2011-12 H.-R. Chuang EE NCKU 5-1 CHAPTER 5 Impedance Matching and Smith Chart * Pozar MW (Ch 5), Impedance Matching and Tuning * Pozar RF (Ch 2), Itransmission Lines & Microwave Networks *Ludwig, (Ch 3, Ch 8), Matching and Biasing Networks *Rogers, (Ch 4), Radio Frequency Integrated Circuit Design - Matching with Lumped Elements - L Network - T & t Networks - Lumped Elements for MIC : Chip R, L, C. - Microstrp Single-Stub and Double-Stub Tuning - Quarter-Wave Transformer - * The Bode-Fano Criteria Appendix Smith Chart TransmitterLCTZAZMZ 2011-12 H.-R. Chuang EE NCKU 5-2RFsignalXRFsignalXRFsignalX I mpedance matching (or tuning) is important for the following reasons : Z0) /( ) ( ) (0 0 11Z Z Z Z Sin in in + = I or: Loss) Return (or t coefficien ReflectionMatchingNetworkLZincident(orinput)reflectioninZLoad - minimum power loss in the feed line & maximum power delivery - linearizing the frequency response of the circuit - improving the S/N ratio of the system for sensitive receiver components (low-noise amplifier, etc.) - reducing amplitude & phase errors in a power distribution network (such as antenna array-feed network) * Factors in the selection of matching networks - complexity -bandwidth requirement (such as broadband design) - adjustability - implementation (transmission line, chip R/L/C elements ..) RF Choke /4 microstrip RF Choke 42-4-20.5 0.250ZShort-Cirucited (S.C.)0/ Z Xscl* At high freq., capacitance is like Short-cirucited 2011-12 H.-R. Chuang EE NCKU 5-3Matching Network Types: L-/T-/t-section Networks L-section Networks (Two-component ): Lumped elements: L/C SZLZLC) (aSZLZ1L2L) (eSZLZ1L2L) ( fSZLZLC) (bSZLZ2C) (c1CSZLZ1C) (d2CSZLZL) (gCSZLZL) (hC In any particular region on the Smith chart, several matching circuits will work and others will not. The figure shows what matching networks will work in which regions. How does one choose? There are a number of popular reasons for choosing one over another. 1. Sometimes matching components can be used as dc blocks (capacitors) or to provide bias currents (inductors). 2. Some circuits may result in more reasonable component values. 3. Personal preference. Sometimes when all paths look equal, you just have to shoot from the hip and pick one. 4. Stability. Since transistor gain is higher at lower frequencies, there may be a low-frequency stability problem. In such a case, sometimes a highpass network (series capacitor, parallel inductor) at the input may be more stable. 5. Harmonic filtering can be done with a lowpass matching network (series L, parallel C ). This may be important, for example, for power amplifiers (PA). 2011-12 H.-R. Chuang EE NCKU 5-4L-section Network (1) complex ZL to real Z0 matching jXZLjB(a)ZLjB(b)jX0Zadmittanceadmittance0Z How to determine jX & jB ? Let zL = ZL / Zo = (RL + jXL) / Zo = r + jx 1. Analytic Solutions or 2. Smith Chart Solution (1) if ) 1 (0 > > z Z RL [zL is inside the (1 +jx) circle ] =>choose (a) why? for impedance matching (to Z0) => jXjB R jXZL L++ + =110( ) = = B XR X Z R ZX BX BZ R XL L LL L L( )( )0 001 => + = + + =) ( ) ( ) 1 (0 02 202 20L L LL LL L L L LR B Z R Z X B XX RR Z X R Z R XB (2) if ) 1 (0 < < z Z RL [zL is outside the (1 +jx) circle ] =>choose (b) why? for impedance matching (to Z0) => jBR j X X ZL L++ + =1 10( ) + = + =BZ X X Z RX X BZ RL LL L0 00( )( ) => = =0 00) () (Z R R Z BX R Z R XL LL L L (1+jx) circler >1r Z0 =100 O (zL is inside the (1 +jx) circle) We choose (a) The solutions are RL 200 XL 100 Zo 100B1XLRLZoRL2XL2ZoRL. .RL2XL2B1 2.899103.=X11B1XLZoRL.ZoB1RL.X1 122.474 =B2XLRLZoRL2XL2ZoRL. .RL2XL2B2 6.899103.=X21B2XLZoRL.ZoB2RL.X2 122.474 = At f =500 MHz O =1000Z38.8nH0.92pFO = 100 200 jZLSolution 1 (low pass)O =1000Z2.61pF46.1nHO = 100 200 jZLSolution 2 (high pass) BW% 60 5 . 0 / 3 . 0 3 . 05 . 9 2 ) 1 . 0 ( 33 . 02= ~= ~ ~ I ~ IGHz BWdB RL SWR bandwidth (1) Solution & (low pass) (high pass) 2011-12 H.-R. Chuang EE NCKU 5-6Smith Chart Representation of the Matching Process O =1000Z38.8nH0.92pFO = 100 200 jZLSolution 1 (low pass) O =1000Z38.8nH0.92pFO = 100 200 jZL0.92pF38.8nHoo L o L Lo L LZ Z Z Zj Z Z z6 . 26 45 . 0) /( ) (2 / Z = + = I = = O = 100 200 jZL 2011-12 H.-R. Chuang EE NCKU 5-7 (2) Complex to complex conjugate matching (Ludwig, RF Circuit Design P401) (Conjugate Matching for maximum power transfer ) versa vice &0 ) ( of ) ( argument " " choose > - - + O = > O = ) 75 ( ) 150 (A TR RGHz fj Zj ZAT215 7575 150=O + = O + =TZ AZ*A MZ Z =TransmitterZLjBjXadmittance0Zcomplex ZL to real Z0 matching complex ZA to complex ZT conjugate matching 2011-12 H.-R. Chuang EE NCKU 5-8 - + = + = = + = + = = O =2 . 0 1 75 / ) 15 75 ( /1 2 75 / ) 75 150 ( /75000j j Z Z zj j Z Z zZA AT TLet GHz fj Zj ZAT215 7575 150=O + = O + =TZ AZ*A MZ Z =Transmittercomplex ZT to complex ZA conjugate matching 2011-12 H.-R. Chuang EE NCKU 5-9(3) General L-section matching network (complex to complex) GHz fj Zj ZLs250 2525 50=O = O + =+ = = + =1 5 . 0 *1 5 . 05 . 0 1j zj zj zLLsLz*LzsZ LZ*L sZ Z =Transmittercomplex Zs to complex ZL : conjugate matching *LD C B Asz z paths) (four , , , ACB D 2011-12 H.-R. Chuang EE NCKU 5-10Ex: L-section Lumped-Elements & Microstrip Matching Networks Conjugately Matched Amplifier Design (Pozar MW EX11-3 or RF EX6-3 ) Design an amplifier for maximum gain at 4.0 GHz using single-stub matching sections. Calculate and plot the input return loss & the gain from 3 to 5 GHz. The GaAs FET has the following S parameters (Z0=50 O): f S S S S (GHz) 11 21 12 22 30 080 89 99 003 56 076 4140 072 116 76 003 57 073 5450 066 142 54 003 62 072 68. . . .. . . .. . . . 2.86 2.60 2.39 Z Z Z Z Z Z Z Z Z Z Z Z FET S-parameters Touchstone file: Poz_11-3.s2p ! poz_11-3.s2p : Pozar Ex. 11-3 transistor S parameters ! Typical s-parameters at minimum attenuation setting, Ta=25C #ghz s ma r 50 3.00 0.800 -89.0 2.860 99.0 0.030 56.0 0.760 -41.0 4.00 0.720 -116.0 2.600 76.0 0.030 57.0 0.730 -54.0 5.00 0.660 -142.0 2.390 54.0 0.030 62.0 0.720 -68.0 It cab be derived that (see chapter of RF Amplifier Design) Z = I = I Z = I = I Z = I Z = IoL outoS inLs61 87 . 0123 87 . 061 876 . 0123 872 . 0** & 2011-12 H.-R. Chuang EE NCKU 5-11 Microstrip Matching Networks 0120 . 50O50O50O0206 . 0206 . 0206 . 50O50O50OoutIsIinILI (f =4 GHz) ) 50 (5 . 83 68 . 1297 . 26 43 . 461 87 . 0123 87 . 061 876 . 0123 872 . 00**O = = = Z = I = I Z = I = I Z = I Z = IZj Zj ZLinoL outoS inLs & Lumped Elements Matching Networks 1.63nH2.54pF4.19nH1.32pF50O50O3 12 4000outIsIinILI 2011-12 H.-R. Chuang EE NCKU 5-12* By Smith-Chart tool DP-Nr. 1(4.4 - j27.0)Ohm Q =6.1 4.000 GHz DP-Nr. 2(4.4 +j14.1)Ohm Q =3.2 4.000 GHz DP-Nr. 3(49.4 - j0.2)Ohm Q =0.0 4.000 GHz DP-Nr. 1(4.4 - j27.0)Ohm Q =6.1 4.000 GHz DP-Nr. 2(3.6 +j13.0)Ohm Q =3.6 4.000 GHz DP-Nr. 3(50.4 +j1.4)Ohm Q =0.0 4.000 GHz rtransmission-line matching network(open-circuited stub) 2011-12 H.-R. Chuang EE NCKU 5-13 DP-Nr. 1(4.4 - j27.0)Ohm Q =6.1 4.000 GHz DP-Nr. 2(3.6 - j12.7)Ohm Q =3.5 4.000 GHz DP-Nr. 3(48.0 - j0.0)Ohm Q =0.0 4.000 GHz Transmission-line matching network (shorted-circuited stub) 2011-12 H.-R. Chuang EE NCKU 5-14 Forbidden Regions for L-type Matching Networks with O = = 500Z Zs =>The shaded areas denote values of load impedance that cannot be matched to 50 2011-12 H.-R. Chuang EE NCKU 5-15Design Example: Forbidden Regions for L-type Matching Networks 50) = (with network - L of regions forbidden from (d) or (c) choose =>1 > 1.6 = Since for SLLL LZzGHz fZj zX R1) 50 (2 . 1 6 . 160 8000= O = = O = O = 2011-12 H.-R. Chuang EE NCKU 5-16 Quality factor & Bandwidth (BW) (there are much more to be discussed!) PPssn P P P s s sGBRXQ jB G Y jX R Z| |or | | or = + = + = Lo o nLQfBWBWf QQ = =|.|

\|= 2 * T Matching Network (discussed next) 2011-12 H.-R. Chuang EE NCKU 5-17* T & t Matching Network: The.addition of 3rd element into the two-element (L) matching network introduces an additional degree of freedom in the i!uit, and allows us to control the value of QL (to be discussed)by choosing an appropriate intermediate impedance. =>wider (matching) bandwidth T Matching Network t Matching Network GHz fj Zj ZLin130 6020 10=O = O + =GHz fj Zj ZLin130 6020 10=O = O + = 2011-12 H.-R. Chuang EE NCKU 5-18Comparison between L-, T - & t - network Design a match circuit at the center frequency of 100 MHz * Prof. C.-F. Chang course note (NCCU) 51 O 0.1 H 10 pF 510 O L t T 4-element ladder 2011-12 H.-R. Chuang EE NCKU 5-19 Microstrip Line Matching Networks (Ludwig P431) In the mid-GHz and higher frequency range, the wavelength becomes sufficiently small and the distributed components are widely used. Also, the discrete R/L/C lumped elements will have more noticeable parasitic effects (see chapter 2) and let to complicating the circuit design process Distributed componenets (such as transmission line segments) can be used to mix with lumped elements From Discrete Components to Microstrip Lines Avoid using inductors (if possible) due to higher resistive loss (& higher price) In general, one shunt capacitor & two series transmission lines is sufficiently to transform any load to any input impedance. EX: transform load LZ to an input impedance inZ Identify input & load SWR circles Choose A (yA=1-j0.6) & transform zL to A by a series TL (l1) =>Transform A to B (on the input SWR circle) by a parallel C1 =>Transform B to zin by a series TL (l2) GHz fj z j Zj z j Zin inL L5 . 16 . 1 2 . 1 80 602 . 0 6 . 0 10 30=O + = O + = O + = O + =zL +series-TL (l1) =>A +shunt C1 =>B +series-TL (l2) =>zin 2011-12 H.-R. Chuang EE NCKU 5-20Single-Stub Matching Networks - 4 adjustable parameters: ) , , (, 0 , 0 L L s sZ l Z l O + = = + = = = = O =O + = O =2 . 1 1 /6 . 0 8 . 0 / 16 . 0 8 . 0 /7590 7545 60000j Z Z zj z yj Z Z zZj Zj Zin inL LL LinLg =0.8 conductance circle I nput SWR circle associated with zin has two intersected points (A & B) with g =0.8 conductance circle yA=0.8 +j1.05 yB=0.8 - j1.05 zL to A (yA=0.8 +j1.5) by adding a shunt open-circuited (O.C.)TL lSA The corresponding susceptance for the stub : jbSA=yA- yL =(0.8 +j1.05)-( 0.8 +j0.6)=0.45 O.C. point (g=0) to the point of ibSA =0.45 is lSA =0.067 A to zin is lLA =0.266 g =0 (O.C.) ibSA =0.45 2011-12 H.-R. Chuang EE NCKU 5-21 |.|

\|tt=|.|

\|tt== =>ssBssBsB sB ssBlllll l ll2tan21tan22tan 2 tan2//)11 : stub circuit - short : stub circuit - open ( design stub Balanced 2011-12 H.-R. Chuang EE NCKU 5-22Double-Stub Matching Networks = = = = =O + = O = =051 . 0 074 . 08 / 3 8 /50 50501 13 2 10s sLinl ll l lj ZZ Z 2011-12 H.-R. Chuang EE NCKU 5-23Quarter-Wave Transformer () ( only useful for pure-resistance matching ) 20 000000tan&24 000 0) () (tantan) (Z Z ZZZZjZ ZZ j ZZl jZ Zl Z j ZZ Z ZLL LLllLLin' = =>'' = + ' ' +' = | + ' | ' +' = = = |t= |= Ex: A microstrip quarter-wave trasformer that matches a 50 O miscrostrip line to a 20 O load at f =4 GHz (substrate: cr=2.5, thickness h =0.75 mm) 2.13[mm] 4.03[mm]12.73[mm]] [ 50O] [ 62 . 31 O] [ 20O * Double Quarter-Wave Transformer for wider (matching bandwidth) ) (0Z) (0Z'LZ) (1z V+transmission line 1 4 / = l0 ) (1 =z VXinZquarter-wavelengthtransmission line 2 2011-12 H.-R. Chuang EE NCKU 5-24* (Matching) Bandwidth (Af ) of a Quarter-Wave Transformer Pozar, Mcrowave & RF Design of Wireless Systems Approximate behavior of the reflection coefficient magnitude of a quarter-wave transformer near the design frequency It can be proved that 2001sec212(((

u+ =I Z ZZ ZLLm (((

t =tu = == =AI I00110 0002cos424222) ( 2) (2Z ZZ Zffff fBWffLLm m mmm I ncreased BW for Smaller load mismatch (ZL/Z0) 2011-12 H.-R. Chuang EE NCKU 5-25