Applied Electronics III · 120 h · Topic 5 of 10

Low-level and tuned amplifiers

The internal feedback of an RF transistor is small, but it decides whether it oscillates. Going from |S12| = 0.05 to 0.08 leaves the same transistor potentially unstable.

S parameters Rollett factor Maximum gain Neutralization Low-noise amplifier

01Describing an RF amplifier: the S parameters

At radio frequency, voltages and currents cannot be measured with open or short circuits: transistors oscillate and cables have lengths comparable to the wavelength. That is why devices are described with scattering parameters, which relate the incident and reflected waves at each port with all ports terminated in 50 Ω. They are the RF version of the two-port networks of Circuit Theory II:

Parameter	Meaning
S₁₁	Input reflection coefficient
S₂₁	Forward gain, from input to output
S₁₂	Reverse transmission: the internal feedback of the transistor
S₂₂	Output reflection coefficient

They are measured with a vector network analyzer and manufacturers publish them as a function of frequency and bias. With them the input and output matching networks are designed.

02Stability: the Rollett factor

Because of the internal feedback \( S_{12} \), a transistor can present negative resistance at one port for certain loads at the other, and oscillate. It is unconditionally stable if it does not oscillate with any passive load, a condition that is checked with two numbers:

\[ K = \frac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2\,|S_{12}S_{21}|} > 1 \] \[ |\Delta| = |S_{11}S_{22} - S_{12}S_{21}| < 1 \] If they are met, the input and the output can be matched at the same time, and the gain obtained is the maximum available gain, \( MAG = |S_{21}/S_{12}|\,(K - \sqrt{K^2 - 1}) \). If not, there are forbidden loads, which show up as stability circles on the Smith chart, and the limit is the maximum stable gain \( MSG = |S_{21}/S_{12}| \).

Lab · stability and gain of an RF transistor

A transistor with S₁₁ = 0.6∠−160°, S₂₁ = 3∠80°, S₁₂ with phase 45° and S₂₂ = 0.5∠−60°. The magnitude of the internal feedback S₁₂ is varied. The blue curve is K as a function of |S₁₂|; the green one, the maximum available gain where it exists.

|S₁₂| =

|S₂₁| =

03Tuned and low-noise amplifiers

A tuned amplifier uses resonant circuits as its load, to amplify only one band and reject the others: they are the intermediate-frequency stages of receivers. Its bandwidth is set by the loaded Q of the tanks, and with several synchronous stages the band narrows, as seen in multistage amplifiers. The collector–base capacitance, which in the S parameters is \( S_{12} \), can make them oscillate; it is cancelled with a neutralization or avoided with a cascode.

In a low-noise amplifier (LNA), the first stage of a receiver, the input is not matched for maximum gain but for the source impedance that gives the minimum noise figure, \( \Gamma_{opt} \), which the manufacturer publishes. The two do not coincide, and the design is a compromise that is resolved with constant-gain and constant-noise circles on the Smith chart. The noise figure and its measurement were covered in amplifier measurements.

04In the lab

Activity 1 · S parameters

Measure with a vector network analyzer the S parameters of a transistor mounted on its evaluation board, and compute K and the maximum gain as a function of frequency.

Activity 2 · Oscillation

Find a frequency where K < 1, present at the input a load inside the instability circle and observe the oscillation with a spectrum analyzer.

Activity 3 · Stabilizing

Add a resistor in series with the base or in parallel with the output and measure how K rises and how much the gain drops.

05Common mistakes

Checking stability only at the operating frequency. The transistor can oscillate far from the band, where it has more gain.
Matching an LNA for maximum gain. Noise figure is lost, and that is what matters in the first stage.
Using K without looking at |Δ|. Both conditions are needed.
Forgetting the board parasitics. They change the S parameters relative to those in the datasheet.

06Self-assessment

What does S₂₁ = 3∠80° mean?

A wave voltage gain of 3 times, 9.5 dB, with 80° of phase, between 50 Ω ports.

What is the maximum stable gain?

\( |S_{21}/S_{12}| \): the limit when the transistor is potentially unstable, attainable by stabilizing it with resistors.

Why is an LNA not matched to the conjugate of its input?

Because the minimum noise figure is obtained with another source impedance, Γ_opt.

What does neutralization do?

It cancels the internal collector–base feedback with an external feedback of opposite sign.

07Further reading

Guillermo Gonzalez. Microwave Transistor Amplifiers: Analysis and Design. 2nd ed., Prentice Hall, 1997. Design with S parameters: stability, gain and noise.
David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2012. S parameters and the design of microwave amplifiers.
Keysight Technologies. S-Parameter Design. Application Note 154, Keysight. A practical introduction to amplifier design with S parameters.