# The RF Stability Factor Explained

### Key Takeaways

• Accurate stability factor analysis is vital for preventing oscillations and achieving optimal performance in RF amplifiers and related systems.

• The K-factor and µ-factor are crucial for evaluating the stability of RF circuits, particularly amplifiers, across various operating conditions.

• Unconditionally stable circuits (K > 1, µ > 1) are ideal, while conditionally (K < 1, µ > 1) or potentially unstable circuits (K < 1, µ < 1) require careful design considerations.

Assessing the RF stability factor of systems that contain amplifiers is important to ensure circuit function

The RF stability factor refers to the numerical value used to assess the stability of RF circuits — particularly amplifiers. This factor indicates whether a circuit is unconditionally stable, conditionally stable, or potentially unstable. This can be determined by a K-factor value, µ-factor value, or a combination of both.

In the table below, we see how the K-factor and µ-factor can provide information on a circuit’s stability.

### RF Stability Factor Value and Associated Circuit Condition

 K Factor Value µ Factor Value Circuit Stability Meaning Example K < 1 µ < 1 Potentially Unstable The circuit may exhibit oscillations or instability due to insufficient stability margins at both input and output. An amplifier with K < 1 and µ < 1 might oscillate or fail to function properly under certain conditions. K < 1 µ > 1 Output Potentially Unstable While the input is stable, the output may still be unstable, leading to potential oscillations or performance issues. A filter with a stable input but unstable output leads to unpredictable performance. K > 1 µ < 1 Input Potentially Unstable The output is stable, but the input may be unstable, affecting overall circuit behavior. An oscillator with a stable output but an input that is sensitive to impedance mismatches. K = 1 µ = 1 Marginally Stable The circuit is on the edge of stability at both input and output. A circuit that operates stably under specific conditions but is sensitive to variations in parameters. K > 1 µ > 1 Unconditionally Stable The circuit is stable under all linear operating conditions at both input and output. A robust amplifier design that maintains stability across a wide range of operating conditions.

## Why Is an RF Stability Factor Important?

In designing an amplifier, ensuring its stability across a wide frequency range is vital. Ideally, this assessment should span from direct current (DC) to the highest frequency of oscillation, FMAX,  of the device. Specifically, stability is used to describe the amplifier's ability to resist generating unwanted oscillations. These oscillations can range from noticeable large-signal disturbances at full power to more subtle spectral issues that might be detectable using a spectrum analyzer but nonetheless should be accounted for. A lack of stability outside your target frequency band could result in a significant reduction of up to 20 dB in your amplifier's gain within the desired band.

## RF Stability Factor Measurements

When designing amplifiers or selecting one from vendor data sheets, stability can be measured in a couple of ways.

### K-Factor

The most common stability criterion is the K-factor, calculated from the S-parameters in a two-port network. A K-factor greater than 1 indicates unconditional stability. The K-factor can be calculated as:

### Mu Factor

Another approach is the Mu factor, which provides more detailed stability information across a range of frequencies. It provides a different approach to assessing stability, focusing on the input or output stability of the circuit. The µ factor is particularly useful for circuits where unilateral assumptions (i.e., assuming no feedback from output to input) are invalid. The µ factor is defined based on the S-parameters of a two-port network, similar to the K-factor.

### Mu Factor

• S11 and S22 are the input and output reflection coefficients

• S12 and S21 are the reverse and forward transmission coefficients, respectively

• δ = S 11 S22 − S12 S21  is the determinant of the S-parameter matrix

• S*11 is the complex conjugate of S11

The Mu factor provides insight into how the input matching of the circuit affects stability. A circuit with a high µ factor is less sensitive to variations in source impedance, which is crucial for reliable performance in RF systems.

• If µ >1, the circuit is unconditionally stable at the input.
• If µ <1, the input stability of the circuit is potentially compromised, and it may be conditionally stable or unstable, depending on other factors.

Note that the µ factor, like the K-factor, is a linear stability measure. It doesn't account for non-linear effects, which can be significant in RF circuits, especially at high power levels, and only assesses the input stability. To fully evaluate a circuit's stability, both the input (µ factor) and output stability should be considered.

## Stability Types

Evaluating stability in amplifiers is a process that must account for all frequencies where potential oscillation could occur. Here's a breakdown of how stability is assessed.

### Unconditional Stability

A circuit falls into this category if it shows no tendency to oscillate under any passive load or source impedance. Unconditional stability implies that the network can handle any impedance visible on the Smith chart, ranging from the center to the edge (up to gamma=1.0), at any phase angle. Notably, a gamma value (the reflection coefficient) of less than 1 indicates a positive real part of the impedance.

• K-Factor Criterion: To qualify as unconditionally stable, a circuit must have a K-factor greater than 1 across its entire operational frequency range.
• Implications: These circuits are the gold standard in design, exhibiting minimal risk of oscillation or instability under various external conditions.

### Conditionally Stable Circuits

A circuit is considered conditionally stable if it maintains stability only under specific load and source impedance conditions. This type of stability means the network remains stable as long as the input and output are matched to the intended characteristic impedance (typically 50 ohms, sometimes 75 ohms). However, mismatches can lead to a range of impedances that may provoke oscillations.

• K-Factor Criterion: For conditionally stable circuits, the K-factor can vary, being greater than 1 depending on the frequency.
• Implications: Circuits of this kind demand meticulous impedance matching and careful control of load and source conditions to ensure stability.

### Potentially Unstable Circuits

A circuit is potentially unstable if there is a significant risk of oscillation under normal operating conditions.

• K-Factor Criterion: This category is characterized by a K-factor that consistently falls below 1 throughout the operational frequency spectrum.
• Implications: Circuits with this level of instability are generally viewed as problematic and risky, often requiring significant redesign or the implementation of additional stabilization strategies.

Software tools are an increasingly essential method for assessing circuit stability. Discover how Cadence's AWR software can elevate your design process. AWR's advanced features, including linear and nonlinear stability analysis and EM simulation, offer unparalleled support for RF engineers. Tap into the power of AWR to ensure your circuits are optimally stable and ready for the challenges of modern RF design.

Leading electronics providers rely on Cadence products to optimize power, space, and energy needs for a wide variety of market applications. To learn more about our innovative solutions, talk to our team of experts or subscribe to our YouTube channel.

# About the Author

Cadence PCB solutions is a complete front to back design tool to enable fast and efficient product creation. Cadence enables users accurately shorten design cycles to hand off to manufacturing through modern, IPC-2581 industry standard.