# Barkhausen’s Stability Criteria for Stability in Oscillators

Oscillators can take numerous topologies, such as simple crystal oscillators or transistor-based circuits with reactive components. One option for building or modeling oscillators based on active circuitry is to use an amplifier circuit, such as an op-amp-based oscillator. Examples include a comparator-based circuit with or without hysteresis. These circuits rely on feedback to set the oscillation frequency to a specific value such that a stable sine or square wave is produced at the output.

In the converse case, there is a need to avoid or prevent an oscillation, such as the unstable oscillation sometimes seen in op-amps, active filters, or amplifiers, and you will have to understand the criteria for stability. The borderline between instability and stability can be defined using Barkhausen’s criteria.

## Stability in Oscillator Design

Oscillators are circuits where we desire an output that has a periodic waveform, such as a square wave or a sine wave. No matter what the output waveform is, we want the repetition to occur at a single frequency, and possible with a particular duty cycle in the case of an oscillator with hysteresis. We would then prefer that all other frequencies not required to construct the output waveform do not exhibit any sustained oscillation at all.

The Barkhausen stability criteria is one tool to predict whether an oscillation will be stable

• Stable convergence: The oscillator exhibits a decaying oscillation if excited briefly at a particular frequency, and the output eventually decays to zero.

• Stable oscillation: The oscillator exhibits the desired output frequency with no growth or decay.

• Unstable oscillation: The output oscillation grows and becomes unbounded, eventually railing out the system.

The block diagram below shows the basic architecture of an oscillator and its feedback loop. The feedback loop has a transfer function 𝛽, and the total loop gain is given by 𝛽A.

The condition on the phase shift and loop gain required to produce a sustained oscillation are:

|𝛽A| = 1 and 𝜑 = 2n𝜋 (n = integer)

In other words, a specific frequency that exhibits |𝛽A| = 1 and 𝜑 = 2n𝜋 will exhibit a sustained oscillation at its input power value. For example, if the circuit is excited with a broadband source, that source can excite the frequency where |𝛽A| = 1.

This condition is ideal for sustained oscillations in an oscillator circuit and it marks the borderline between stable/unstable behavior:

• If |𝛽A| < 1, the oscillations will decay and eventually stop, indicating the system is stable.

• If ∣Aβ∣ > 1, the oscillations will grow in amplitude over time, revealing the system is unstable.

## How to Engineer a Stable Oscillator

Engineering a stable oscillator to operate at a target frequency typically involves iterating through the various component values in portions of the feedback loop. In an off-the-shelf amplifier circuit, nothing on the chip will be configurable, so the designer must attempt to create oscillatory behavior by adding some passive components into the feedback loop (essentially what is done in high-frequency transimpedance amplifiers).

If you look at the application circuit and gain-phase plot for an off-the-shelf component, you may see particular frequencies where you could attempt to create a stable oscillation if desired:

Gain-phase plot for a high-frequency op-amp from Texas Instruments (OPA846ID)

Is it possible for an amplifier/oscillator to output a stable oscillation even if the feedback loop does not exhibit any reactance in its phase shift? The answer is “yes” and an example comes from discrete RF amplifier circuits. In these circuits, the phase is very sensitive to PCB parasitics because these create excess reactance along the feedback loop. This comes from the pads and traces that connect the feedback loop along the component.

Typically this is observed around Bluetooth and higher frequencies. This is one reason why, in high GHz components, everything is placed on the semiconductor die or in-package.

## A Note on Terminology

The terms “stable” and “unstable” are used differently when looking at different types of systems where stable or unstable behavior are expected/desired. In the Barkhausen case, where we want to have a stable oscillation, the resulting oscillation at |𝛽A| = 1 is considered either unstable or stable behavior, depending on who you ask. When speaking of an amplifier circuit (e.g., high-frequency op-amp or transimpedance amp), the oscillation would be considered unstable.

In another type of system, such as coupled dynamical systems of two or more equations, we normally use Poincare-Bendixson stability to understand oscillatory behavior between two quantities. In this case, the decaying convergence to a periodic oscillation would be considered a stable limit cycle.

The field of stability in electronics and engineering in general is very rich, but take care to understand the terminology involved when verifying stable/unstable behavior.

Whenever you want to build and analyze your oscillator circuits using Barkhausen’s criteria, make sure you simulate your designs with the complete set of tools in PSpice from Cadence. PSpice users can access a powerful SPICE simulator as well as specialty design capabilities like model creation, graphing and analysis tools, and much more.