An Introduction to Broadband Impedance Transformation for RF Power Amplifiers

By Anthony J. Bichler RF Micro Devices, Inc.

his paper discusses obtained by simply reversing the sign of the This tutorial article reviews broadband impe- imaginary part. Here Z* denotes the complex impedance matching Tdance-transform- conjugate of Z; thus, for linear systems the principles and techniques, ing techniques specific for condition for maximum power transfer is

as they are applied to radio frequency power when ZLoad = ZSource*, or: ZL = ZS*. power device matching amplifiers. Single and As the frequency of operation changes for

in amplifier circuits multiple Q matching ZS, relative to its parasitics, the value of the techniques are demon- resistive component can substantially change strated for broadband performance; here the as well as the value of the imaginary compo- reader will understand the importance of a nent. Transforming a standard system load impedance trajectory relevant to load impedance to present a driving point load

pull contours. impedance ZL that maintains a complex conjugate relationship to the source impedance Introduction change over frequency is the most challenging When analytically defining radio frequen- aspect of broadband design. cy circuits, a common approach incorporates Note: The linear condition for maximum admittance or impedance. Admittance, which power transfer is often traded for other per- is symbolized by Y, is defined in terms of con- formance parameters such as efficiency or ductance G and an imaginary susceptance gain. For this tradeoff the load impedance will component, jB. Admittance is often useful not hold a conjugate relationship; however, the when defining parallel elements in a network challenge of maintaining a load for this per- and is expressed by the complex algebraic formance parameter over a broadband will equation Y = G + jB. generally remain the same. Impedance, the mathematical inverse of admittance, is symbolized by Z and consists of A Review of Smith Chart Fundamentals a resistive component R in units of ohms and a Philip H. Smith introduced the Smith reactive or imaginary component jX. Together Chart in Electronics Magazine on January in a series complex expression they define 1939, revolutionizing the RF industry [1, 2]. impedance as Z = R + jX. Impedance in this This chart simplified complex parallel to rectangular form is often used in industry to series conversions graphically and, for the define a power device’s optimal source or load. first time, provided intuitive transmission line For linear systems, the condition for maxi- solutions. mum power transfer is obtained when the The Smith Chart is a graphical reflection impedance of the circuit receiving a signal has coefficient system with normalized conformal an equal resistance and an opposite reactance mapping of impedance or admittance coordi- of the circuit sending the signal. In the math- nates, as shown Figure 1 and 2, respectively. ematics of complex variables, this relationship Reflection coefficient is often referred to as is known as the complex conjugate. The com- gamma and is symbolized by the Greek letter plex conjugate of a complex number is Γ. Gamma in its simplest form is defined as

34 High Frequency Electronics High Frequency Design BROADBAND MATCHING

Figure 1 · The Impedance Smith Figure 2 · The Admittance Smith Figure 3 · The Immittance Chart Chart. Chart. with an SWR circle (green line) defined by [Γ] radius. the ratio of the root of the incident will conveniently reference to 50 power wave versus the root of the ohms or another value as defined by the circumference is equivalent to 90 reflected power wave [3]: the user. degrees of wavelength rotation. The Smith Chart’s horizontal cen- A Smith Chart that has combined W Γ= r terline is known as the resistive line. the Admittance and Impedance It is scaled left to right, zero to infin- Smith Charts for simplicity is known Wi ity, with the normalized impedance as the Immittance Chart [5]. With where Wr is the reflected power and (Zo) centered in the middle of the the Immittance Chart a network con- Wi is the incident power. Gamma can chart. On the lower half of the chart sisting of capacitors or inductors in also be defined in terms of impedance below the resistive line, are the shunt or in series can be easily cas- where capacitive coordinates and above the caded without rotating the chart. line are the inductive coordinates. In Figure 3, an Immittance Chart, ZZ− Γ = LS The circles that are tangent to the the Gamma vector magnitude defines + ZZLS right side of the impedance Smith the radius of a constant standing Chart are the circles of constant wave ratio (SWR) circle. Standing For a large impedance mismatch resistance. Above and below the right waves are a phenomenon of the volt- Γ would approach unity, and for a side at R = infinity are the semicir- age or current waves from the sum- near perfect match Γ would approach cles of constant reactance. mation of an incident power wave and zero. On the Admittance Smith Chart, reflected power wave on a transmis- Impedances are often normalized the circles that are tangent to the left sion line from a mismatched load. when plotted on the Smith Chart. side are the circles of constant con- SWR is the ratio of the maximum ver-

Normalizing ZL to the center of the ductance and the semicircles above sus the minimum voltage or current Smith Chart where Z = 1 gives one and below to the left where R = 0 are on a standing wave and is commonly access to the chart’s maximum reso- the circles of constant susceptance. referred to as VSWR or ISWR, for lution. Note that normalizing is the The Admittance Smith Chart is voltage or current respectively. [6, 7] division of an impedance by a refer- simply a mirror image of the SWR can be expressed in terms of ence [4]. For example, normalizing Impedance Smith Chart where the Gamma’s magnitude by with the 50-ohm system load Impedance Smith Chart can be rotat- 1 + []Γ impedance as the reference (Z0 = 50), ed by 180 degrees to serve as an SWR = − []Γ a source impedance of ZS = 100 + j50 Admittance Chart. This duality of the 1 would normalize to ZS /Z0 = 2 + j1 = Smith Chart is exploited for admit- zs, where the lower case z is used for tance to impedance conversions by or directly by the mismatch normalized impedances. Most com- simply rotating both the reflection impedances puter aided design or Smith Chart coefficient vector by 180 degrees and Z programs have simplified the nor- then the chart itself by 180 degrees. SWR = L Z malization process. In the case of Note that since the Smith Chart is a S normalizing, the center of the chart reflection system, 180 degrees around or

36 High Frequency Electronics Z center frequency (F ) divided by the resistive transformation ratio is = S C SWR 3 dB bandwidth and is expressed as given by [12] ZL

2 where the equation choice is dictated FC 1 +=QR Q = ratio by which one provides a quantity Unloaded BW greater than unity. For increasing bandwidth by SWR circles are used throughout For simple resonant tank net- increasing the number of n-sections the following Smith Chart illustra- works, unloaded Q can be substituted having equal Q the relationship tions to quantify the mismatch over with loaded Q in bandwidth calcula- becomes frequency. tions [11]. When the resonant fre- +=2 n quency is equal to the center frequen- 1 QR()ratio The Importance of Quality cy, then unloaded Q can define the Factors bandwidth by Note: Using the guideline bound- It is important to understand the aries above in this reference does not F quality factors Q, as they are integral BW = C yield the optimal broadband design. to bandwidth. Q factors are used to QUnloaded Other topologies will be discussed define the quality of a reactive ele- such as the Chebyshev response ment by its ability to store energy, to or with substitution transformation, which has a signifi- fundamentally define bandwidth, cant bandwidth advantage over the F and to define the ability of a loaded BW = C single Q matching technique. Single Q network to store energy.To ease some XR/ advantages to be considered are trans- of the confusion with these Q factors formation efficiency with smaller com- they have been assigned the terms The Q of the load is often used to ponent values and design simplicity. unloaded Q, loaded Q, and Q of the define a loaded network, which typi- In practice using more than a load respectively [8]. cally consists of ideal (lossless) match- four-section matching network will Unloaded Q is fundamentally ing elements. The network is not loss- not yield greater bandwidth. defined as the ratio of stored energy less since energy is propagated to and Also, Q of the load should not be versus dissipated energy [10] or absorbed by the load. It is defined as substituted with unloaded Q or load- before with the unloaded Q as a ratio ed Q. For example, in the following reactive power IX2 Q ==of the reactance to resistance multiple section illustrations, which Unloaded real power IR2 are bounded by a Q of the load curve X = 1.75 (for a 50 to 3 ohm transforma- Q = which reduces to R tion), yield more 3 dB bandwidth than defined by the loaded Q of 1.75. X Q = or in terms of vectors; the imaginary Q of the load will be referred to Unloaded R component magnitude versus the throughout the remainder of this dis- resistive component. cussion as the single letter Q. For capacitors, unloaded Q is Plotting Q of the load as constant expressed as a ratio of capacitive ratio on the Smith Chart will define a Computer-Aided Design (CAD) reactance to equivalent series resis- constant Q curve. These Q curves are and Other Smith Chart Programs tance (ESR) [9] or often used as guideline boundaries Smith Chart programs such as for broadband transformations and the early Motorola Impedance X = C will be used throughout the following Matching Program (MIMP) provide a QUnloaded RESR illustrations to define the transform- useful tool by automating the repeti- ing networks. As a rule in broadband tive graphical computations [13]. and for inductors unloaded Q is transformations, maintaining a lower Considering the frequency point cal- expressed by Q curve for a given transformation by culations required for resolution of a increasing the number of n-sections broadband matching network, this is X will yield a higher bandwidth. a tedious task at best. Smith Chart Q = L Unloaded R For a single section transforma- programs quickly and accurately plot tion where the resistive line and a the required trajectories and circles where R is the series resistance from constant Q of the load curve bound allowing the designer to focus on the the windings of the coil. the transformation, the relationship design and not the mechanics of gen- Loaded Q is defined by the band’s between the Q of the load and the erating a display. Other Smith Chart

January 2009 37 High Frequency Design BROADBAND MATCHING

Figure 4 · ZO = 50, ZS = 3, Q = 1.75, SWR = 1.4; N1-2 Series L = 0.9 pH, N2-3 Shunt C = 25 pF; N3-4 Series L = 3.8 nH, N4-5 Shunt C = 6.5 pF. programs followed MIMP such as transformation is reversed. Trans- winSmith [14], LinSmith [15], and forming from the 50-ohm system Smith32 [16]. Although these pro- impedance the load trajectory is illus- grams leveraged an engineer’s intu- trated relative to a laterally diffused itive creativity with symmetrical Q metal oxide semiconductor device matching solutions, they fall short of (LDMOS) load pull performance con- the sophistication that CAD systems tours in Figure 5. provide. Systems such as Applied The trajectory intersects several Wave Research’s Microwave Office contours in gain and linearity; how- [17] and Agilient’s Advanced Design ever, the contours represent perfor- System (ADS) [18] offer electromag- mance for single frequency operation netic simulation of arbitrary struc- (2-tone 880 MHz). These contours tures, complex network synthesis, will follow a trajectory of their own and optimizers that provide a fully relative to the parasitic capacitance automated solution. Modern CAD of an LDMOS device. Moreover and systems now offer a complete simula- important to note, the trajectory of Figure 5 · Trajectory relative to tion toolset with non-linear synthesis these contours will track opposite 60WPEP LDMOS Load Pull contours and layout functions. (counterclockwise) to the driving [19]: Zo = 3, 28VDC, 900 MHz; Max When the design challenge is point load trajectory thus further Gain = 23.6 dB @ Z = 1.0 + j1.3; Min more fundamental, and when the degrading broadband performance. IMD3 = -32.7 dBc @ Z = 1.0 + j0.0; best solution is intuitively derived, Figure 6 illustrates a model of a Max Eff = 66.75% @ Z = 0.9 + j1.8. Smith Chart programs are well suit- LDMOS power amplifier with plotted ed for the task. The following demon- complex conjugate load impedance strations were plotted with the points at 800 MHz, 900 MHz, and data or when the data sheet is not author’s preference, Smith32. 1000 MHz. For this model, load pull applicable to the design an approxi- contours would track counterclock- mation can be derived [20]. Transformation and Performance wise as with the indicated conjugate The purely resistive component of Figure 4 illustrates a 2-section (4- trajectory points [4]. An ideal coun- the optimal load (RL) can be approxi- element) transformation from a 3- terclockwise load trajectory would be mated from the operational RF out- ohm driving point impedance to a 50- a challenge to any broadband design- put power and supply voltage from ohm load, a 16.7:1 transformation er; the popular compromise is a com- the equation ratio. Confined to a constant Q = 1.75 pressed and or folded trajectory 2 curve and the resistive line, the design. = V RL Gamma from the 800 - 1000 MHz tra- The optimal output load impe- 2Pout jectory (in red) is quantified with a dance of RF transistors as generally SWR circle of 1.4 (center green circle). published in manufacture data sheets With the transistor biased off the To predict the performance includes all capacitive and package output parasitic capacitance can then response from the trajectory, the lead parasitics. In the absence of this be measured directly with a capaci-

38 High Frequency Electronics High Frequency Design BROADBAND MATCHING

Figure 7 · Z0 = 50, Q = 1.75; N1-2 Figure 8 · Z0 = 50, Q = 1.75; N1-2 Shunt C = 6 pF. Series L = 15nH. Figure 6 · A simple LDMOS model with indicated complex conjugate loads: Z0 = 5, RL = RS = 13, CS = 47.5 ty, the following transformations are In Figure 9 a two-element low- pF, LS = 250 pH. confined to a <25% factional band- pass network is charted on a Z0 = 25 width (800 MHz to 1000 MHz) with normalized Smith Chart. The nor- 900 MHz set as the reference. malized impedance of 25 ohms is cal- tance meter. The output inductance In Figure 7 shunt capacitance culated from the geometric mean of can be derived from package and rotates the trajectory clockwise from the system load and source wire-bond mechanical dimensions. ZL = 50 with increasing frequency or impedance, 50 to 12.5 ohms respec- In this packaged 28V, 30W, with capacitance value following the tively [5]. LDMOS model admittance equation =∗ ZZZGeo L S ==2 ( ) = Ω 1 RRS Model L 28/ 2 30 13 ==ω jBC j C jXC The constant Q curve of 1.75 is For 900 MHz the capacitive para- derived from the resistive ratio of sitic (CS) and lead inductance (LS) Note the trajectory from 800 MHz 50/12.5 from the equation transform RS Model to ZS Model = 1 – 2j to 1000 MHz (in red) is co-angular +=2 Other transistor technologies such with the shunt capacitive reactance 1 QRratio as gallium arsenide (GaAs) and galli- (in blue) following the constant con- um nitride (GaN) have greatly ductance circle. Note that the impedance trajecto- reduced capacitance for broadband In Figure 8, series inductance ry is no longer co-angular to the con- performance. For example RFMD’s rotates the trajectory clockwise along stant resistance arc of the series GaN1C process having higher current a constant resistance circle with inductance reactance (nodes 2-3). density is only 0.05 - 0.1 pF/W where increasing frequency or with an LDMOS has roughly 0.75 pF/W of increase of inductance following the High-Pass Lumped Elements and output capacitance. The bandwidth reactance equation the High-Pass L-Network achievable is highest for the GaN fol- In summary, shunt C and series L Shunt inductive reactance as lowed by GaAs, and LDMOS. The disperse a trajectory with increasing demonstrated in Figure 10 rotates tradeoff of the lower capacitance tech- frequency. In other words when using clockwise along a constant conduc- nologies is monetary with LDMOS these matching elements in a low- tance circle with increasing frequen- having the best economical value. pass network, the higher frequencies cy following the susceptance equation will rotate and transform more than 11 Transformation with the Low-Pass the lower frequencies, which spreads jB == L ω L-Network the trajectory relative to frequency in jXL j L For standardization and uniformi- a clockwise direction.

40 High Frequency Electronics Figure 9 · Z0 = 25, Q = 1.75; N1-2 Shunt Figure 10 · Z0 = 50, Q = 1.75; N1-2 Figure 11 · Z0 = 50, Q =1.75; N1-2 C = 6.1 pF; N2-3 Series L = 3.8 nH. Shunt L = 5 nH. Series C = 2 pF.

This element is different than the low-pass L-network, the higher fre- two matching elements discussed quencies are transformed less than previously such that shunt induc- the lower frequencies. If the low-pass tance susceptance decreases with trajectory of Figure 9 were overlaid increasing frequency. onto Figure 12, the two trajectories Series capacitance is similar; how- would form the letter X. Exploiting ever, its reactance is plotted on a con- this relationship by combining these stant resistance circle in Figure 11 dispersion effects can leverage a following the reactance equation broadband transformation.

1 jX = Compressing Trajectory C jCω Dispersion A broadband band-pass network Series capacitive reactance is illustrated in Figure 13, a 50 to 3 rotates clockwise with increasing fre- ohm transformation similar to the quency and decreases with increas- one in Figure 4. With the Smith ing frequency. Chart normalized to the geometric Shunt L and series C disperse an mean, it is easy to see that low pass Ω impedance trajectory in a clockwise nodes 1-2-3 are symmetrical in Q to Figure 12 · Z0 = 25 , Q = 1.75; N1- direction with frequency, but the the high pass nodes 3-4-5. Combining 2 Shunt L = 5.1 nH; N2-3 Series C = reactance will be decreasing with fre- these two networks’ halves folds and 8.2 pF. quency. Hence, high-pass matching compresses the trajectory into a con- networks consisting of shunt induc- densed 3-ohm driving point load. tors and series capacitors will trans- Compare this transformation, over-rotated well beyond the resistive form the lower frequencies more than which has a mismatch SWR of 1.08, line at node 3, which compresses the the higher frequencies. to that of Figure 4 where the mis- upper frequency dispersion. Again, In Figure 12, a two-element high- match SWR is 1.4. compare this network of Figure 14 to pass L-network transformation from A Chebyshev broadbanding tech- that of Figure 4; a 3-ohm SWR band- 50 to 12.5 ohms is demonstrated on a nique is illustrated in Figure 14. As width of 1.12 versus 1.4. 25-ohm normalized Smith Chart. discussed earlier, when using low- The transformation is mostly Note that the trajectory is no longer pass networks the higher frequencies symmetrical with two Q curves, an co-angular to the constant resistance transform and rotate more. Here the outer curve (Q1 green) and an inner circle of (nodes 2-3) and that unlike a frequencies higher than 800 MHz are curve (Q3 magenta). However, node 5

January 2009 41 High Frequency Design BROADBAND MATCHING

Figure 13 · 50 to 3-ohm transformation; Z0 = 12.5, Q = 1.75; N1-2 Shunt C = 6.10 Table 1 · Q curves per transforma- pF; N2-3 Series L = 3.85 nH; N3-4 Shunt L = 1.32 nH; N4-5 Series C = 32.3 pF. tion ratio (2-section network). The Q curves are numbered from the

outer most Q1 towards the inner Q3.

Figure 14 · Z0 = 12.3, SWR = 1.12 @ ZS = 3; N1-2 Shunt C = 6.9 pF; N2-3 Series L = 4.4 nH; N3-4 Shunt C = 30.4 pF; N4-5 Series L = 0.99 nH. falls at a higher impedance than the that of the other Q curves. 3-ohm target to center the fish In Figure 15, a three-section Table 2 · Q curves per resistive shaped trajectory at Z = 3 + j0 and so transformation, the trajectory fits transformation ratio (3-section net- therefore a third Q curve (Q2, cyan) is into a 3-ohm 1.01 SWR circle. Three work). The Q curves are numbered defined at node 4. Q curves are adquate for defining the from the outer most Q1 towards the With the complexity of the multi- three section network since the tra- inner Q3. ple Q curve network, deriving a jectory is small and circular in shape, design from a Smith Chart alone unlike in Figure 14; here no would not be an intuitive process. The impedance offset is needed at node 7. formations. Figure 16 demonstrates Q curves are in overlapping fractions As mentioned above, Table 2 was an immediate approach to a high Q of the resistive transformation and derived from optimization. Here Q transformation from a purely resis- do not hold the relationship with the curves are provided for resistive tive impedance of Z0 = 50, to a load transformation ratio as before with transformation ratios of 1.67:1 (50 impedance with where ZL = 20 + j50. single Q networks. This network and ohms to 30 ohms) to 100:1 (50 ohms A broadband match in this case is Tables 1 and 2 were derived by opti- to 0.5 ohms). seemingly impossible to design, espe- mization with an ADS simulator uti- cially when considering the source lizing a gradient optimizer. Complex Transformations impedance dispersion from the large

Note that the inner Q curve (Q3) All transformations discussed corresponding parasitics. However, as a function of the transformation previously have been purely resistive the transformation can be forbearing; ratio holds an inverse relationship to to resistive (50-ohm to 3-ohm) trans- Figure 17 includes an additional

44 High Frequency Electronics Conclusion Multiple frequency point load pull contours demonstrate the necessity for a compressed and/or folded impedance trajectory for optimized broadband power amplifier design. Single Q matching where the resistive line and Q curve serve as guideline boundaries are too often present- ed as the mainstream broadband design technique. Here we have shown that multiple Q curve transformations although more complex in their derivation have superior bandwidth over the single Q matching technique although the single Q matching technique is easily demonstrated we recommend that designers consider a multiple Q transformation. Furthermore, where device and package parasitics disperse the Figure 15 · Z0 = 12.2, SWR = 1.01 @ ZS = 3; N1-2 Shunt C = 4.30 pF, N2-3 Series L = 6.22 nH; N3-4 Shunt C = 16.57 pF, N4-5 Series L = 2.42 nH; N5-6 source impedance counter to a broad Shunt C = 42.05 pF, N6-7 Series L =0.63 nH. band transformation, the use of multiple Q curve transformations is per- haps a categorical.

Acknowledgements The author is indebted to many friends who reviewed this document for accuracy. Special thanks to John B Call for many broadband network discus- sions. Special thanks to Kal Shallal for the LDMOS load pull contours, and for the device modeling discus- sion.

References: 1. “Philip H. Smith: A Brief Biography” by Randy Rhea, Noble Publishing 1995. 2. Smith® Chart is a registered trademark and is the property of Analog Instrument Company, New Providence, NJ. Ω Ω Figure 16 · Z0 = 50 , Q = 2.5. Figure 17 · Z0 = 50 , Q = 2.5; N1-2 3. Michael Hiebel, Fundamentals Shunt C = 4.0 pF, N2-3 Series L = 2.8 of Vector Network Analysis, Rohde & nH; N3-4 Shunt L = 5.0 nH, N4-5 Schwarz 2007, pg. 14. shunt capacitor that is proportioned Series L = 7.0 nH. 4. Chris Bowick, RF Circuit for the complex target impedance. Design, Newnes imprint of Butter- This additional element re-orders the worth-Heinemann, 1982, Ch. 4 - 5. dispersion effects of the transforming how the dispersion effects from 5. Herbert L. Krauss, Charles W. network; hence improving broadband lumped elements can be leveraged to Bostian, Fredrick H. Raab, Solid performance. It is another example of compress and fold the trajectory. State Radio Engineering, Zhuyi

January 2009 45 High Frequency Design BROADBAND MATCHING

Publishing of Taiwan 1980. 14. Agilent Technologies, win- Author Information 6. Donslf W. Dearholt, William R. Smith 2.0, Noble Publishing 1998. Anthony Bichler is a design engi- McSpadden, Electromagnetic Wave 15. linSmith, John Coppens, 1999- neer with RFMD in Chandler Propagation, McGraw-Hill Inc. 1973 , 2008, www.jcoppens.com/soft/linsmith. Arizona. His 25 Ch. 5.4 - 5.5. 16. Pederson, Ib F., Smith32, years of RF 7. Joseph F. White, High Denmark 2002. experience Frequency Techniques / An Introduc- 17. Applied Wave Research, Inc. includes power tion to RF and Microwave Engin- Microwave Office®, El Segundo, CA. amplifier design eering, John Wiley & Sons 2004. 18. Agilent Technologies, for RFID, cellu- 8. Randy Rhea, “Yin-Yang of Advanced Design System. lar handsets, Matching: Part 2- Practical Matching 19. Load Pull contours courtesy of and base sta- Techniques,” High Frequency Khalid Shallal, RFMD. tions. Presently, Electronics, April 2006. 20. B. Becciolini, Impedance Tony is working on quad-band trans- 9. The RF Capacitor Handbook, Matching Networks Applied To R-F mit modules for GSM, PCS, and DCS American Technical Ceramics Corp. Power Transistors, Motorola AN-721, handsets. Interested readers may 1994. Motorola Inc., 1974. contact him at [email protected] 10. 1989 ARRL Handbook for the Radio Amateur, American Radio Article Archives Relay League, pp 2-27 thru 2-29. 11. Thomas L. Floyd, Electronics Remember, all of our technical articles and columns are Fundamentals: Circuits, Devices, and available as PDF files for download from our Web site! Applications, 2nd ed. McMillan Publishing Co. 1991, Chapters 14.4 - Articles in the current print and Online Edition become 14.7 available in the Archives upon publication of the next issue. 12. J. F. White, High Frequency Techniques / An Introduction to RF and Microwave Engineering, John Wiley & Sons 2004, pp. 70-71. 13. Dan Moline, Motorola Impedance Matching Program, www.highfrequencyelectronics.com Motorola Inc., April 6, 1992.

46 High Frequency Electronics