A P P L I C AT I O N S M O B I L E CO M M U N I C AT I O N S

Fundamentals of oscillators for the microwave range

Foto: SES/ASTRA

Maximum stability

The author, GERHARD LOHNINGER, Dipl.-Ing., is project manager for silicon MMICs and responsible for product support for fourth-generation bipolar transistors (SIEGET). He is also responsible for product definition of discrete silicon RF semiconductors.

Oscillators are important components of electronic circuits throughout the frequency spectrum. They may be divided into fixed-frequency oscillators and tunable voltagecontrolled oscillators (VCOs). The former are commonly used to drive mixers or as transmitters with or without an amplifier output stage. Prime examples are satellite reception in the ASTRA band, where a 10 GHz local oscillator down-converts the carrier to an IF of 0.95 to 1.75 GHz, or a 2.45 GHz Doppler radar module whose core is a multilayer epoxy board containing a stabilized oscillator. VCOs are mainly used in phase-locked loop (PLL) systems, which make different frequency settings possible within a defined frequency band. Examples are VCO modules in mobile phones or TV and radio receivers. Although the basics of both oscillator types are the same, this article is confined to the fixed-frequency variety.

(1/98.3.1)

In communications, instrumentation and computer applications, stable oscillators for signal generation and control are essential to the functionality and performance of countless electronic circuits,modules and systems. Silicon bipolar transistors BFP 405, BFP 420 and BFR 92P, and bias controller BCR 400 are key components of oscillators for these vital functions. Feedback or negative resistance Oscillators can be described as circuits based on the principle of feedback or negative resistance. Feedback oscillators operate with active components, i.e. transistors, whereas negative resistance oscillators use Gunn, IMPATT (impact ionization and avalanche transit time) or BARRIT (barrier injection transit time) diodes. The negative resistance and thus the application potential of these components as active oscillators can be derived from their voltage-current characteristic. The noise signal Ein acts as the input signal to the feedback circuit. It is distributed over various components. The output signal E0 is calculated from the relationship

This yields

the phase along the loop must be a multiple of the angle ϕ, i.e. 2πn. When the signal increases due to excitation by noise, nonlinear effects subsequently reduce the loop gain until the stationary condition GD(e)H = 1 and phase ϕ = 2πn are satisfied. Another method of calculating the requirement for oscillation is provided by the negative resistance model. Here the resistance -RD must represent a defined variable to excite an oscillation. The noise voltage Ein is not present until the system’s operating voltage is applied and therefore has the characteristic of a conditional jump. The resistance -RD and the capacitance C are assigned to the active part (device), while the inductance L and resistance RL belong to the circuit load. With the aid of a Laplace transform, the temporal variation of E0 as a function of Ein in the presence of a conditional jump can be calculated as follows:

To generate an oscillation, the loop gain must be greater than 1 when the operating voltage is applied, and

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A P P L I C AT I O N S M O B I L E CO M M U N I C AT I O N S

where Ein (t) in the time domain ⇔ Ein (s)/s in the frequency domain, and L.di(t)/dt ⇔ L.s.I(s), where

Feedback and negative resistance circuits

The time function E0 can be determined from the image function by transformation into the original function with the aid of the conversion formulas as follows:

(1/98.3.2)

I(s)/ Cs and R.i(t) ⇔ R.I(s).

The feedback structure (left) consists of a linear amplifier of gain G, a nonlinear component D(e) and a feedback gain H.The equivalent circuit diagram of a negative resistance oscillator is shown on the right.

Colpitts oscillator Given that

,

For inductive loads this configuration produces a Colpitts oscillator.The steepness of the slope gm depends on the operating current of the transistor.

This yields the voltage E0 in the time domain as follows:

(1/98.3.3)

corresponds in the time domain to

Smith diagram

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Load impedance Z(ω) (load line) and impedance Z(A) (device line) of a typical circuit in the Smith diagram.

Small signal

(1/98.3.4)

So if the magnitude of resistance -RD is greater than RL, an oscillation will start to develop from the noise. If a quality term (Q = ω0L/RL) is introduced, a smaller frequency shift occurs from the beginning of the oscillation up to the stationary condition at the frequency ω0. If the definitions Z(A) = -ZD = RD-jXD and Z(ω) = ZL = RL+jXL apply to the impedances, then Z(A) = Z(ω) must apply to the stationary oscillator condition, i.e. RL = RD and XD = XL. The output voltage E0 is then Ein sin(ω0t) / Q if nonlinear effects are neglected. Both types of calculation, negative resistance and feedback, may be used for feedback oscillators. If a bipolar transistor is simulated by a simple equivalent circuit and recalculated with the aid of the complex calculation, the resistance RD is

The following apply to point P2: Z(A) = R(A) - jX(A) = 25Ω + j 10 Ω ZD =-Z(A) = -25Ω - j 10 Ω Z(A) = Z(W) In the stationary case Z(ω) = 25 Ω + j 10 Ω

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A P P L I C AT I O N S M O B I L E CO M M U N I C AT I O N S

(1/98.3.5)

Amplitude and phase conditions

obtained as follows: RD = -gm/ω2C1·C2 (Colpitts oscillator) Operating point and load impedance Assuming that variation of the impedance Z(A) with frequency is obtained by adjusting the oscillator’s tuning screw, there will be a stable, low-noise operating point at P1, as shown in the Smith diagram. The conditions for this are: 1. The angle ϕ (phi) must lie between 0 and 180°. The angle ϕ is defined as the tangent at Z(A) [at point P1 pointing away from the small-signal impedance] turning clockwise toward the tangent at Z(ω) [at point P1 in Oscillators are used for the direction of rising triggering mixer components, as transmitters in satellite communi- frequency]. cations and in other demanding 2. Minimum phase applications. They can be intenoise occurs when grated into circuits with feedback ϕ = 90°. or negative resistance. Fixedfrequency oscillators must satisfy 3. When the operating special conditions in each circuit voltage is applied, the type. stable operating point is the one that first describes an angle ϕ from 0 to 180° starting from small-signal operation. At an operating frequency f B of 9.35 GHz (point P2), for example, this means that a load impedance is required with a real component of 25 Ω and an inductive component of 0.17 nH. This guarantees a

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The load impedance Z(ω) is the critical parameter of a stable oscillator function.The two reflection factors must satisfy defined conditions.

stable, low-noise operating point. If the oscillator is operated along the curve Z(A) with the relevant operating points P3, P4 and P5, then point P4 is set after the operating voltage has been applied. P5 is not a stable operating point because the angle ϕ is greater than 180°. If the oscillator’s operating voltage is applied (P1) and the distance between resonator and tuning screw reduced, then point P2 is reached first, followed by P3 up to P6. No stable point exists there, and a frequency jump to P7 occurs. In the reverse case, P7 is a stable operating point starting at a high frequency. As the distance between tuning screw and resonator increases, the region between P4 and P5 is reached, where the frequency jumps between P3 and P2 (hysteresis). Resonant load curves must therefore be avoided in the vicinity of the operating point, as the oscillation may become unstable if the temperature fluctuates (Z(A)changes). A rule of thumb for optimum load impedance is R(A) ≅ R(A0)/2. where Z(A0) represents the input impedance for the oscillator in small-signal operation, i.e. at the first moment that the operating voltage is activated. A derivative of these relationships shows that the oscillator delivers more or less its

maximum power. As soon as a first design is available, however, it is much more efficient to perform measurements with a tuner (load pulling). This is an RF instrument with which practically all the impedances of a Smith chart – except for very high or low values – can be implemented. Various measurement and calibration procedures make it possible to measure the impedance set on the tuner at a particular oscillator output power. Measurement can be simplified by using automatic tuning systems with suitable programming. A somewhat different approach is of particular interest for VCOs and serial feedback oscillators. To meet the oscillation requirements, the reflection factors ΓD and ΓL must correlate as follows: ΓD ·ΓL = 1 The equation can therefore be resolved into two partial conditions: 1. Amplitude condition: |ΓD| · |ΓL| ≥1 2. Phase condition: ϕ (ΓD) + ϕ (ΓL) = 2πn (where n = 0, 1, 2,...) This means that the oscillation conditions can be checked at every oscillator port and the circuit properties optimized to initiate oscillations. The second part of this article covering applications with silicon bipolar RF transistors and the BCR 400 bias controller will appear in the next issue of Components.

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