The fact proves, graph of Shi Mi Si Yuan remains the main tool of impedance of firm transmission line.

 

When the actual application issue that handles RF system, total meeting encounters the job of a few special difficulty, undertaking matching to the different impedance of circuit of each part cascade is among them one of. Usually, the circuit that need undertakes matching includes aerial and low noise amplifier (between LNA) match, power amplifier is outputted (between RFOUT) and antenna match, between LNA/VCO output and first detector input match. The objective that match is to assure signal or energy to convey effectively from ” of “ signal source “ laden ” .

 

In high frequency end, parasitism component (the resistance that joins the capacitance between on-line inductance, ply and conductor for instance) apparent to matching a network to have, unpredictable influence. Frequency is when tens of million hertz above, theory is calculated and emulate cannot have satisfied a requirement far, to get proper final result, the RF test that still must consider to undertake in the lab, hand-in-hand travel is proper and harmonious. The structural kind that needs to be worth certain circuitry with computation and corresponding object element are worth.

 

Have the method that a lot of kinds of impedance match, include

 

● computer emulates: Because this kind of software is,for what different function designs is to be used at impedance to match not merely, use so rise more complex. The architect must be familiar with input numerous data with proper form. Design personnel still needs to have the technical ability that finds useful data from inside many output result. Additional, unless the computer is made technically for this utility, otherwise circuitry emulation software is installed impossibly beforehand go up in the computer.

 

● handiwork calculates: This is a kind of extremely trival method, because need is used to longer (a few kilometers of “ the computational formula of ”) , and the data that is processed is complex number more.

 

● experience: Had worked in RF domain only old person ability uses this kind of method. Anyhow, it agrees with only elder expert.

 

Si Yuan of ● Shi Mi pursues: The article wants the content of key discussion.

 

The main purpose of the article is the structure that revises Shi Mi Si Yuan to pursue and cultural background, and sum up it using a method mediumly actually. Discussion subject includes the actual model of parameter, find out the numerical value that matchs network component for instance. Of course, graph of Shi Mi Si Yuan can be found out for us not only most high-power transmits match a network, still can help an architect optimize noise coefficient, the influence of certain quality factor and undertake stability is analysed.

 

Graph 1. Impedance and foundation of graph of Shi Mi Si Yuan

 

ABC

 

In introductory Smith the circle pursues before using, had better review RF environment to fall (the electromagnetic wave that is more than 100MHz) IC to connect a line transmits a phenomenon. The join between transmission line of this pair of RS-485, PA and antenna, LNA and next transducer / the application such as the join between first detector is effective.

 

Everybody knows, should make signal source conveys laden power is the greatest, signal source impedance must be equal to laden conjugate impedance, namely:

 

RS + JXS = RL – JXL

 

Graph 2. The equivalent of expression RS + JXS = RL – JXL pursues

 

Below this condition, the energy that transmits to load from signal source is the biggest. Additional, to transmit power effectively, satisfy this condition to be able to avoid energy to reflex signal source from load, transmit in such as video especially, the environment of high frequency application of RF or microwave network is more such.

 

Shi Mi Si Yuan pursues

 

Shi Mi Si Yuan pursues is by a lot of circumferential a graph that interweaves together. Correct use it, can not making any calculative premise a next getting apparently view very complex system match impedance, what need exclusively is read along circumferential line take and dog data.

 

Shi Mi Si Yuan pursues is reflection coefficient (gamma, express with symbolic Γ ) extremely coordinate graph. Reflection coefficient also can define parameter of scattering of the port that it is sheet from maths, namely S11.

 

Shi Mi Si Yuan pursues is the laden generation that matchs through impedance of test and verify. We do not consider impedance directly here, use reflection coefficient ΓL however, the character that reflexes coefficient to be able to reflect load (like admittance, gain, cross guide) , the ΓL when the issue that handling RF frequency is more useful.

 

We know to reflex coefficient definition to be those who reflex voltage of wave voltage and incident wave to compare:

 

Graph 3. Laden impedance

 

The intensity of laden reflex signal depends on of signal source impedance and laden impedance break match rate. The expression definition that reflexes coefficient is:

 

 

Because impedance is plural, reflexing coefficient also is plural.

 

To reduce the amount of sealed parameter, the parameter that solidify often appears and often can use in application. Characteristic impedance of here Z0 () it is constant normally and it is real, it is commonly used normalization mark fiducial value, be like 50Ω , 75Ω , 100Ω and 600Ω . Then we can define the laden impedance of normalization:

 

 

Accordingly, write the formula that reflexes coefficient afresh for:

 

 

From on we can see pattern the immediate impact between laden impedance and its reflex coefficient. But this relation is a complex number, use falsely so. We can pursue Shi Mi Si Yuan regard as the graph of afore-mentioned equation expresses.

 

To establish round plan, equation is indispensible rearrange in order to accord with the form of standard geometry graph (be like circle or ray) .

 

Above all, by equation 2.3 beg solution to go out;

 

 

And

 

 

The real department of 2.5 mixes your equality empty ministry is equal, gain two independent relation:

 

 

Rearrange equality 2.6, through equality 2.8 to 2.13 get final equation 2.14. This equation is to be in answer planar (Γr, on Γi) , round parameter equation (X – A)² + (y – B)² = R² , it with [R/(r + 1) , 0] is the centre of a circle, radius is 1/(1 + R) .

 

 

More detail refers to graph 4a

 

Graph 4a. Circumferential the impedance that the dot that go up expresses to have same fact department

 

For example, the circle of R = 1, with (0.5, 0) is the centre of a circle, radius is 0.5. It included a delegate to reflex the origin 0 o’clock (0, 0) (load and characteristic impedance photograph match) . With (0, 0) represents laden short circuit for the circle of 1 for the centre of a circle, radius. When laden open a way, the circle degrades to be nodded for (with 1, 0 for the centre of a circle, radius is 0) . As corresponding as this is the biggest reflection coefficient 1, namely all incident wave are reflexed to come back.

 

When making Shi Mi Si Yuan pursue, have the problem that a few need note. A few the most important facets are below:

 

● all circumferential have only identical, exclusive node (1, 0) .

● delegate 0Ω , do not have resistor namely (the circle of R = 0) is the biggest circle.

The circle of the resistor correspondence of ● infinity degrades to be nodded for (1, 0)

● is actual in the resistance that did not lose, if appear,negative block is worth, produce vibration likely.

● chooses a correspondence at new resistance circumferential be equal to chose a new resistance.

 

Constructive

 

Through equality 2.15 to the commutation of 2.18, 2.7 type are OK derivation gives another parameter equation, equation 2.19.

 

 

Same, 2.19 also be to be in answer planar (Γr, the round parameter equation on Γi) (X – A)² + (y – B)² = R² , its the centre of a circle is (1, 1/x) , radius 1/x.

 

More detail refers to graph 4b

 

Graph 4b. Circumferential the impedance that the dot that go up expresses to have X of same theory ministry

 

For example, ×the circle of = 1 with (1, 1) is the centre of a circle, radius is 1. All circles (X is constant) include a feature (1, 0) . Circumferential and as different as real department is, x can be plus already also can be negative number. Originate filter public platform reminds this demonstrative answer plane issues half is on its half mirror. All round the centre of a circle pass the 1 o’clock perpendicular on cross axle to go up in.

 

Complete round plan

 

To finish Smith the circle pursues, we two bunch circumferential put together. All round meetings that can discover tuft is circumferential and all circle with another circumferential bunch intersect. If foregone impedance is R + Jx, need to find correspondence to be able to get corresponding reflection coefficient at two of R and X circumferential node only.

 

But interchangeability

 

Afore-mentioned processes are reversible, if foregone reflex coefficient, can find two circumferential node to read the cost that takes corresponding R and × thereby. The process is as follows:

 

● decides impedance in Smith the correspondence on round figure is nodded

● finds as corresponding as this impedance reflection coefficient (Γ )

Impedance of ● foregone character and Γ , find out impedance

● is impedance changeover admittance

● is found out equivalent impedance

● finds out as corresponding as reflection coefficient component to be worth (the component that matchs a network especially, see a picture 7)

 

Inference

 

Because of Smith the circle pursues is a kind of solution that is based on a graph, the accuracy of earnings result relies on graphical precision directly. The RF application example that with Smith round chart shows one is below:

 

Exemple: Foregone and characteristic impedance is 50Ω , laden impedance is as follows:

 

 

Undertake normalization to the value above and designation pursues medium in the circle (5) seeing a picture:

 

Graph 5. The dot on graph of Shi Mi Si Yuan

 

Can adopt a plan now the round figure of 5 solves a reflection coefficient Γ directly. The picture gives impedance to nod (wait for impedance circle and the nodical) that wait for reactance circle, should numerate only what they go up in axis of rectangular coordinates level and vertical axis is umbriferous, got Γi (of Γr and empty ministry sees graph 6) reflexing the real department of coefficient.

 

The likelihood in this example is put in 8 kinds of circumstances, in the graph 6 show the reflection coefficient Γ that Smith can get correspondence directly on round figure:

 

Graph 6. Numerate directly from X-Y axis the real department that reflexes coefficient Γ and empty ministry

 

Express with admittance

 

Shi Mi Si Yuan pursues is to use impedance (resistor and reactance) build. Once made Smith,the circle pursues, can analyse the parameter below series connection and shunt-wound circumstance with it. This article is to turn from filter public platform, it reminds say to be able to add new series connection element, the influence that gains component newly certainly needs to arrive along circumferential shift only their corresponding numerical value can. However, increasing process of analyse of shunt-wound component time is not so simple, need considers the parameter of other. Normally, use admittance to tackle shunt-wound element more easily.

 

We know, the basis defines Y = 1/Z, z = 1/Y. The unit of admittance is mho or Ω-1 (now the unit of admittance is Xi Menzi or S) . And, if Z is plural, criterion Y also is plural certainly.

 

So Y = G + JB (2.20) , among them the “ conductance ” that G makes component, b calls “ susceptance ” . In figure when should scrupulous, according to assume logically it seems that, can reach: G = 1/R reachs B = 1/X, however actual condition is not such, such computation can cause result error.

 

When expressing with admittance, the first thing that should do is normalization, y = Y/Y0, reach Y = G + Jb. But how to calculate reflex coefficient? Undertake derivation through the posture below:

 

 

The expression symbol that is G as a result and Z are contrary, have Γ(y) = -Γ(z) .

 

If know Z, can find instead through taking the sign of Γ with (0, of 0) equidistant but nod in what return way. Rotate around origin 180° can get similar result (7) seeing a picture.

 

Graph 7. 180° spends the result after rotating

 

Of course, apparently seeing new feature seem is a different impedance, z and 1/Y are actually denotive it is same a component (the aspect that this new worth appears to differ for on round figure, and reflection coefficient is not identical also, ordinal analogize) . This article is to turn from filter public platform, it reminds the graph that speaking the account that shows this kind of case is us itself is an impedance graph, and what new feature represents is an admittance. The numeric unit that because this is on round figure,numerates is Xi Menzi.

 

Although use this kind of method to be able to have transition, but when the problem that solving circuit of a lot of shunt-wound component still not applicable.

 

Admittance circle pursues

 

In the discussion in front, we see impedance circle graph go up each the dot can be passed with Γ answer planar origin rotates for the center get after 180° to it corresponding admittance dot. Then, whole impedance circle the graph rotates 180° got admittance circle pursues. This article is to turn from filter public platform, it reminds say this kind of method is very convenient, it makes we need not establish a new plan. All and circumferential node (etc conductance circle and wait for susceptance circle) appear naturally in the dot (- 1, 0) . Use admittance circle pursues, make add shunt-wound component to become very easy. On maths, admittance circle graph constructs by the formula below:

 

 

Solve this equation:

 

 

Next, the real department of 3.3 mixes your equation empty ministry is equal, we get two new independent concerns:

 

 

From equality 3.4, we are OK derivation gives the posture below:

 

 

It also is answer planar (Γr, the round parameter equation on Γi) (3.12) of equation of X – A)² + (y – B)² = R² (, with [- G/(g + 1) , 0] is the centre of a circle, radius is 1/(1 + G) .

 

From equality 3.5, we are OK derivation gives the posture below:

 

 

Get likewise (X – A)² + (y – B)² = R² parameter equation (equation 3.17) .

 

Beg solution equivalent impedance

 

When should solving what be put in series connection and shunt-wound component at the same time to mix circuit, can use same a graph of Shi Mi Si Yuan, undertake arriving from Z in need Y or rotate the graph from the Y changeover to Z.

 

Consideration graph shows a network 8 times (among them component undertook normalization) with Z0 = 50Ω . Series connection reactance (X) is right plus is inductance component, to negative number is capacitance component. And susceptance (B) is right plus is capacitance component, to negative number is inductance component.

 

Graph 8. Circuit of a many component

 

This circuit needs to undertake simplifying (9) seeing a picture. From most right begins, have a resistor and an inductance, numerical value is 1, we can be in of R = 1 circumferential with I ＝ the circumferential nodical place of 1 gets equivalent of a series connection is nodded, nod A namely. Next component is shunt-wound component, we turn to admittance circle to pursue (will whole plane rotates 180°) , right now before need general that the dot becomes admittance, write down for A” . We rotate plane now 180° , then we add shunt-wound element below admittance mode, along conductance circle anticlockwise direction (negative worth) mobile distance 0.3, get nodding B. It is component of a series connection next. We return impedance circle to pursue again now.

 

Graph 9. the graph the component in 8 networks ravels undertake an analysis

 

Before returning impedance circle to pursue, indispensible still change a moment ago spot into impedance (admittance) is before this, the point that gains after commutation is written down for B” , this article is to turn from filter public platform to use afore-mentioned methods, will round figure rotates 180° recurs impedance pattern. Along resistor circumferential shift is apart from 1.4 get C added element of a series connection nodding, the attention is anticlockwise shift (the) that lose a value. Undertake similar operation can add next element (undertake planar change coming back changes admittance) , along wait for conductance round clockwise (because be,be worth) the distance that shift appoints (1.1)—— is nodded this write down for D. Finally, we return impedance mode to add the last element () of series connection inductance. Then we get needs value, z, be located in 0.2 resistor circle and the node with 0.5 round reactance. So far, reach Z = 0.2 + J0.5. If characteristic impedance of the system is 50Ω , have Z = 10 + J25Ω(See graph 10) .

 

Graph 10. In the network component that Smith draws on round figure

 

Undertake impedance matchs stage by stage

 

Another good that Shi Mi Si Yuan pursues is to undertake impedance matchs. This and the equivalent impedance that find out a foregone network are contrary process. Right now, two end (it is signal source and load normally) impedance is fixed; If pursue,11 are shown. Our target is to be in both between the network with insert a design good already achieved appropriate impedance to match.

 

Graph 11. The typical circuit with impedance is foregone and sealed component

 

Be like it seems that first and not more equivalent than finding impedance is complex. But the problem is being planted indefinitely at having the combination of component can make match a network to have similar effect, public platform reminds filter to still need to consider other factor (for instance the structural type of filter, quality factor is mixed finite optional component) .

 

The method that achieves this one goal is to be in Smith adds series connection and shunt-wound element ceaselessly on round figure, till the impedance that gets we want. Originate public platform reminds filter to look from the graph, find a way to join Shi Mi Si Yuan pursues namely the dot that go up; Same, the best way that shows this kind of method is to give out an example.

 

Our target is source impedance matchs below 60MHz job frequency (ZS) and laden impedance (ZL) (sees graph 11) . Network structure has been been certainly low, l (also can rating the issue is the impedance that how makes laden change is equal to ZS into numerical value, namely) of ZS answer conjugate. The process of solution is below:

 

Graph 12. Graph the network of 11, its corresponding stipple is on graph of Shi Mi Si Yuan

 

Wanting the first thing that do is each impedance is worth normalization. If did not give out characteristic impedance, choose with load / the numerical value of signal source is worth in the impedance of same amount level. Hypothesis Z0 is 50Ω . Then

 

ZS = 0.5 – J0.3, z*S = 0.5 + J0.3, ZL = 2 – J0.5.

 

Next, on the graph line out is nodded twice this, a represents ZL, d represents Z*S

 

Next differentiate and laden connective the first component (shunt-wound capacitance) , first admittance of ZL translate into, get nodding A” .

 

The next after C of firm link capacitance chooses the place that appears on circular arc. Because do not know the value of C, so we do not know specific place, however the way that we know to move really. Originate the capacitance that filter public platform reminds paralell connection should be on admittance circle graph edge clockwise shift, till the numerical value that finds correspondence, get nodding) of B (admittance. Next component is series connection component, need so go up B changeover to impedance plane, get B” . B” needs and D is located in same on resistor circle. Look from the graph, have a way only to D from A” , but if want to be nodded through the B among (namely B”) , try with respect to what need a course for many times and examine. After finding bit of B and B” , we can measure A” to arrive to B and B” the arc length of D, former the normalization susceptance value that is C, latter is the normalization reactance value of L.

 

A” is B = 0.78 to the arc length of B, criterion B = 0.78×Y0 = 0.0156S.

 

Because of ωC = B, so C = B/ω= B/(2πf) = 0.0156/[2π(60×106)] = 41.4pF.

 

B” is to the arc length of D×= 1.2, then X = 1.2×Z0 = 60Ω . By ωL = X, get L = X/ω= X/(2πf)= 60/[2π(60×106)] = 159nH.

 

Graph 13. Circuit of MAX2472 type job

 

The output that the 2nd example is MAX2472 matchs circuit, match at 50Ω laden impedance (ZL) , working frequency shows) 14 times for 900MHz (graph. This network uses as identical as MAX2472 data data configuration structure, the graph gave out to match a network on, originate filter public platform reminds include a shunt-wound inductance and series connection electric capacity, gave out to match what network component is worth to search a process below.

 

Graph 14. The graph shows a network 13 times in Smith the corresponding job on round A graph is nodded

 

Change S22 scattering parameter into above all equivalent normalization source impedance. The Z0 of MAX2472 is 50Ω , s22 = 0.81/-29.4° is changed into ZS = 1.4 – J3.2, ZL = 1 and ZL* = 1.

 

Next, locate on round figure two are nodded, ZS mark is A, ZL* mark is D. Because with signal source connective it is the first component it is shunt-wound inductance, source impedance changeover becomes admittance, get nodding A’ .

 

The next after LMATCH of firm link inductance nods the circular arc of the place, because do not know the numerical value of LMATCH, the position that because this cannot decide circular arc,terminates. But, after we understand join LMATCH and its changeover becomes impedance, what source impedance should be located in R = 1 is circumferential go up. From this, the impedance that gets after series connection capacitance should be Z = 1 + J0. It is a center with origin, 180° rotates on the circle in R = 1, reflection coefficient circle and the dot of nodical union A’ that wait for susceptance circle can get) of B (admittance. B nods corresponding impedance to be B’ dot.

 

After finding B and B” , can measure the length of circular arc A”B and circular arc B”D, the first measured value can get LMATCH. The normalization of susceptance is worth, the 2nd measured value gets the normalization of CMATCH reactance is worth.

 

The measured value of circular arc A”B is B = -0.575, b = -0.575×Y0 = 0.0115S. Because of 1/ωL = B, criterion LMATCH = 1/Bω= 1/(B2πf) = 1/(0.01156×2×π×900×106) = 15.38nH, it is 15nH approximately.

 

The measured value of circular arc B”D is×= -2.81, x = -2.81×Z0 = -140.5Ω . Because – 1/ωC = X, criterion CMATCH = -1/Xω= -1/(X2πf) = -1/(-140.5×2×π×900×106) = 1.259pF, it is 1pF approximately.

 

These computational values did not consider parasitism inductance and parasitism capacitance, gets numerical value is adjacent as numeric as what give out in data data: LMATCH = 12nH and CMATCH = 1pF.

 

Summary

 

Having the software with powerful function and high speed, high-powered computer today: Meeting suspicion is solving people circuit is basic problem when whether to still need a kind of such foundations and simple method.

 

Actually, a true engineer should have academic knowledge not only, should have the capacity that uses all sorts of resource to solve a problem more! A few numbers are joined to reach the result is an easy thing really next in the program, the solution when the problem is very complex, and not exclusive when, let the computer make such job go to the lavatory especially. However, if can understand the basic theory that the working platform place of the computer uses and principle, know their origin, such engineer or architect can be become more comprehensive with the expert that is worth reliance, gotten result is more reliable also.