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FIGURE 323
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N OT E : All signs ( ) must be maintained for all calculated numbers 1 S will subtract 1
from the real term of S, and will change the sign of the imaginary term (or 1 (02 + j01) = 08 j01) S equals the complex conjugate of S If is outside a bracket, then the answer to everything within the bracket must be converted into the complex conjugate
Small-Signal Design and Vector Algebra
When required to utilize full complex numbers (Z q ) in our amplifier calculations, we can perform the required mathematical functions by the following methods: To multiply polar quantities: First, multiply the magnitudes; then add the phase angles To divide polar quantities: First divide the magnitudes; then subtract the phase angles To subtract polar quantities: First, convert to rectangular notation (R + jX; see To Convert Polar Into Rectangular Form below); then subtract R1 R2 = RT, and jX1 jX2 = jXT ; then convert the rectangular answer back to polar
Use full vector algebra in S-parameters that are so marked Employ only the S-parameter s magnitudes if not so marked
Amplifier Design
To add polar quantities: Perform as in subtraction; but add the rectangular values (RT = R1 + R2; jXT = jX1 + jX2)
Small-Signal Amplifier Stability
A typical amplifier must be unconditionally stable across all frequencies and input/ output impedances An amplifier may oscillate at anywhere from low frequencies (where the port terminations are not matched to 50 , and the transistor s gain is naturally high, until it is limited by the matching/coupling network) all the way up to the maximum frequency at which the amplifier still shows greater than unity gain Also, when the transistor is not stable and begins to oscillate, it can shift the bias point of the stage, consuming more current and increasing internal device dissipation, and possibly causing its destruction What does instability in an amplifier look like Figure 324 displays an oscillating amplifier as would be seen in the frequency domain on a spectrum analyzer, showing the possible areas of oscillations Normally only one or two of these oscillations will pop up, but they can easily be distinguished from other spurs by placing your finger in the area of the (low voltage and low current!) circuit, and then closely seeing if these spurs begin to shift in frequency If so, then you are viewing instabilities in the amplifier, and these must be eliminated by stabilizing the circuit (Even though most unstable amplifiers will oscillate at where the gain is highest, which is at the low frequencies, these low-frequency oscillations will mix with the in-channel frequencies, causing visible oscillations at and near our passband of interest) Some engineers erroneously believe that an amplifier need only remain stable when presented with 50 , since that is what the input of the following stage will be or at least is expected to be However, unconditional stability across all frequencies and impedances is critical, since the next or prior stage, which would normally be a filter or another matched (narrowband) amplifier, will be presenting
0 REF 10 AMPLITUDE 20 30 40 50 60
FREQUENCY
DESIRED TONE
OSCILLATIONS
SPECTRAL HUMP
OSCILLATION
L CIL AT ION
FIGURE 324
In the output of an unstable ampli er one or more of these spectra will be seen
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anything but 50 when looking outside of their own passbands This means that the amplifier may actually be seeing a VSWR of 18:1, or even higher, at certain out-ofband frequencies, and thus must be stable across the entire area that it possesses any appreciable gain Conditionally stable transistors may still be used in amplifier designs, but only if the amplifier stage is never presented with any impedance that would cause it to become unstable Nevertheless, selecting a transistor that is unconditionally stable will give us one less thing to worry about But since such a transistor may not be all that common, we may have to take a conditionally stable device and force it into unconditional stability Such a conditionally stable transistor will have some frequencies that have conditional points of stability and, through inductive emitter degeneration, collector and base resistive loading, or negative feedback, we can make this conditionally stable transistor into a fully stable amplifier at all frequencies Stability of an amplifier stage is dependent on a transistor s temperature, bias, signal level, and HFE spread, as well as the active device s positive internal feedback mechanism, excessively high gain outside of the desired bandwidth (usually at lower frequencies), and external positive feedback caused by support components, PCB layout, or an RF shield s box modes (see RF Shielding Resonances, Sec 1332) Since amplifier instability will be the number-one major headache of any circuit design engineer, it will benefit us to delve back again into this important issue As stated, virtually all active devices are quite stable when presented with a perfect 50- source and load over the entire frequency range that the device exhibits gain Indeed, most instability problems occur when the circuit designer does not take into account the elevated low-frequency gain of a normal amplifier (Fig 325), and the transistor s possible instabilities when presented with anything other than a 50- termination This lack of a 50- termination as the frequency is decreased can be due to two main causes: (1) The amplifier s matching circuits are good only for a narrow band of frequencies, so they
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