An Introduction to Broadband Impedance Transformation for RF Power Amplifiers
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High Frequency Design From January 2009 High Frequency Electronics Copyright © 2009 Summit Technical Media, LLC BROADBAND MATCHING 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 con- jugate 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.