Asimple analytic method for transistor
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1 A Simple Analyti Method for ransistor Osillator Design his straightforward mathematial tehnique helps optimize osillator designs By Andrei Grennikov Institute of Miroeletronis, Singapore Asimple analyti method for transistor osillator design has en developed. his tehnique defines expliit expressions for optimum values of feedbak elements and load through bipolar transistor z-parameters. Suh an approah is useful for pratial optimization of a series feedbak mirowave bipolar osillator. Mirowave osillator design in general represents a omplex problem. Depending on the tehnial requirements for designing an osillator, it is neessary to define the onfiguration of the osillation sheme and a transistor type, to measure the small-signal and large-signal parameters of a transistor-equivalent iruit and to alulate eletrial and spetral harateristis of the osillator. his approah is very suitable for implementing CAD tools if a transistor used in mirowave osillator iruits is represented by a two-port network. here are two ways to evaluate the basi parameters of the transistor equivalent iruit; one is by diret measurement and the other is by approximating based on experimental data with reasonable auray in a wide frequeny range [1-3]. Furthermore, the equivalent iruit model an easily integrated into a F iruit simulator. In large-signal operation, it is neessary to define the appropriate parameters of the ative two-port network and the parameters of external feedbak elements of the osillator iruit. herefore, it is desirable to have an analyti method to design a single-frequeny optimal mirowave osillator. his helps to formulate the expliit expressions for feedbak elements, load impedane and maximum put power in terms of transistor-equivalent iruit elements and their urrent-voltage harateristis [4]. Suh an approah an derived based on a Figure 1. he series feedbak bipolar osillator equivalent iruit. two-step proedure. First, the optimal ombination of feedbak elements for realizing a maximum small-signal negative resistane to permit osillations at the largest amplitude is defined. Seond, for a given osillator iruit onfiguration with maximal put power, by taking into aount the large-signal nonlinearity of the transistor equivalent iruit elements, the realized small-signal negative resistane will haraterized to determine the optimum load. eent progress in silion bipolar transistors has signifiantly improved frequeny and power harateristis. In ontrast to the field-effet transistors (FEs), the advantages of redued low-frequeny noise and higher transondutane make bipolar transistors more appealing for osillator design up to GHz. A simple analyti approah used to design a mirowave bipolar osillator with optimized feedbak and load will speed up the alulations of the values of feedbak elements and simplify the design proedure. 36 APPLIED MICOWAVE & WIELESS
2 General approah Generally, in a steady-state large-signal operation, the design of the mirowave bipolar osillator is ahieved by defining the optimum bias onditions and the values of feedbak elements as well as the load that orresponds to the maximum power at a given frequeny. Now, lets look into the generalized two-port iruit of the transistor osillators as shown in Figure 1, where Z i i + j i, i 1,, Z L L + j L. Suh an equivalent osillation iruit is used mainly by mirowave and radio frequeny osillator design. he dotted-lined box (as shown in Figure 1) represents the small-signal SPICE Ers- Moll model of the bipolar transistor in the normal region of operation. his hybrid-p model an aurately simulate both DC and high-frequeny havior up to the transition frequeny f g m /πc e [5]. For generi mirowave bipolar osillator design, the osillation will arise under apaitive reatane in an emitter iruit ( <), indutive reatane in a base iruit ( 1 >), and either indutive ( L >) or apaitive ( L <) reatanes in a olletor iruit. For a single frequeny of osillation, the steady-state osillation ondition an expressed as Z ( I,)+ Z ( ) L where Z L () L ()+j L () and Z (I, ) (I, )+ j (I, ). he expression of put impedane, Z, an written as ( + ) Z Z Z Z Z Z + Z + Z + Z + Z (1) () By substituting 1 and into equation (), the optimal real and imaginary parts of the put impedane Z an defined as follows: ( 1 1) + + Small-signal osillator iruit design At radio and mirowave frequenies, the ondition r >>1/ C e is usually fulfilled. Besides, it is possible to ignore the effet of base-width modulation (the so-alled Early effet) with a signifiant derease of the final result auray, and to onsider the resistane r o as an infinite value. he parasiti lead indutanes and substrate apaitane an taken into aount in the external feedbak iruit. By doing so, the internal bipolar transistor in ommon-emitter small-signal operation an haraterized by the following real and imaginary parts of z-parameters: a + r b gm + ( 1 1) ( o ) (4) (5) where Z ij (i, j 1, ) are z-parameters of the hybrid transistor model. o optimize the osillator iruit, the negative real part of the put impedane Z has to maximized. Based on expression (), it is possible to find optimal values for 1 and under whih the negative value is maximized by setting a C a rb g m (6), 1 (3) a + C 1 11 he optimal values 1 and based on ondition (3) an expressed with the impedane parameters of the ative two-port network in the following manner [4]: a 1 1+ where / πf By substituting the expressions for real and imaginary parts of the transistor z-parameters from the system of equations (6) to equations (4) and (5), the optimal 38 APPLIED MICOWAVE & WIELESS
3 (a) Figure. Amplitude dependenies of large-signal transondutane, onstant bias olletor urrent and fundamental amplitude. values of imaginary parts of the feedbak elements 1 and an rewritten as follows: 1 1 rb C 1 re C In addition, the real and imaginary parts of optimum put impedane Z + j (7) (b) Figure 3. Amplitude dependenies of (a) optimum iuit parameters and (b) the real and imaginary parts of put resistane and put power of the bipolar osillator. an expressed as: From equation (8), it follows that as frequeny inreases, the absolute value of the negative resistane will redued and the maximum osillation frequeny f max omes zero. With onsidering the parasiti series resistors r e and r, the expression for f max is Large-signal osillator iruit design Generally, at least three steps are required for designing a large-signal osillator iruit. First, it is neessary to hoose an appropriate iruit topology, seond, to determine the devie large-signal harateristis and, third, to optimize the osillator iruit to ahieve the desired performanes. o analyze and design the fundamental negative resistane bipolar osillator, it is advisable to apply a quasi-linear method, based on the use of the ratios tween the fundamental harmonis of urrents and voltages as well as the representation of nonlinear elements by equivalent averaged fundamental linear ones. he derivation of equivalent linear elements in terms of signal voltages is based on stati and voltageapaitane bipolar transistor harateristis. In a omf rb r r r r e ac max b f 8πrC b e 11 a 1 r + r + C b e 11 e 1 ( r ) C e (8) (9) Equation (9) mathes with the well-known expression for f max of the bipolar transistor, on whih a maximum power amplifiation fator is unity, and the steady-state osillation ondition is arried solely for the lossless osillation system [6]. 4 APPLIED MICOWAVE & WIELESS
4 mon ase, all elements of the transistor equivalent iruit are nonlinear and depend signifiantly on operation mode, espeially for transondutane g m and baseemitter apaitane C e. However, in pratie, it is enough to limited to the nonlinear elements g m, Τ and C, as the base resistane r b poorly depends on a bias ondition. Besides, to alulate the onstant bias and fundamental olletor urrents, it is suffiient to use a linear approximation of transition frequeny Τ and the small-signal value C at operating point [4]. Under the assumption of a suffiiently low olletor-emitter bias value, the applied bipolar transistor model does not represent the effet of the forward-retified urrent aross the olletor-base juntion and the olletor voltagebreakdown phenomenon. In this ase, it is assumed that the transition from a soft-exited osillation mode to a steady-state stationary mode is aused mainly by the nonlinear harateristi of the transondutane g m in ontrast to the FE osillator. Here, under large-signal operation both the osillator impedane and power predition are primary due to a hange in differential drainsoure resistane [4]. When the base-emitter urrent is suffiiently small, the following exponential model an applied to approximate the family of the experimental transfer I-V harateristis: (1) where V is the base-emitter juntion voltage, I s is the reverse olletor saturation urrent, and V is the temperature voltage. By restriting to the fundamental frequeny, V an expressed as V E b E e I o e + v os t, where I o is a onstant bias olletor urrent, E b and E e are the base and emitter bias voltages, and e is a selfbias resistor. aking into aount that the voltage drops on olletor and emitter transistor equivalent iruit resistors r and r e are negligible, the values of the large-signal transondutane g m1 and the onstant bias olletor urrent I o as the funtions of the juntion fundamental voltage amplitude v an then defined as follows: g 1 v Io Is I [ ] I I exp ( V / V ) 1 s E E I exp v v I s b e o e m1 1 E E I exp b e o e (11a) (11b) where I (v /V ), I 1 (v /V ) are the modified first order Bessel funtions [7]. For the steady-state stationary osillation mode, it is 1 neessary to define the analyti relations tween the load urrent amplitude and fundamental amplitude of olletor voltage with input voltage amplitude v. hese relations are expressed in terms of the transistor z-parameters and osillator iruit parameters in the following manner: I V l e Z11 + Z + Z1 Z Z Z Z + Z V Z ( Z + Z + Z ) Z Z + Z Z ( Z + Z ) Z Z (1) In a steady-state operation, the put power of the osillator is P I e Z / l L By replaing load resistane Z L from equation (1) and using the parameters of transistor equivalent iruit from (6) to (8), P an expressed as ml b e P ag r + r + 11 r + r b v (13) esults An analyti approah was applied to the mirowave bipolar osillator design. he bipolar transistor has the following parameters of its equivalent iruit: f 6 GHz, C.5 pf, g m 1.6 A/V, r b 4 ohms, r e.3 ohm, r 1.75 ohms, L e L.5 nh, L b.3 nh. he numerial alulation was performed using the values of the osillation frequeny f 4 GHz, biases E b V and E e V, self-bias resistor e 1 ohms and reverse saturation olletor urrent I s 1 µa. he numerial results obtained are shown in Figures and 3. Figure shows the amplitude dependene of large-signal transondutane g m1, onstant bias olletor urrent I o, and fundamental olletor amplitude v e. Figure 3 shows the amplitude dependene of (a) optimum iruit parameters, 1 and (b) the real and imaginary parts of the put resistane, and put power P. Aording to Kurokawa [8], as the negative resistane redues with the inrease of the base-emitter juntion and olletor voltage amplitudes, the stable osillations are established in the osillator. In that ase, the value of the large-signal transondutane g m1 is signifiantly redued as shown in Figure. In order to realize the maximal or required put power in a linear operation region with the saturation effet, it is neessary to hoose the appropriate load value or to use the diode in parallel to load for restrition of olletor voltage amplitude. he applied model of the mirowave bipolar APPLIED MICOWAVE & WIELESS
5 transistor does not inlude a olletor-base diode, as suh an operation region is not reommended for the osillator due to the signifiant deterioration of its noise properties. By means of the ondition v e E, the appropriate amplitude restrition at the numerial alulation an onsidered. From numerial alulations, it follows that v e and I o are inreased linearly with the inrease of base-emitter juntion amplitude v, starting from some signifiantly small values as shown in Figure. herefore, analytially it is interesting to onsider the influene of the emitter self-bias resistor e on the harater of amplitude dependene of the fundamental put urrent I 1. he fundamental put urrent an expressed as I l (14) By differentiating, equation (11b) an rewritten based on d I (v /V )/d(v /V ) I 1 (v /V ) in the form of di dv Eb Ee Eoe v s I 1 exp 1 o Io + Is V + I + I o s e (15) From the definition of modified first kind Bessel funtions, it follows that for values (v /V ) 5 the following inequality is to defined: 9. I ( v / V )/ I ( v / V ) 1 1 ( ) I1 v / V I v / V Hene, as it follows from (15), the value of onstant bias olletor urrent I o varies pratially linearly with the inrease v under ondition v 5V, and the slope of the dependene I o (v ) is defined by value e. By substituting expression (11b) into equation (14), it allows us to rewrite the expression for fundamental put urrent I 1 as ( ) I1 v / V Il( v) [ Io( v)+ Is] I v / V (16) From equations (15) and (16) it follows that for a bipolar osillator with an emitter self-bias resistor, the olletor onstant bias urrent I o and the fundamental put urrent I 1 depend linearly on the base-emitter juntion voltage amplitude v under the ondition v /V 5. hus, the linearizing influene of the emitter self-bias resistor for the transistor osillator is similar to the influene of the negative feedbak resistor on a linearization of the amplifier gain-transfer harateristis. From Figure 3b, it follows that it is possible to realize a suffiiently high level of put power under the diret onnetion of standard load L 5 ohms with use of a speial load mathing iruit. Conlusion A simple analyti method of mirowave and F bipolar osillator design has en developed, allowing the definition of expliit expressions for optimum values of feedbak elements and load through bipolar transistor z- parameters. A negative resistane onept is used to design series feedbak mirowave and F bipolar osillators with optimized feedbak elements and maximum put power in terms of transistor impedane parameters. Based on its small-signal z-parameters and DC harateristis, a simplified large-signal model for a bipolar transistor is derived. he linearizing effet of the external emitter self-bias resistor is analytially shown. eferenes 1. C. Chang, M. Steer, S. Martin and E. eese, Computer-Aided Analysis of Free-unning Mirowave Osillators, IEEE rans. Mirowave heory ehniques, J. rew, MESFE Models for Mirowave Computer-Aided Design, Mirowave Journal, May D. Warren, J. Golio and W. Seely, Large and Small Signal Osillator Analysis, Mirowave Journal, May A. Grennikov and V. Nikiforov, An Analyti Method of Mirowave ransistor Osillator design, Int. J. Eletron., P. Antognetti and G Massobrio, Semiondutor Devie Modeling with SPICE. MGraw-Hill, New York, Carson, High-frequeny Amplifiers, John Wiley & Sons, New York, A. Angot, Complements De Mathematiques, Paris, K. Kurokawa, Some basi harateristis of broadband negative resistane osillator iruits, Bell Syst.eh. J., Author Information Andrei Grennikov reeived a Dipl-Ing. degree from the Mosow Institute of Physis and ehnology and a Ph.D degree from Mosow ehnial University of Communiation and Informatis in 198 and 1991, repetively. In 1983, he gan working as a researh assistant at the Sientifi esearh Department of Mosow ehnial University of Communiation and Informatis. Sine 1998, he has worked with the Institute of Miroeletronis, Singapore. He may reahed via telephone at , via fax at , and by at andrei@ime.org.sg. 44 APPLIED MICOWAVE & WIELESS
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