Microwave oscillator design generally represents a complex problem. De- pending on the technical require- ments of the oscillator, it is necessary to define.
TECHNICAL FEATURE
MICROWAVE TRANSISTOR OSCILLATORS: AN ANALYTIC APPROACH TO SIMPLIFY COMPUTER-AIDED DESIGN
M
icrowave oscillator design generally represents a complex problem. Depending on the technical requirements of the oscillator, it is necessary to define the configuration of the oscillation system and transistor type, evaluate and measure smalland large-signal parameters of the transistor equivalent circuit and use an appropriate nonlinear simulator to calculate the oscillator electrical and spectral characteristics. When a transistor is used, representing the circuit as a two-port network is the most suitable approach for computer-aided design of free-running microwave oscillators. In this case, the basic parameters of the equivalent circuit either can be directly measured or approximated on the basis of experimental data with sufficient accuracy over a wide frequency range.1–3 In large-signal operation, it is necessary to define the appropriate parameters of the active two-port network as well as the parameters of the oscillator circuit’s external elements. Initially, the values of the external circuit elements are unknown. In addition, it is difficult to directly choose the values for a given microwave oscillator with a required oscillation frequency without any preliminary calculation. This process can be sufficiently timeconsuming and, in a typical case, calls for much simulation. Therefore, it is convenient to use an analytic method of optimizing microwave oscillator design that incorporates explicit expressions for feedback elements and load impedance in terms of the transistor
equivalent circuit elements and its static characteristics.4 A conventional analytic approach to derive explicit expressions for the optimum values of an oscillator circuit first was applied to a bipolar transistor oscillator using an ideal transformer for feedback.5 Later, an analytic approach was used with a GaAs FET oscillator using a simplified transistor model where the influence of internal feedback from the gatedrain capacitance had not been taken into account.6 Such an approach involves a two-step procedure. First, the optimum combination of feedback elements required to realize a smallsignal maximum negative resistance to permit oscillations at the largest amplitude is defined. Then, taking into account the large-signal nonlinearity of the transistor equivalent circuit elements, the realized small-signal negative resistance is characterized to determine the optimum load impedance when maximum output power is generated for a given oscillator circuit configuration. A GENERAL ANALYTIC APPROACH In general, the purpose of the microwave bipolar oscillator design procedure is to define the optimum bias conditions and the values of feedback elements and load that correspond to the maximum power at a given frequency in
ANDREY V. GREBENNIKOV Institute of Microelectronics Singapore
TECHNICAL FEATURE According to the optimization analytic procedure presented previously,4 the optimum values of X1 and X2 at which the negative value of Rout is maximal are determined by solving
Cco + Vµ b
Lb
rµ
rb +
−
rπ
rc
Cci
rce Cs
Cπ
Vπ
∂Rout ∂Rout = 0 and =0 ∂X 1 ∂X 2
Lc
The optimal values of X01, X02 and X0out depend on the impedance parameters of the active two-port network such that
− Z1
ZL
gmFVπ − gmRVµ, τ
Re
X 01 = Le e
The series-feedback bipolar transistor oscillator’s equivalent circuit. Fig. 2 The series-feedback oscillator with a simplified transistor equivalent circuit. ▼
Lb
Cc
rb1
rc
Lc c
rb2 gmVπ Vπ
Cπ ZL
Z1
re Le e Z2
Zout
steady-state, large-signal operation. Figure 1 shows a generalized two-port circuit for transistor oscillators used mostly at RF and microwave frequencies where Zi = Ri + jXi, i = 1, 2 and ZL = RL + jXL. The equivalent SPICEGummel/Poon model of the bipolar transistor, accurately simulating both its DC and high frequency behavior up to the transition frequency fT = gm/2πCe, is also shown.7 The steady-state oscillation condition for a single oscillation frequency is expressed as Zout(I, ω) + ZL(ω) = 0
(1)
where Zout(I, ω) = Rout(I, ω) + jXout(I, ω) is the equivalent one-port network output impedance looking into the collector of the transistor as a series-tuned circuit, and ZL(ω) = RL(ω) + jXL(ω) is the frequency-dependent load impedance. To optimize the oscillator circuit in terms of the maximum value of negative real output impedance of a one-port network, the expression for output impedance Zout is written as Zout = Z 22 + Z 2 –
(Z12 + Z 2 )(Z 21 + Z 2 ) Z11 + Z 2 + Z1
(2R 2 + R12 + R 21 )(R 21 – R12 ) – X 12 + X 21 2 2( X 21 – X 12 )
X 0out = X 02 + X 22 –
▲ Fig. 1
b
R 21 – R12 R12 + R 21 X + X 21 – R11 – R1 – X 11 + 12 2 2 X 21 – X 12
X 02 =
Zout
Z2
(3)
(2)
(
R 21 – R12 0 Rout – R 2 – R 22 X 21 – X 12
)
(4)
SMALL-SIGNAL OSCILLATOR CIRCUIT DESIGN Figure 2 shows another, simplified version of the bipolar transistor’s equivalent circuit where Cc = Cco + Cci, rb1 = rbCci/Cc and rb2 = rbCco/Cc. This equivalent bipolar transistor circuit is possible because the condition rb > 1/ωCe simplifies the analytic and numerical calculations substantially without a significant decrease in the final result’s accuracy at RF and microwave frequencies. As a result, the internal bipolar transistor in a commonemitter small-signal operation is characterized by the real and imaginary parts of its Z parameters as R11 = R12 2 ω 1 = a + rb 2 gm ωT
R 21 = R 22 = R11 +
a ω TCc
X 11 = X 12 = –a
ω 1 – rb 2 ω T gm
X 21 = X 11 + X 22 = X 11 –
a ωC c
a ω ωC c ω T
2
(5)
TECHNICAL FEATURE I1 V1 Yload
Y1
▲ Fig. 3
The principle of nonlinear circuit simulation.
TABLE I
Cco (pF)
0.35
Cci (Vµ = 0) (pF)
0.34
Cπ (Vπ = 0) (pF)
2
Le (nH)
0.3
Lb (nH)
0.3
Lc (nH)
0.5
re (Ω)
0.3
rb (Ω)
4
rc (Ω)
1.75
τ (ps)
10
Y1 (mA)
TRANSISTOR EQUIVALENT CIRCUIT PARAMETERS
a
=
FREQUENCY (GHz)
where 1 ω 1+ ωT
2
ω T = 2πf T Substituting the expressions for real and imaginary parts of the Z parameters from Equation 5 into Equation 4, the optimum values of imaginary parts of the feedback elements X10, X20 and X0out expressed through the parameters of the bipolar transistor equivalent circuit are given by X 01 = X 02 = –
1 ω – r b1 2ωC c ωT
1 ω – rb 2 + re 2ωC c ωT
X 0out = –
(
)
1 ω – Roout 2ωC c ωT
(6)
Thus, it follows from Equation 6 that the oscillations in the optimal microwave oscillator arise as a result of the capacitive reactance in the emitter circuit (X20 < 0), inductive reactance in the base circuit (X10 > 0), and inductive (XL > 0) or capacitive (XL < 0) reactance in the collector circuit depending on the value of R0out.
Pout (dBm)
Yosc
Pout (dBm)
OSCILLATOR CIRCUIT
LARGE-SIGNAL SIMULATION 20 nH 8 pF One of the most P1 35 Ω 50 nH popular approaches to nonlinear freerunning oscillator FREQ SINGLE TONE analysis is to use the 0.8 pF 1.2 nH + + harmonic balance nHarm: 5 BIAS − − BIAS Freq: ?3.5GHz 4.5GHz? equations developed for the circuit V = −2 V=9 and consider the oscillation frequency as an additional optimization variable.1 ▲ Fig. 4 The simulated series-feedback bipolar transistor oscillator equivalent circuit. Such an algorithm is Fig. 5 The simulated starting used in the new veroscillation conditions. ▼ sion of Microwave Harmonica as a part of the Serenade 7.5 circuit sim20 ulator.8 The basic idea in this method 0 can be explained starting from the oscillation conditions when an oscillator −20 circuit is considered as a one-port cir−40 cuit. To determine the resonant frequency, the program computes the −60 circuit loop gain by imposing a small −80 test voltage source on the circuit, as 3.4 3.6 3.8 4.0 4.2 4.4 4.6 shown in Figure 3. The fundamental FREQUENCY (GHz) source current I1 = Re(I1) + jIm(I1) is a function of zero phase peak voltage V1 and frequency f. If f is the circuit 4.05 25 self-resonant frequency, then the 4.00 20 phase of the current I1 is zero and 3.95 15 3.90 10 Im(I1) = 0 for a nonzero Re(I1). For a 3.85 5 nonzero voltage V 1 and a small Im(I 1), the steady-state oscillation 20 30 40 50 60 70 80 conditions are present for Re(I1) = 0 Re (Ω) when the circuit feedback loop gain is ▲ Fig. 6 Output power and oscillation equal to unity. Starting oscillation frequency vs. emitter bias resistance. conditions are found at Re(I 1) < 0 and Im(I1) = 0. ▼ Fig. 7 Output power vs. load resistance. To verify the accuracy of the analytic approach used to determine the 22 oscillator external feedback parame21 ters, a medium power microwave 20 bipolar transistor was chosen. To de19 termine the value of output resis18 0 tance R out for the chosen value of 17 load resistance RL, the amplitude bal50 60 70 80 20 30 40 0 ance equation R out + rc + RL = 0 must RL (Ω) be used. For the preliminary chosen cillation conditions were determined oscillation frequency f = 4 GHz, the by sweeping the frequency f of the optimal oscillator feedback parameexternal test source from 3.5 to 4.5 ters according to the theoretical preGHz. The curves satisfy starting oscildictions given by Equation 6 must be lation conditions under linear smallL = 1.2 nH and C = 0.8 pF, respecsignal operation where Re(I 1 ) tively. The transistor equivalent cir< 0 and Im(I1) = 0 at 4.24 GHz, as cuit parameters are listed in Table 1. shown in Figure 5. A nonlinear circuit simulation was Large-signal oscillation conditions performed for the microwave bipolar with various values of emitter bias resistransistor series feedback circuit ostor Re, load RL and feedback elements cillator; the equivalent circuit is L and C are shown in Figures 6, 7 shown in Figure 4. The starting os-
TECHNICAL FEATURE Pout (dBm)
25 20 15 10 5 0.7
1.20
1.0
1.3
0.80
1.6 1.9 L (nH) 0.60 0.50 C (pF)
2.2
0.45
▲ Fig. 8
Output power vs. circuit feedback elements.
and 8, respectively. Despite some sufficiently serious preliminary simplification, the experimental results indicate good accuracy of the predicted oscillation frequency by means of an analytic calculation of the feedback parameters (taking into account the standard loading conditions with RL = 50 Ω). The exact value of the steady-state oscillation frequency f = 4 GHz is realized at Re = 28 Ω when the output power Pout is close to the maximum value. Furthermore, the output power Pout = 21.5 dBm is very close to the maximum value 21.9 dBm at optimal load RL0 = 45 Ω where complete phase compensation is realized. To verify the validity of the chosen feedback parameters L = 1.2 nH and C = 0.8 pF, some circuit simulations with other sets of feedback parameters that satisfy the oscillation conditions at the resonant frequency f = 4 GHz have been performed. From the results, it follows that a deviation of the feedback parameters from the optimal set leads first to a deteriora-
tion of the phase conditions and then to oscillation failure. (The broken lines in the Pout vs. L and C plot define the borders of the oscillation region.) The constant bias collector current did not exceed 32 mA during the simulation procedure. CONCLUSION A simple analytic method of microwave bipolar transistor oscillator design has been developed that defines explicit expressions for optimum values of feedback elements through the bipolar transistor Z parameters. A negative resistance concept is utilized to design a series-feedback microwave bipolar transistor oscillator with optimized feedback elements and maximum output power in terms of the transistor impedance parameters. Such an approach significantly simplifies the nonlinear circuit simulation procedure. Final simulation results indicate the attractiveness and advisability of the circuit parameter analytic evaluation used for nonlinear computer-aided design. ■ References 1. C.R. Chang, M.B. Steer, S. Martin and E. Reese, “Computer-aided Analysis of Freerunning Microwave Oscillators,” IEEE Transactions on Microwave Theory and Techniques, Vol. 39, October 1991, pp. 1735–1745. 2. R.J. Trew, “MESFET Models for Microwave Computer-aided Design,” Microwave Journal, Vol. 33, No. 5, May 1990, pp. 115–128. 3. D.A. Warren, J.M. Golio and W.L. Seely, “Large- and Small-signal Oscillator Analysis,” Microwave Journal, Vol. 32, No. 5, May 1989, pp. 229–246.
4. A.V. Grebennikov and V.V. Nikiforov, “An Analytic Method of Microwave Transistor Oscillator Design,” International Journal of Electronics, Vol. 83, December 1997, pp. 849–858. 5. D.F. Page, “A Design Basis for Junction Transistor Oscillator Circuits,” Proceedings of IRE, Vol. 46, November 1958, pp. 1271–1280. 6. M. Madihian and T. Noguchi, “An Analytical Approach to Microwave GaAs FET Oscillator Design,” NEC Research and Development, No. 82, July 1986, pp. 49–56. 7. P. Antognetti and G. Massobrio, Semiconductor Device Modeling with SPICE, McGraw-Hill, New York, NY, 1993. 8. Serenade 7.5, Reference Volume, Ansoft Corp., New Jersey, 1998. Andrey V. Grebennikov received his Dipl Ing degree from the Moscow Institute of Technology and his PhD degree from the Moscow Technical University of Communication and Informatics in 1980 and 1991, respectively. He joined the scientific research department of the Moscow Technical University of Communication and Informatics as a research assistant in 1983. Since October 1998, Grebennikov has been with the Institute of Microelectronics, Singapore. His research interests include the design and development of power FM broadcasting and VHF-UHF television transmitters; hybrid ICs of narrowand wideband RF and microwave low and high power high efficiency and linearity amplifiers; and single-frequency and voltage-controlled oscillators using bipolar and FET devices.
