Periodic steady-state analysis

Periodic steady‑state analysis (PSS analysis) computes the periodic steady‑state response of a circuit at a specified fundamental frequency, with a simulation time independent of the time constants of the circuit. The PSS analysis also determines the circuit's periodic operating point which is required starting point for the periodic time‑varying small‑signal analyses: PAC, PSP, PXF, and Pnoise. The PSS analysis works with both autonomous and driven circuits. PSS is usually used after transient analysis.

What is Periodic steady‑state analysis?

Periodic steady‑state analysis, abbreviated PSS analysis, is a specialized simulation technique used in electronic circuit design. Its primary purpose is to determine how a circuit behaves once all transients have died out and the circuit is oscillating or switching in a perfectly periodic manner. Unlike a conventional transient simulation, which marches forward in time until the circuit settles, PSS directly computes the steady‑state waveform at a specified fundamental frequency.

The method is deliberately engineered so that the simulation time does not depend on the circuit’s intrinsic time constants (e.g., RC or LR values). This property makes PSS especially valuable for circuits whose time constants span many orders of magnitude, such as high‑frequency switching regulators or RF mixers, where a transient run could require an impractically long simulation window.

In addition to the waveform itself, PSS determines the circuit's periodic operating point. This operating point is a prerequisite for a suite of periodic time‑varying small‑signal analyses that examine how the circuit reacts to small perturbations around its steady‑state condition.

Fundamental Concepts

Periodic steady‑state response

A circuit that is periodically driven (e.g., by a clock or a sinusoidal source) or that self‑oscillates (autonomous) will, after an initial transient phase, settle into a repeatable pattern that repeats every period T = 1/f₀, where f₀ is the fundamental frequency chosen for the analysis. The periodic steady‑state response is the exact shape of voltage and current waveforms that repeat every T without further change. PSS analysis solves directly for this repeatable waveform rather than simulating each individual cycle.

Simulation‑time independence

Traditional transient simulation integrates the circuit’s differential equations step‑by‑step, and the total simulation time scales with the longest time constant present. In contrast, PSS decouples simulation effort from those time constants. By formulating the problem in the frequency domain (or using harmonic balance techniques), the solver converges to the periodic solution without marching through every nanosecond of physical time. The result is a simulation time that depends primarily on the number of harmonics retained and the complexity of the nonlinear equations, not on how slowly the circuit might charge or discharge.

Where PSS fits in the Simulation Flow

PSS is usually used after transient analysis. A typical design verification flow looks like this:

1. DC Operating Point (OP) Analysis – establishes the static bias of the circuit.
2. Transient Analysis – runs a time‑domain simulation long enough to observe whether the circuit settles into a periodic pattern.
3. Periodic Steady‑State (PSS) Analysis – takes the periodic behavior observed (or the designer‑specified frequency) and computes the exact steady‑state waveforms.
4. Periodic Small‑Signal Analyses – PAC, PSP, PXF, Pnoise, etc., use the periodic operating point from PSS as their starting condition.

By inserting PSS after transient, designers can verify that the circuit indeed reaches a periodic regime and then obtain a high‑accuracy description of that regime for downstream analyses.

The periodic operating point

The periodic operating point is the set of node voltages and branch currents that repeat every period. In linear circuits, this point reduces to a simple sinusoidal steady‑state solution. In nonlinear circuits—such as switching power converters, class‑D amplifiers, or phase‑locked loops—the operating point is a periodic orbit in the circuit’s state space.

PSS analysis determines this orbit automatically, providing a concrete numerical baseline for subsequent time‑varying small‑signal studies. Without an accurate periodic operating point, the later analyses would start from an inconsistent or non‑representative state, leading to incorrect predictions of gain, stability, or noise performance.

Small‑signal analyses that rely on the PSS result

Once the periodic operating point is known, designers can explore how small perturbations affect the circuit. Four key analyses are built on the PSS foundation:

All four require the periodic operating point as a starting condition because the small‑signal equations are linearized around a time‑varying operating trajectory, not around a static DC point.

Autonomous vs. driven circuits

PSS is applicable to both autonomous and driven circuits:

• Autonomous circuits generate their own periodic behavior without an external periodic source. Classic examples include crystal oscillators, ring oscillators, and relaxation oscillators. In these cases, the fundamental frequency is not imposed externally; the PSS solver must locate a self‑consistent periodic solution that satisfies the circuit’s nonlinear equations.

• Driven circuits receive a periodic stimulus from an external source, such as a clock, a sinusoidal voltage, or a pulse‑width‑modulated (PWM) driver. The fundamental frequency is typically known a priori (e.g., the clock frequency), and the PSS analysis computes how the circuit responds to that imposed periodic excitation.

The ability to handle both categories makes PSS a versatile tool across a broad spectrum of analog, mixed‑signal, and RF designs.

Why engineers use PSS

1. Efficiency for stiff circuits

When a circuit contains components with vastly different time constants (e.g., a high‑speed switch coupled to a large output filter), a transient simulation may need to run for milliseconds of real time while stepping through picoseconds of simulated time. PSS bypasses that inefficiency by converging directly to the periodic solution.

2. Accurate small‑signal data in the presence of large‑signal dynamics

Standard AC analysis assumes a linear, time‑invariant (LTI) operating point. In circuits where large‑signal switching dominates (e.g., DC‑DC converters), that assumption breaks down. By providing a periodic operating point, PSS enables periodic AC (PAC) and related analyses that capture the interaction between large‑signal switching and small‑signal modulation.

3. Noise prediction under realistic switching conditions

Noise generated by a switching node is modulated by the switching waveform. Periodic noise (Pnoise) analysis, built on the PSS result, yields a realistic noise spectrum that accounts for the periodic nature of the circuit, which is impossible to obtain from a pure DC operating point.

4. Design of control loops and filters

Control‑loop designers need to know how the loop gain varies with frequency while the power stage is actively switching. Periodic transfer function (PXF) analysis supplies that information, allowing accurate stability margins to be calculated.

5. Verification of periodicity

By running a transient simulation first, engineers can verify that the circuit indeed settles into a periodic regime. PSS then provides a mathematically exact representation of that regime, which can be archived and reused for parametric sweeps or Monte‑Carlo studies.

Typical workflow and example

Below is a representative workflow that illustrates how PSS fits into a real‑world design cycle. The example uses a buck DC‑DC converter, a common switching power supply, but the steps are generic to any periodic circuit.

1. Define the circuit topology – Include the input source, MOSFET switch, diode, inductor, output capacitor, and control circuitry.

1. Run a DC operating point (OP) analysis – Establish the bias currents and voltages for the control loop.

1. Perform a transient simulation – Apply the intended duty cycle and let the simulation run for several switching periods. Observe the voltage across the inductor and the output capacitor. Verify that after a few microseconds the waveforms repeat with a constant shape.

1. Select the fundamental frequency – For a buck converter the switching frequency (e.g., 500 kHz) becomes the fundamental frequency for PSS.

1. Configure the PSS analysis –

• Set the fundamental frequency to 500 kHz.
• Choose the number of harmonics to retain (e.g., 5–7) to capture the essential waveform details.
• Indicate whether the circuit is autonomous (if the control loop generates the switching) or driven (if an external clock forces the switch).

1. Run the PSS solver – The engine iteratively solves the nonlinear equations until the periodic operating point converges. The solver’s runtime is largely independent of the LC filter’s time constant, which may be several microseconds.

1. Validate the result – Compare the PSS waveforms with the last few periods of the transient simulation. They should match closely, confirming that the periodic solution is accurate.

1. Launch periodic small‑signal analyses –

• PAC to obtain the gain and phase of the control‑to‑output transfer function while the switch is toggling.
• Pnoise to predict output voltage noise caused by the switching action.
• PXF to extract the loop gain for stability assessment.

1. Parametric sweeps – Vary component values (e.g., inductor L, capacitor C, or duty cycle) while re‑running PSS automatically. Because each PSS run is fast, designers can explore large design spaces efficiently.

1. Monte‑Carlo analysis – Combine PSS with statistical variations of component values to assess yield and robustness under manufacturing tolerances.

Through this workflow, the designer obtains high‑fidelity, frequency‑aware data that would be impractical to gather with pure transient or DC analyses.

Practical tips for successful PSS runs