A Simple Analytic Method for Transistor Oscillator Design

1 A Simple Analytic Method forTransistor Oscillator DesignThis straightforward mathematical technique helps optimize Oscillator designsBy Andrei GrebennikovInstitute of Microelectronics, SingaporeAsimple Analytic Method for transistoroscillator Design has been developed. Thistechnique defines explicit expressions foroptimum values of feedback elements and loadthrough bipolar Transistor z-parameters. Suchan approach is useful for practical optimizationof a series feedback microwave bipolar Oscillator Design in general repre-sents a complex problem. Depending on thetechnical requirements for designing an oscilla-tor, it is necessary to define the configuration ofthe oscillation scheme and a Transistor type, tomeasure the small-signal and large-signal para-meters of a Transistor -equivalent circuit and tocalculate electrical and spectral characteristicsof the Oscillator . This approach is very suitablefor implementing CAD tools if a Transistor usedin microwave Oscillator circuits is representedby a two-port network.

2 There are two ways toevaluate the basic parameters of the transistorequivalent circuit; one is by direct measurementand the other is by approximating based onexperimental data with reasonable accuracy in awide frequency range [1-3]. Furthermore, theequivalent circuit model can easily be integrat-ed into a RF circuit large-signal operation, it is necessary todefine the appropriate parameters of the activetwo-port network and the parameters of exter-nal feedback elements of the Oscillator , it is desirable to have an analyticmethod to Design a single-frequency optimalmicrowave Oscillator . This helps to formulatethe explicit expressions for feedback elements,load impedance and maximum output power interms of Transistor -equivalent circuit elementsand their current-voltage characteristics [4].Such an approach can be derived based on atwo-step procedure. First, the optimal combina-tion of feedback elements for realizing a maxi-mum small-signal negative resistance to permitoscillations at the largest amplitude is , for a given Oscillator circuit configura-tion with maximal output power, by taking intoaccount the large-signal nonlinearity of thetransistor equivalent circuit elements, the real-ized small-signal negative resistance will becharacterized to determine the optimum progress in silicon bipolar transistorshas significantly improved frequency and powercharacteristics.

3 In contrast to the field-effecttransistors (FETs), the advantages of reducedlow-frequency noise and higher transconduc-tance make bipolar transistors more appealingfor Oscillator Design up to 20 GHz. A Simple ana-lytic approach used to Design a microwave bipolaroscillator with optimized feedback and load willspeed up the calculations of the values of feed-back elements and simplify the Design APPLIEDMICROWAVE& WIRELESS Figure 1. The series feedback bipolar oscillatorequivalent APPLIEDMICROWAVE& WIRELESSG eneral approachGenerally, in a steady-state large-signal operation, thedesign of the microwave bipolar Oscillator is achieved bydefining the optimum bias conditions and the values offeedback elements as well as the load that correspondsto the maximum power at a given frequency. Now, letslook into the generalized two-port circuit of the transis-tor oscillators as shown in Figure 1, where Zi= Ri+ jXi,i = 1, 2, ZL= RL+ jXL.

4 Such an equivalent oscillationcircuit is used mainly by microwave and radio frequencyoscillator Design . The dotted-lined box (as shown inFigure 1) represents the small-signal SPICE2 Ebers-Moll model of the bipolar Transistor in the normal regionof operation. This hybrid-p model can accurately simu-late both DC and high-frequency behavior up to thetransition frequencyfT= gm/2 Ce[5]. For genericmicrowave bipolar Oscillator Design , the oscillation willarise under capacitive reactance in an emitter circuit (X2<0), inductive reactance in a base circuit (X1>0), andeither inductive (XL>0) or capacitive (XL<0) reactancesin a collector a single frequency of oscillation, the steady-stateoscillation condition can be expressed as(1)where ZL( )=RL( )+jXL( ) and Zout(I, )= Rout(I, )+jXout(I, ).The expression of output impedance, Zout,can bewritten as(2)where Zij(i,j= 1, 2) are z-parameters of the hybridtransistor optimize the Oscillator circuit, the negative realpart of the output impedanceZouthas to be on expression (2), it is possible to find optimal val-ues for X1and X2under which the negative value Routismaximized by setting(3)The optimal valuesX10and X20based on condition (3)can be expressed with the impedance parameters of theactive two-port network in the following manner [4]:(4)By substitutingX10and X20into equation (2), the opti-mal real and imaginary parts of the output impedanceZoutcan be defined as follows:(5)Small-signal Oscillator circuit designAt radio and microwave frequencies, the condition rbe>>1/ Ceis usually fulfilled.

5 Besides, it is possible toignore the effect of base-width modulation (the so-calledEarly effect) without a significant decrease of the finalresult accuracy, and to consider the resistance roas aninfinite value. The parasitic lead inductances and sub-strate capacitance can be taken into account in theexternal feedback circuit. By doing so, the internal bipo-lar Transistor in common-emitter small-signal operationcan be characterized by the following real and imaginaryparts ofz-parameters:(6)By substituting the expressions for real and imagi-nary parts of the transistorz-parameters from the sys-tem of equations (6) to equations (4) and (5), the optimalRR agrRRRaCXX agrXXaCambTTcTmbcTTT11122212211111221112 21111== + ==+== =+ =+ where /==2 fTRRRRR RX XRRRXXXRRXXRRR outoutouto022221221221122112102022211221 1222224=+ ++()+ ()++()=+ ()XRR RXXXX202122121121221222= ++() () +XRRXXRRRR XXX1021122112122111111122122= + ++ RXRX outout1200==,ZZZZZZZZZZout=+ +()+()++2221222121121ZI ZoutL, ()+()=040 APPLIEDMICROWAVE& WIRELESS values of imaginary parts of the feedback elements X10and X20can be rewritten as follows:(7)In addition, the real and imaginary parts of optimumoutput impedance can be expressed as.

6 (8)From equation (8), it follows that as frequencyincreases, the absolute value of the negative resistanceR0out will be reduced and the maximum oscillation fre-quency fmaxbecomes zero. Without considering the par-asitic series resistors reand rc, the expression for fmaxis(9)Equation (9) matches with the well-known expressionfor fmaxof the bipolar Transistor , on which a maximumpower amplification factor is unity, and the steady-stateoscillation condition is carried out solely for the losslessoscillation system [6]. Large-signal Oscillator circuit designGenerally, at least three steps are required for Design -ing a large-signal Oscillator circuit. First, it is necessaryto choose an appropriate circuit topology, second, todetermine the device large-signal characteristics and,third, to optimize the Oscillator circuit to achieve thedesired performances. To analyze and Design the funda-mental negative resistance bipolar Oscillator , it is advis-able to apply a quasi-linear Method , based on the use ofthe ratios between the fundamental harmonics of cur-rents and voltages as well as the representation of non-linear elements by equivalent averaged fundamental lin-ear ones.

7 The derivation of equivalent linear elements interms of signal voltages is based on static and voltage-capacitance bipolar Transistor characteristics. In a com-ffrCTbcmax=8 RrrrrRrRaCarrRCXCR routcbbeeTebeeoutcoutcT0111111001212=+++ ++ ++ = () ZRjXoutoutout00 0=+XCrXCrcbTceT10201212= = Figure 2. Amplitude dependencies of large-signaltransconductance, constant bias collector current andfundamental amplitude. Figure 3. Amplitude dependencies of (a) optimum cicuitparameters and (b) the real and imaginary parts of outputresistance and output power of the bipolar Oscillator .(a)(b)42 APPLIEDMICROWAVE& WIRELESSmon case, all elements of the Transistor equivalent cir-cuit are nonlinear and depend significantly on operationmode, especially for transconductance gmand base-emitter capacitance Ce. However, in practice, it isenough to be limited to the nonlinear elements gm, and Cc, as the base resistance rbpoorly depends on a biascondition.

8 Besides, to calculate the constant bias andfundamental collector currents, it is sufficient to use alinear approximation of transition frequency and thesmall-signal value Ccat operating point [4]. Under theassumption of a sufficiently low collector-emitter biasvalue, the applied bipolar Transistor model does not rep-resent the effect of the forward-rectified current acrossthe collector-base junction and the collector voltage-breakdown phenomenon. In this case, it is assumed thatthe transition from a soft-excited oscillation mode to asteady-state stationary mode is caused mainly by thenonlinear characteristic of the transconductance gmincontrast to the FET Oscillator . Here, under large-signaloperation both the Oscillator impedance and power pre-diction are primary due to a change in differential drain-source resistance [4].When the base-emitter current is sufficiently small, thefollowing exponential model can be applied to approximatethe family of the experimental transfer I-V characteristics:(10)where Vbeis the base-emitter junction voltage, Icsisthe reverse collector saturation current, and VTis thetemperature voltage.

9 By restricting to the fundamentalfrequency, Vbecan be expressed asVbe= Eb Ee IcoRe+ vbecos t,where Icois a constant bias collector current, Eband Eeare the base and emitter bias voltages, and Reis a self-bias into account that the voltage drops on collec-tor and emitter Transistor equivalent circuit resistors rcand reare negligible, the values of the large-signaltransconductance gm1and the constant bias collectorcurrent Icoas the functions of the junction fundamentalvoltage amplitude vbecan then be defined as follows:(11a)(11b)where I0(vbe/VT), I1(vbe/VT) are the modified first orderBessel functions [7].For the steady-state stationary oscillation mode, it isnecessary to define the Analytic relations between theload current amplitude and fundamental amplitude ofcollector voltage with input voltage amplitude vbe. Theserelations are expressed in terms of the Transistor z-para-meters and Oscillator circuit parameters in the followingmanner:(12)In a steady-state operation, the output power of theoscillator is By replacing load resistance ZLfrom equation (1) andusing the parameters of Transistor equivalent circuitfrom (6) to (8), Poutcan be expressed as(13)ResultsAn Analytic approach was applied to the microwavebipolar Oscillator Design .

10 The bipolar Transistor has thefollowing parameters of its equivalent circuit:fT = 6 GHz, Cc= pF,gm= A/V, rb= 4 ohms, re= , rc= ohms, Le= Lc= nH, Lb= nH. Thenumerical calculation was performed using the values ofthe oscillation frequencyf= 4 GHz, biases Eb= 0 V andEe= 2 V, self-bias resistor Re= 100 ohms and reversesaturation collector current Ics= 10 A. The numericalresults obtained are shown in Figures 2 and 3. Figure 2shows the amplitude dependence of large-signaltransconductance gm1, constant bias collector current Ico,and fundamental collector amplitude vce. Figure 3 showsthe amplitude dependence of (a) optimum circuit para-meters, X10and X20(b) the real and imaginary parts ofthe output resistance R0out, X0outand output power to Kurokawa [8], as the negative resistanceR0outreduces with the increase of the base-emitter junc-tion and collector voltage amplitudes, the stable oscilla-tions are established in the Oscillator .