart, we put two clusters of circles together. It can be found that all circles of one cluster of circumferences intersect all circles of another cluster of circumferences. If the impedance is known to be r + jx, it is only necessary to find the intersection of the two circles corresponding to r and x to obtain the corresponding reflection coefficient.

**interchangeability**

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

• Determine the corresponding point of the impedance on the Smith chart

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

• Knowing 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 coefficients (especially the components of the matching network, see Figure 7)

**inference**

Because the Smith chart is a graph-based solution, the accuracy of the result is directly dependent on the accuracy of the graph. The following is an example of an RF application represented by a Smith chart:

Example: Given that the characteristic impedance is 50Ω, the load impedance is as follows:

The above values are normalized and plotted in a circle chart (see Figure 5):

Figure 5. Points on the Smith Chart

The reflection coefficient Γ can now be solved directly from the circle diagram in Figure 5. Draw impedance points (intersection points of equal impedance circles and equal reactance circles), and just read out their projections on the horizontal and vertical axes of the Cartesian coordinates to get the real part Γr and imaginary part Γi of the reflection coefficient (see Figure 6) .

There are eight possible cases in this example, and the corresponding reflection coefficient Γ can be directly obtained on the Smith chart shown in Figure 6:

Figure 6. Direct readout of the real and imaginary parts of the reflection coefficient Γ from the XY axis

**Expressed by admittance**

The Smith chart is built using impedance (resistance and reactance). Once a Smith chart is made, it can be used to analyze the parameters for both series and parallel conditions. New series elements can be added, and the effect of the new elements can be determined by simply moving around the circle to their corresponding values. However, the analysis process is not so simple when adding parallel elements, and other parameters need to be considered. In general, parallel elements are easier to handle with admittance.

We know that by definition Y = 1/Z, Z = 1/Y. The unit of admittance is mho or Ω-1 (the unit of admittance was Siemens or S earlier). And, if Z is complex, Y must also be complex.

So Y = G + jB (2.20), where G is called the “conductance” of the element and B is called the “susceptance”. Care should be taken in the calculation. According to the seemingly logical assumptions, it can be concluded that G = 1/R and B = 1/X, but this is not the case, and the calculation will lead to wrong results.

When expressing admittance, the first thing to do is to normalize, y = Y/Y0, to get y = g + jb. But how to calculate the reflection coefficient? Derive by the following formula:

The result is that the expression for G has the opposite sign of z and has Γ(y) = -Γ(z).

If you know z, you can find a point that is equidistant from (0, 0) but in the opposite direction by inverting the sign of . Rotating 180° around the origin gives the same result (see Figure 7).

Figure 7. Result after 180° rotation

Of course, on the surface the new point appears to be a different impedance, but in fact Z and 1/Z represent the same element. (On a Smith chart, different values correspond to different points and have different reflection coefficients, and so on) The reason for this is that our graph itself is an impedance graph, and the new point represents an admittance . Therefore, the unit of the value read on the pie chart is Siemens.

Although conversion is possible in this way, it is not suitable for solving many parallel element circuits.

**Admittance circle**

In the previous discussion, we saw that each point on the impedance circle can be obtained by rotating 180° around the origin of the Γ complex plane to obtain its corresponding admittance point. Thus, the admittance diagram is obtained by rotating the entire impedance diagram by 180°. This method is very convenient, it saves us from having to create a new graph. The intersection of all circles (isoconductance circles and isosusceptance circles) naturally occurs at the point (-1, 0). Using an admittance diagram makes it easy to add parallel elements. Mathematically, the admittance circle diagram is constructed by the following formula:

Solve this equation:

Next, making the real and imaginary parts of Equation 3.3 equal, we get two new independent relations:

From Equation 3.4, we can derive the following equation:

It is also the parametric equation (x – a)² + (y – b)² = R² (equation 3.12) of the circle on the complex plane (Γr, Γi), with[g/(g + 1)，0]is the center of the circle and the radius is 1/(1 + g).

From Equation 3.5, we can derive the following equation:

The parametric equation of (x – a)² + (y – b)² = R² is also obtained (Equation 3.17).

**Solving for Equivalent Impedance**

When solving a mixed circuit with both series and parallel elements, the same Smith chart can be used to rotate the graph when a transformation from z to y or y to z is required.

Consider the network shown in Figure 8 (where the components are normalized to Z0 = 50Ω). The series reactance (x) is positive for inductive elements and negative for capacitive elements. The susceptance (b) is positive for capacitive elements and negative for inductive elements.

Figure 8. A multi-element circuit

This circuit needs to be simplified (see Figure 9). Starting from the far right, there is a resistance and an inductance, both with a value of 1, we can get a series equivalent point, point A, at the intersection of the circle with r = 1 and the circle with I = 1. The next element is the parallel element, we turn to the admittance circle diagram (rotate the whole plane 180°), at this time we need to turn the previous point into admittance, denoted as A’. Now we rotate the plane 180°, so we add the parallel element in admittance mode and move the distance 0.3 counterclockwise (negative value) along the conductance circle to get point B. And then another series element. Now let’s go back to the impedance circle diagram.

Figure 9. The components of the network of Figure 8 are disassembled for analysis

Before returning to the impedance circle diagram, it is necessary to convert the previous point into impedance (previously it was admittance). The point obtained after the transformation is marked as B’. Using the above method, rotate the circle diagram 180° to return to the impedance mode. Moving along the resistor circle a distance of 1.4 to get point C adds a series element, moving counterclockwise (negative value). Do the same to add the next element (transform plane rotation to admittance) by moving the specified distance (1.1) clockwise (because it is a positive value) along the circle of equal conductance. This point is marked as D. Finally, we go back to impedance mode to add the last component (the series inductor). So we get the desired value, z, at the intersection of the 0.2 resistance circle and the 0.5 reactance circle. So far, z = 0.2 + j0.5. If the characteristic impedance of the system is 50Ω, there is Z = 10 + j25Ω (see Figure 10).

Figure 10. Network elements drawn on a Smith chart

**Impedance matching step by step**

Another use of the Smith chart is for impedance matching. This is the opposite of finding the equivalent impedance of a known network. At this point, the impedances at both ends (usually the signal source and the load) are fixed, as shown in Figure 11. The goal is to insert a designed net between the two to achieve proper impedance matching.

Figure 11. Typical Circuit with Known Impedance and Unknown Components

At first glance it doesn’t seem like it’s more complicated than finding the equivalent impedance. But the problem is that there are an infinite number of combinations of components that can make matching networks have similar effects, and other factors (such as filter construction type, quality factor, and limited optional components) need to be considered.

The way to do this is to keep adding series and parallel elements on the Smith chart until we get the impedance we want. Graphically, it’s finding a way to connect the points on the Smith chart. Again, the best way to illustrate this approach is to give an example.

Our goal is to match the source impedance (ZS) and load impedance (zL) at 60MHz operating frequency (see Figure 11). The network structure has been determined to be low-pass, L-shaped (the problem can also be viewed as how to transform the load into an impedance whose value is equal to ZS, ie ZS complex conjugate). Here is the solution process:

Figure 12. The network of Figure 11, plotting its corresponding points on the Smith chart

The first thing to do is to normalize the impedance values. If no characteristic impedance is given, choose an impedance value that is in the same order of magnitude as the load/source value. Suppose Z0 is 50Ω. So zS = 0.5 – j0.3, z*S = 0.5 + j0.3, ZL = 2 – j0.5.

Next, mark these two points on the graph, A for zL and D for z*S

Then determine the first component (parallel capacitor) connected to the load, first convert zL into admittance, and get point A’.

Determine where the next point will appear on the arc after connecting capacitor C. Since we don’t know the value of C, we don’t know the exact location, but we do know the direction of movement. The capacitors in parallel should move clockwise on the admittance diagram until the corresponding value is found, giving point B (admittance). The next element is a series element, so it is necessary to convert B to the impedance plane to get B’. B’ must be on the same resistance circle as D. Graphically, there is only one path from A’ to D, but if you want to go through the middle point B (that is, B’), you need to go through many tries and tests. After finding points B and B’, we can measure the arc lengths from A’ to B and B’ to D, the former being the normalized susceptance value of C and the latter being the normalized reactance value of L. The arc length from A’ to B is b = 0.78, then B = 0.78 × Y0 = 0.0156S. Since ωC = B, C = B/ω = B/(2πf) = 0.0156/[2π(60 × 106)] = 41.4pF.

The arc length from B to D is × = 1.2, so X = 1.2 × Z0 = 60Ω. From ωL = X, L = X/ω = X/(2πf)= 60/[2π(60 × 106)] = 159nH.

Figure 13. Typical MAX2472 Operating Circuit

The second example is the output matching circuit of the MAX2472, matched to a 50Ω load impedance (zL) and operating at 900MHz (Figure 14). The network uses the same configuration as the MAX2472 data sheet. The figure above shows the matching network, including a shunt inductor and series capacitor. The following shows how to find the value of the matching network components.

Figure 14. Corresponding operating points on the Smith a-chart for the network shown in Figure 13

The S22 scattering parameters are first converted into equivalent normalized source impedances. Z0 of the MAX2472 is 50Ω, S22 = 0.81/-29.4° translates to zS= 1.4 – j3.2, zL = 1 and zL* = 1.

Next, locate two points on the circle graph, marked A for zS and D for zL*. Since the first element connected to the signal source is a shunt inductor, convert the source impedance to admittance and get point A’.

Determine the arc where the next point is located after connecting the inductor LMATCH. Since the value of LMATCH is unknown, the location of the arc termination cannot be determined. However, we understand that after connecting the LMATCH and converting it to impedance, the source impedance should be on the circle at r=1. From this, the resulting impedance with the capacitors in series should be z = 1 + j0. Taking the origin as the center and rotating 180° on the circle with r = 1, the intersection of the reflection coefficient circle and the iso-susceptance circle can be combined with point A’ to get B (admittance). The impedance corresponding to point B is point B’.

After finding B and B’, you can measure the length of arc A’B and arc B’D, and the first measurement can get LMATCH. The normalized value of susceptance, the second measurement gives the normalized value of CMATCH reactance. Arc A’B measures b = -0.575, B = -0.575 × Y0 = 0.0115S. Since 1/ωL = B, then LMATCH = 1/Bω = 1/(B2πf) = 1/(0.01156 × 2 × π × 900 × 106) = 15.38nH, which is approximately 15nH. The measured value of arc B’D is × = -2.81, X = -2.81 × Z0 = -140.5Ω. Since -1/ωC = X, then CMATCH = -1/Xω = -1/(X2πf) = -1/(-140.5 × 2 × π × 900 × 106) = 1.259pF, which is approximately 1pF. These calculations do not account for parasitic inductance and capacitance, and the resulting values are close to those given in the data sheet: LMATCH = 12nH and CMATCH = 1pF.

**Summarize**

In today’s world of powerful software and high-speed, high-performance computers, one wonders whether such a basic and rudimentary approach is needed to solve fundamental problems in circuits.

In fact, a real engineer should not only have theoretical knowledge, but also have the ability to solve problems using various resources. It’s really easy to add a few numbers to a program and get a result, and it’s especially handy to have a computer do the job when the solution to the problem is complex and not unique. However, if they can understand the basic theories and principles used by the working platform of the computer and know their origin, such an engineer or designer can become a more comprehensive and trustworthy expert, and the results obtained are more reliable.

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