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Impedance Matching and the Smith Chart – the Fundamentals

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Impedance Matching and the Smith Chart – the Fundamentals antennas tx/rx Impedance matching and the Smith chart – The fundamentals. Tried and true, the Smith chart is still the basic tool for determining transmission line impedances. designer has to be familiar with the cal illustrations of parameters such as By K-C Chan & A. Harter multiple data inputs that need to be finding matching network componentS entered and the correct formats. They values. Of course, matching for maxi- also need the expertise to find the use- mum power transfer is not the only hen dealing with the practical ful data among the tons of results thing one can do with Smith charts. Wimplementation of RF applica- coming out. In addition, circuit simu- They can also help the designer opti- tions there are always some tasks that lation software is not pre-installed on mize for the best noise figures, insure appear nightmarish. One of these is computers unless they are dedicated quality factor impact, asses stability the need to match the different imped- to such an application. analysis, etc. ances of the interconnected block. • Manual computations–Tedious Typically these include the antenna to due to the length ("kilometric") of the A quick primer low-noise amplifier (LNA), RF ouput equations and the complex nature of Before introducing the Smith chart (RFOUT) to anten- utilities, it would na, LNA output to be prudent to pre- mixer input, etc. sent a short re- The matching task fresher on wave is required for a propagation phe- proper transfer of nomenon for IC signal and energy wiring under RF from a "source" to conditions (above a "load". 100 MHz). This At high radio can be true for con- frequencies, the tingencies such as spurious (wires RS485 lines, be- inductances, inter- 0.5 tween a PA and an 0 1 layers capaci- antenna, between tances, conductors a LNA and down- resistances, etc) converter/mixer, elements have a etc. significant, yet It is well known unpredictable that to get the impact upon the maximum power matching network. transfer from a Above a few tenths source to a load, of MHz, theoretical the source imped- calculations and ance must equal simulations are the complex conju- often insufficient. Fundamentals of impedance and the Smith chart. gate of load imped- In-situ RF lab ance, or : measurements, along with tuning the numbers to be manipulated. work, have to be considered for deter- • Instinct–This can be acquired only Rs + jXs = RL – jXL (1) mining the proper final values. The after one has devoted many years to computational values are required to the RF industry. In short, this is for For this condition, the energy trans- set up the type of structure and target the super-specialist! ferred from the source to the load is component values. • Smith Chart–Upon which this maximized. In addition, for efficient There are many possible ways to do article concentrates. power transfer, this condition is impedance matching. Some are: The primary objective of this article required to avoid the reflection of ener- • Computer simulations–Complex is to refresh the Smith chart’s con- gy from the load back to the source. to use since such simulators are dedi- struction and background, and to sum- This is particularly true for high fre- cated for differing design functions marize practical ways to use it. quency environments like video lines and not to impedance matching. The Topics addressed will include practi- and RF and microwave networks. 52 www.rfdesign.com July 2000 What it is Since the impedances are complex −−ΓΓ22 = 1 ri A Smith chart is a circular plot with numbers, the reflection coefficient will r 22 12+−+ΓΓΓrri a lot of interlaced circles on it. When be a complex number as well. (2.6) correctly used, matching impedances, In order to reduce the number of unknown parameters, it is 2Γi x = useful to freeze the ones 22 Zs 12+−+ΓΓΓrri that appear often and are (2.7) common in the applica- Rs Xs XL tion. Here Zo (the charac- Equation (2.6) is then manipulated, by teristic impedance) is developing equations (2.8) through E ZL often a constant and a (2.13), into to the final equation (2.14). RL real industry normalized This equation is a relationship in the value ie: 50 Ω, 75 Ω, 100 form of a parametric equation (x-a)2 + Ω, 600 Ω, etc. We can then (y-b)2 = R2),in the complex plane (Γr, Figure 1. Diagram of Rs + jXs = RL – jXL define a normalized load Γi), of a circle centered at the coordi- impedance by: nates (r/r+1, 0), and having a radius of with apparent complicate structures, 1/1+r. can be made without any computation. z = ZL/Zo = (R + jX) / Zo = r + jx (2.2) The only effort required is the reading 2222 rr+−+=−−ΓΓΓΓΓrriri21 r r and following of values along the circles. With this simplification, we can (2.8) The Smith chart is a polar plot of rewrite the reflection coefficient for- the complex reflection coefficient (also mula as: 22 22 ΓΓir+−++=−rrr21 ΓΓΓ rii r called gamma and symbolized by Γ). (2.9) Or, mathematically defined as the 1- ZZLO− port scattering parameter s or s11. 22 ZZLO− ()1211+−++=−rrrΓΓir () Γ i r A Smith chart is developed by ΓΓ=+ Γ = = ZO (2.10 Lrj i + examining the load where the imped- ZZLO+ ZZLO ance must be matched. Instead of con- ZO − 222r 1 r sidering its impedance directly, one z −1 rjx+−1 ΓΓΓrri− += Γ = = r +1 1 + r expresses its reflection coefficient L, z +1 rjx++1 (2.11) which is used to characterize a load (2.3) (such as admittances, gain, transcon- 2 ductances, etc). The Γ is more useful Here one can see the direct rela- 2 2r r 2 L ΓΓrr− + + Γ i when dealing with RF frequencies. tionship between the load imped- r +1 ()r +1 2 We know the reflection coefficient is ance and its reflection coefficient. 2 − defined as the ratio between the Unfortunately the complex nature of − r = 1 r reflected voltage wave and the inci- the relation is not practically useful, ()r +1 2 1 + r dent voltage wave : so we can use the Smith chart is a (2.12) type of graphical representation of the above equation. Vrefl Γ= To build the chart, the equation Γi Vinc must be re-written to extract standard r=0 (short) geometrical figures (likes circles or r=1 Z stray lines). O First, equation 2.3 is reversed to Γ give: 0 0.5 1 r Vinc r=∝ (open) 1 + ΓL 1 ++ΓΓrij Vrefl ZL zrjx=+ = = 1 − ΓL 1 −−ΓΓrij (2.4) Figure 2. Impedance at the load. and, Figure 3. The points situated on a circle are all the impedances characterized by a same real The amount of reflected signal from impedance part value. For example, the circle, R the load is dependent on the degree of −−ΓΓ22 = 1, is centered at the coordinates (0.5, 0) and has = 1 ri a radius of 0.5. It includes the point (0, 0) which is mismatch between the source imped- r 22 12+−+ΓΓΓrri the reflection zero point (the load is matched with ance and the load impedance. Its (2.5) the characteristic impedance). A short-circuit, as expression has been defined as follows: a load, presents a circle centered at the coordi- By setting the real parts and the nate (0, 0) and has a radius of 1. For an open-cir- cuit load, the circle degenerates to a single point Vrefl ZZLO− imaginary parts of (equation 2.5) (centered at 1, 0 and has a radius of 0). This cor- ΓΓΓ== =+rIj equal, we obtain two independent new responds to a maximum reflection coefficient of Vinc ZZLO+ (2.1) relationships: 1, at which all of the incident wave is totally reflected. 54 www.rfdesign.com July 2000 2 − tered at the coordinates (1, 1/x) ΓΓ− r +=2 1 r ri and having a radius of 1/x. r +1 1 + r Γi 2 + r = 1 Get the picture? ()1 + rr22()1 + To complete our Smith chart, 2.13) we superimpose the two circle’s families. It can then be seen Γ that all of the circles of one fam- 0 0.5 1 r 2 2 ily will intersect all of the circles ΓΓ− r +=2 1 ri of the other family. Knowing the rr+1 1 + 2.14) impedance, in the form of: r + jx, (See Figure 3 for further details) the corresponding reflection coefficient can be determined. It When developing the Smith chart, is only necessary to find the there are certain precautions that intersection point of the two cir- should be noted. Among the more cles, corresponding to the values Figure 4. The points situated on a circle are all the imped- important are: r and x. ances characterized by an identical imaginary impedance • All the circles have one same, part value x. The circle x = 1 is centered at coordinate (1, 1) unique intersecting point at the coor- It’s reciprocating too and has a radius of 1. Furthermore, x can be positive or negative, which explains the duplicate mirror circles at the dinate (1, 0). The reverse operation is also bottom side of the complex plane. Note that the zero-reac- • The zero Ω circle where there is possible. Knowing the reflection tance circle (a pure resistive load) is just the horizontal no resistance (r = 0) is the largest one. coefficient, find the two circles axis of the complex plane. The infinite reactance has • The infinite resistor circle is intersecting at that point and degenerated to one point situated at (1, 0).
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