designing highly-linear microwave power amplifiers based on large

DESIGNING HIGHLY-LINEAR MICROWAVE POWER AMPLIFIERS BASED ON LARGE-SIGNAL IMD SWEET-SPOTS José C. Pedro1, Nuno B. Carvalho1 and José A. Garcia2 1 – Institute of Telecommunications – University of Aveiro – Aveiro, Portugal 2 – Communications Department – University of Cantabria – Santander, Spain

Corresponding Author: José Carlos Pedro Instituto de Telecomunicações Universidade de Aveiro Campus Universitário 3810 – 193 Aveiro Portugal e-mail: [email protected] Tel: 351 234 377900 Fax: 351 234 377901 1st Co-Author:

Nuno Borges Carvalho Instituto de Telecomunicações Universidade de Aveiro Campus Universitário 3810 – 193 Aveiro Portugal e-mail: [email protected]

2nd Co-Author:

José Ángel Garcia Departamento de Ingenieria de Comunicaciones Universidade de Cantabria Avda Los Castros s/n 39005 Santander Spain e-mail: [email protected]

DESIGNING HIGHLY-LINEAR MICROWAVE POWER AMPLIFIERS BASED ON LARGE-SIGNAL IMD SWEET-SPOTS José C. Pedro1, Nuno B. Carvalho1 and José A. Garcia2 1 – Institute of Telecommunications – University of Aveiro – Aveiro, Portugal 2 – Communications Department – University of Cantabria – Santander, Spain

EXTENDED ABSTRACT This paper shows how intermodulation distortion, IMD, sweet spots can be used as a practically meaningful technology to design highly linear microwave and wireless amplifiers. It starts by carefully identifying the physical origins of these points of large signal to nonlinear distortion ratio, distinguishing the ones uniquely determined by the active device’s quiescent point – small-signal IMD sweet spots – from the ones that are dependent on the excitation level – large-signal IMD sweet spots. Since small-signal in-band IMD level is determined by the 3rd order system model parameters, bias points where these parameters are reduced, or even nullified, can be used to design high-dynamic range small-signal circuits as radio receivers’ low noise amplifiers. Moreover, despite the device may not be biased for a small-signal IMD sweet-spot, and so present a non-optimized lowlevel distortion, it may even show very good IMD figures at particular Pin levels located in the higher zones of operation range. These are the large-signal IMD sweet-spots. The way such a curious behavior can be explained requires that the 3rd order model above referred is replaced by another of higher order. That is due to the strong nonlinearities now involved, whose contributions interact with the small-signal 3rd order ones to create these IMD deeps at particular Pin levels Unfortunately, since the use of these IMD sweet-spots as a practical PA design tool requires a precise understanding of the mechanisms that control them, this work stands on a feedback nonlinear model to then analyze their sensitivity to the active device type, the quiescent point, the impedance terminations, the thermal and low-frequency dispersive effects or even the statistical characteristics of the input stimuli. Consequently, design guidelines are given to direct the selection of the PA circuit parameters.

Conference Topics: 1 – Solid-State Devices and Circuits 10 – Computer Aided Design

DESIGNING HIGHLY-LINEAR MICROWAVE POWER AMPLIFIERS BASED ON LARGE-SIGNAL IMD SWEET-SPOTS José C. Pedro1, Nuno B. Carvalho1 and José A. Garcia2 1 – Institute of Telecommunications – University of Aveiro – Aveiro, Portugal 2 – Communications Department – University of Cantabria – Santander, Spain

Abstract—This paper shows how intermodulation distortion, IMD, sweet spots can be used as a practically meaningful technology to design highly linear microwave power amplifiers. It starts by reviewing the physical origins of these points of large signal to nonlinear distortion ratio, distinguishing the ones uniquely determined by the active device’s quiescent point – small-signal IMD sweet spots – from the ones that are dependent on the excitation level – large-signal IMD sweet spots. Then, it evaluates the main amplifier circuit parameters that are known to control these sweet spots, giving this way design guidelines for low level high dynamic range/class A amplifiers or for highly efficient and linear power amplifiers.

I. INTRODUCTION

II. WHAT ARE IMD SWEET-SPOTS

Modern digital transmission systems are continuously demanding for increasingly better microwave and RF circuits. These require very disparate performance figures such as output power (Pout), power added efficiency (PAE) and intermodulation distortion (IMD). Moreover, it is widely known that some of these constitute conflicting requirements whose simultaneous optimization asks for detailed modeling activities and accurate nonlinear design procedures. The object addressed in this paper is the simultaneous fulfillment of PAE and IMD characteristics of microwave power amplifiers (PAs) subject to real telecommunications signals. Although this is in general not possible, there are some particular PA features that provide a means to escape from this apparent dead end. One which has recently been receiving a great deal of attention is the so-called IMD sweet-spots [1]. These are peculiar points of the IMD versus input power (Pin) characteristic where only a few dBs of output-power back-off (and thus a few percent of efficiency degradation) can lead to astonishing high levels of IMD reduction. Unfortunately, the precise understanding of these IMD sweet-spots is still blurred by their nonlinear nature, which has prevented their wide spread through the PA industry. In fact, although their sensitivity to the active device type [2], the quiescent point [1], the impedance terminations [3], the thermal and lowfrequency dispersive effects [4] or even the input signal statistics [5] has been many times reported, its understanding is still so unclear that sometimes they are simply avoided [6]. The objective of this paper is to clarify some of these issues and thus contribute to the dissemination of this promising highly-linear microwave PA design technology.

IMD sweet-spots are points of very good signal to IMD ratio that can be directly related to the active device nonlinear characteristics and to its interactions with the surrounding linear networks. Although various sweets-pots could be defined for each of the distortion products, the band-pass behavior of radio systems determines that the sweet-spots that refer to odd-order distortion dominate this field. So, from now on, these will be the ones of our concern. Briefly, these have been divided into the so-called small-signal and large-signal IMD sweet-spots, depending on the input signal levels at which they operate. Small-signal IMD sweet-spots are special IMD sweet-spots that correspond to points of bias where the IMD observed at small-signal levels is very low [7]. Since small-signal distortion can be modeled by a simple 3rd degree Taylor or Volterra series model [8], which has only one contributing term - Gm3 or H3(ω1,ω2,ω3) - small-signal IMD sweet-spots are the points of bias where Gm3=0 or H3(ω1,ω2,ω3)=0. So, these IMD sweet-spots are independent of input drive level (provided this is kept in the quasi-linear or small-signal region) and constitute important design tools for high dynamic range low-level amplifiers as the front-end or IF amplifiers of radio receivers. As illustrated in Fig. 1, despite the device may not be biased for a small-signal IMD sweet-spot, and so present a non-optimized low-level distortion, it may even show very good IMD results at particular Pin levels located in the higher zones of operation range. These are the large-signal IMD sweet-spots. The way such a curious behavior can be explained requires that the 3rd order model above referred is replaced by another of higher order. That is due to the strong nonlinearities now involved, whose contributions interact with the small-signal 3rd order ones to create these IMD deeps at particular Pin levels [1].

system in which a memoryless nonlinearity, iDS(vGS,vDS,T), depends on an input variable, vGS(t), and two more variables, vDS(t) and T(t), which, themselves, vary in a linear dynamic way with iDS(t). Obviously, this constitutes a nonlinear feedback model in which a memoryless nonlinearity, e.g. here represented by a polynomial:

Two-tone excitation -10

-20

-30

Pout [dBm]

-40

-50

-60

-70

VGS = -0.35V VGS = -0.41V VGS = -0.53V

-80

-90 -15

-10

-5

0

N

yO (t ) = ∑ Gn x(t ) n

5

n =1

Pin [dBm]

Fig. 1. Different IMD versus Pin patterns showing small-signal and large-signal IMD sweet-spots for a HEMT device.

III. AN IMD NONLINEAR FEEDBACK MODEL In order to derive a set of results on the way these IMD sweet-spots are generated, and thus to what type of components or parameters they are sensitive, let us assume the simplified equivalent circuit electrothermal model of a general microwave PA shown in Fig. 2. VGS

VDD

ZL(ω)

R0 vS(t)

Linear Dynamic Matching and Bias Network Mi(ω)

vGS(t)

iDS Thermal

(2)

vDS(t)

Linear Dynamic Matching and Bias Network Mo(ω)

H1 (ω ) = TM i (ω )

R0 vO(t)

Using such a simplified model it is assumed that the PA is composed of an active device (in the present case a FET) and two matching networks Mi(ω) and Mo(ω) represented by there frequency-domian transfer functions, TMi(ω) and TMo(ω). Furthermore, it is also assumed that the major source of nonlinearity is the drain-source current which is simultaneously dependent on two voltages, vGS and vDS, besides the channel temperature, T. (In fact, T represents only the variation of the device temperature around an average – or quiescent – constant temperature). Although the quasi-static approximation implies that iDS(vGS,vDS,T) is a memoryless nonlinearity, the dependence of vDS(t) on iDS(t) via the dynamic load impedance ZL(ω) determines that the overall iDS(t) will be a nonlinear and dynamic function of the input control voltage. Thus, it can be expressed as: (1)

where the output voltage can be given by Vds(ω) = ZL(ω)Ids(ω). As usual, the device temperature is a function of the dissipated power p(t), and thus of iDS(t), and of ZTH(ω), the thermal impedance imposed by the particular device’s thermal structure: T(ω) = ZTH(ω)P(ω). A curious note on this model is that it describes a

G1 TM o (ω ) D (ω )

(3)

where TMi(ω) and TMo(ω) are the transfer functions of the input and output matching networks, respectively, and D(ω)=1-a1F(ω); a 3rd order transfer function for the 3rd order in-band distortion: H 3 (ω1 , ω 2 , ω3 ) =

Fig. 2. Nonlinear dynamic feedback model of a microwave power amplifier.

Ids(ω) = fNL[Vgs(ω), Vds(ω), T(ω)]

is subject to a linear dynamic feedback path, F(ω), such that: XF(ω) = F(ω).Yo(ω), where XI(ω) = XS(ω) + XF(ω), and xS(t) is the PA input excitation [9]. Using this feedback model, the PA in-band characteristics can be expressed by a 1st order frequency-domain transfer function for the signal:

TM i (ω1 ) TM i (ω 2 ) TM i (ω3 ) TM o (ω1 + ω 2 + ω3 ) D (ω1 ) D (ω 2 ) D (ω3 ) D (ω1 + ω2 + ω3 )

⎧⎪ 2 2 ⎡ F (ω1 + ω2 ) F (ω1 + ω3 ) F (ω2 + ω3 ) ⎤ ⎫⎪ + + ⎨G3 + G2 ⎢ ⎥⎬ 3 ⎪⎩ ⎣ D (ω1 + ω2 ) D (ω1 + ω3 ) D (ω2 + ω3 ) ⎦ ⎪⎭

(4)

and so on [9]. Using this model, it comes out that a small-signal IMD sweet-spot is the active device quiescent point that, by controlling G2 and G3 makes H3(ω1,ω2,ω3) null, while a large-signal IMD sweet-spot is the excitation level (for a certain quiescent point) for which the addition of all odd-order IMD contributions make a certain distortion component (typically the adjacent channel distortion, IM3) null. Moreover, the model also tells us that depending on the relative value of the odd and even order terms of (2), and the magnitude of the feedback path F(ω), these sweet-spots can be either determined by only the active device or by it and the external circuitry. IV. CONTROLLING LARGE-SIGNAL IMD SWEETSPOTS A. Sweet-Spot Dependence on Excitation Type As small-signal IMD sweet-spots are identified as a null of H3(ω1,ω2,ω3), they are completely determined by the active device I/V (or also Q/V) characteristics, quiescent point and external circuitry. Hence, they are verified for any signal type. On the contrary, as large-signal IMD sweet-spots are consequence of low and high order IMD interactions, they are only seen at very specific excitation levels.

N

(5)

yO ( x ) = ∑ Gn xI n n =1

Thus the coefficients of (2) are constants imposed by the derivatives of nonlinear function and the expansion point, x0: Gn ( x) =

∂ n yO ( x) ∂ xn

(6) x = x0

Instead, for large-signal levels, where a Taylor expansion presents an unacceptable error or simply does not converge, the best polynomial approximation is given as an orthogonal expansion: N

yO ( x) = ∑ Gnψ n ( x)

search for a reduction in the conduction angle, 2θ, the large-signal sweet-spot would appear for a higher input power level. In Fig. 3, the evolution of a HEMT sweet-spot is presented in terms of VGS, VDS and Pin, under a twotone stimulus. For small excitation levels, the minimum IMD appears at a fixed voltage, corresponding to the H3(ω1,ω2,ω3) null. However, for large input values, the position of this optimum point is significantly dependent on the power level as previously described. The influence of the drain-tosource voltage in the small-signal sweet-spot position shift may be related to the widely known pinch-off modulation produced by VDS in this kind of technology. VDS = 2V

∞

Pin [dBm]

n =1

Gn = ∫ψ n ( x ) yO ( x) pdf x ( x)dx

10

Two-tone IS95 QPSK

(7)

whose base functions ψn(x) are n’th degree polynomials and their coefficients depend on the probability density function of the input stimulus, pdf [10]:

VDS = 4V

10

Two-tone IS95 QPSK

5

5

0

0

-5

-5

Pin [dBm]

In mathematical terms this means that, for very small-signal levels, the best polynomial approximation of the device’s nonlinearity is the Taylor series:

-10

-10

-15

-15

-20

-20

(8)

−∞

The physical significance of this is that a PA designed to present a large-signal IMD sweet-spot when excited e.g. with a two-tone signal, whose signal envelope is a sine, may not show that sweet-spot under a CDMA or a Gaussian stimulus. Moreover, because these more realistic signals present a very complex statistical amplitude distribution, it may even happen that the sharp IMD nulls typical of two-tone IMD tests, are substituted by much smoother IMD valleys. In conclusion, the use of large-signal IMD sweetspots in highly linear PA design requires that even if the circuit may be studied under simplified CW or two-tone excitations during the first steps of the design phase, it must be afterwards optimized under the true operating signal. B. Sweet-Spot Dependence on Active Device Quiescent Point Being associate to interactions between the 3rd order small-signal characteristic and the strong nonlinearities, large-signal sweet-spots are dependent on the device quiescent point. In fact, their existence is mainly determined by the selection of a quiescent point in class C or AB operating regions [2]. For a class C operating point, the IMD contribution of its positive G3 is cancelled when the input voltage excursion reaches the saturation-to-linear zone knee (active-to-saturation knee for bipolars). In this way, it could be expected that lowering the gate voltage in the

-25 -0.8

-0.7

-0.6

-0.5

VGS [V]

-0.4

-0.3

-25 -0.8

-0.7

-0.6

-0.5

-0.4

-0.3

VGS [V]

Fig. 3 Sweet-spot evolution with VGS and Pin for two VDS values.

Properly controlling the quiescent point, a largesignal sweet-spot could be produced at the desired excitation value, at least for these input formats. Solutions for automatically adjusting the quiescent point with the RMS value of the excitation envelope, based on this behavior, could result in a Pin-Pout characteristic with a wide or multiple optimum points. Although only small differences seem to exist for the signals employed in this example, it should be taken into account that the linearity improvement obtained with the IS95 signal is much smoother. C. Sweet-Spot Dependence on Impedance Terminations As seen from the above feedback system model, the dependence of iDS on vDS should have its own impact on the IMD sweet spot, via the output matching network. So, to study the effects of impedance terminations on IMD characteristics, consider the ideal resistive active device of Fig. 2 whose typical output I/V curves are depicted in Fig. 4. Large-signal IMD sweet spots have been attributed to interactions between iDS(vGS) mild nonlinearities and iDS(vGS,vDS) strong nonlinearities, arising when output signal excursion enters the triode region (the saturation region in case of bipolar devices) [1]. Accordingly, as load-line slope determines the signal level for which the saturation-to-triode zone knee is

reached (see Fig. 4), it should be expected that the Pin for which the IMD sweet spot is observed will be strongly dependent on load termination RL. This is illustrated in Fig. 4a) and b), where a shift of the simulated IMD sweet spot position is evident when RL is varied. So, the first conclusion we may draw for this simplified resistive case is that, to every intrinsic resistive load corresponds an IMD sweet spot, although for a different Pin. 20 0 -20

RL

-40

PIMD (dBm)

iDS

IMD sweet-spots [4]. Although their detailed analysis stands beyond the scope of this text, Eq. (4) clearly indicates that any undesired feedback path (in the present case via the load termination or self-heating) provided at the second harmonic, F(ωi+ωj), or the base-band, F(ωi-ωj), components, can jeopardize the sought IMD cancellation. Depending on being inphase or out-of-phase with the primary IMD components, they are known to possibly shift the sweet spot position, reduce its signal to IMD ratio and create PA long-term memory effects or even IMD sideband asymmetries [4].

IMax

-60

CONCLUSIONS

-80 -100

RL

-120 -140 -160 -180

Q VK

-40

VDC

-30

-20

vDS

VBR

-10

0

10

20

Pin (dBm)

a) b) Fig. 4 Impact of resistive PA load resistance, RL, on large-signal IMD sweet spots.

If now parasitic effects were added to our device model, intrinsic load impedance would, in general, present a reactive part. vds(t) would no longer be in phase with ids(t) and the previous small-signal largesignal IMD interaction would cease to produce the previous perfect cancellation. So, optimizing signal to IMD ratio using a large-signal IMD sweet spot requires that the optimum extrinsic load termination should be the one that, after de-embedding the parasitics, reestablishes the pure intrinsic resistive load. As seen from the simulation results shown in Fig. 5, this corresponds to a slight rotation in the load impedance smith chart x-axis [3]. j1

j0.5

j2 30

35

j0.2

35

40

50 1 45 45 50

50

50

0.5

45

0.2 40

45

0

50 55

45

45

35 45 4 0

50

45

2 40 50 45 40

30

35 30

40

-j0.2

35

40

-j0.5

-j2

-j1

Fig. 5 Load pull simulation of signal to IMD values at the IMD sweet spot.

In summary, a second possible conclusion is that maximum IMD sweet spot effects will be obtained for output impedance values that are seen by the transistor’s nonlinear iDS(vGS,vDS) source as pure resistances. That is illustrated by Fig. 5, where the inband impedance presented by the Linear Dynamic Matching and Bias Network of Fig. 2 was changed. Further conclusions could also be drawn for the impact of out-of-band terminations on the large-signal

Despite the complexity of the analysis required, and the intricate dependence on various circuit parameters showed by IMD sweet-spots, recent advances on nonlinear distortion modeling are turning them into a very promising highly linear PA design tool. ACKNOWLEDGMENTS The authors would like to thank the EC for the Network of Excellence TARGET, Portuguese FCT for MEGaN Project and Acção Integrada Luso-Espanhola E-72/03, under which this work was partially done. REFERENCES [1]

N. Carvalho and J. Pedro, “Large and Small Signal IMD Behavior of Microwave Power Amplifiers”, IEEE Trans. on Microwave Theory and Tech., vol. MTT-47, , pp. 2364-2374, Dec. 1999. [2] P. Cabral, N. Carvalho and J. Pedro, “An Integrated View of Nonlinear Distortion Phenomena in Various Power Amplifier Technologies”, Focused Session: High-Linearity Power Amplifiers, European Gallium Arsenide and Other Compound Semiconductors Application Symposium, Munich, Oct. 2003. [3] J. Pedro and N. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits, Artech House, 2003. [4] N. Carvalho and J. Pedro, "A Comprehensive Explanation of Distortion Sideband Asymmetries", IEEE Trans. on Microwave Theory and Tech., vol. MTT-50, pp. 2090-2101, Sep. 2002. [5] M. Burgos-Garcia, F. Perez-Martinez, “Simple Procedure for Optimum Linearisation of Amplifiers in Multicarrier Applications”, IEE Electronics Lett., vol. 30, pp.114-115, Jan. 1994. [6] F. Palomba, M. Pagani, I. De Francesco, A. Meazza, A. Mornata, G. Procopio and G. Sivverini, “Process-Tolerant High Linearity MMIC Power Amplifiers”, in Proc. Gallium Arsenide Applications Symposium Proc., Munich, pp. 73-76, Oct. 2003 [7] J. Pedro, “Evaluation of MESFET Nonlinear Intermodulation Distortion Reduction By Channel Doping Control”, IEEE Trans. on Microwave Theory and Tech., vol. MTT-45, pp.1989-1997, Nov. 1997. [8] J. Pedro and J. Perez, “Accurate Simulation of GaAs MESFET's Intermodulation Distortion Using a New DrainSource Current Model”, IEEE Trans. on Microwave Theory and Tech., vol. MTT-42, pp.25-33, Jan. 1994 [9] J. Pedro, N. Carvalho and P. Lavrador, “Modeling Nonlinear Behavior of Band-Pass Memoryless and Dynamic Systems”, 2003 IEEE Intern. Microwave Symp. Dig., pp.2133-2136, Philadelphia, Jun. 2003. [10] M. Schetzen, “Nonlinear System Modeling Based on the Wiener Theory”, Proc. of the IEEE, vol. 69, pp.1557-1573, Dec. 1981.