## Improving Convergence in Eta Designer

*Michael Seeman, Ph.D., July 30, 2018*

**Eta Designer uses a periodic steady-state (PSS) simulation as a basis for many of its analyses. Efficiency, loss and control simulations all rely on a valid PSS simulation result to function correctly. In complex and isolated power converter topologies, the PSS simulation may not converge easily, and additional efforts must be made to utilize Eta Designer's advanced analysis functions. This article provides some guidelines and advice to achieve steady-state convergence in your designs.**

Eta Designer is built on a high-performance simulation engine integrated with a periodic steady-state (PSS) simulator. The PSS engine is used for all simulations except for the *Transient Simulation*, so ensuring convergence is necessary to fully utilize Eta Designer.

Eta Designer runs a background PSS simulation each time the schematic is changed. This simulation is run with the current Design Variable values. You can check the status of this simulation via the icon at the bottom-right corner of the schematic window. A green check mark means the PSS simulation converged successfully, while a yellow triangle indicates that convergence was not achieved. A red error symbol shows there's a problem with the circuit preventing any simulation from running.

Efficiency and loss simulations require PSS convergence across the operating points. For example, the Magnetics Designer examines the design variable corners to find the worst-case magnetics loss. If the PSS solution cannot be obtained at any relevant corners, the Magnetics Designer shows an error.

Examine the convergence of the PSS simulation over design variable corners by using the *Steady-State Waveforms* simulation. Use the *Sweep Variables* function to examine the convergence at the desired extremes of the design variables. If a corner does not converge, a *Transient Simulation* may highlight the problem

The following sections highlight the two PSS methods used in Eta Designer, one targeting speed and accuracy, and the other targeting converters where the first method does not converge, as well as AC/DC topologies. Finally, methods to improve the convergence of the PSS simulation are discussed.

Before examining the details of the two periodic steady-state algorithms, it is worthwhile discussing what a PSS algorithm is. Power converters operate in a periodic or pseudo-periodic manner, and with constant input and output conditions, should reach steady-state operation once any transients decay. Efficiency, control stability and other converter metrics are typically calculated at steady state.

Periodic steady-state methods find the set of initial state values such that after one (or multiple) periods, the set of state values equals the set of initial state values. A state can represent a capacitor voltage, inductor current, or controller integrator state.

Eta Designer uses two periodic steady-state algorithms. The first one is a dynamic optimization method to find the steady-state conditions without relying on an extended transient simulation. It will find the set of initial conditions as close as possible to the stationary solution through an iterative approach. The second method uses a transient simulation, in conjunction with a stable controller, over a user-specified number of periods to reach an approximation of steady-state conditions.

The desired PSS method can be selected from the *Steady-State Algorithm* dialog in the *Simulation Settings* submenu in the *Simulate* menu. The *Long-Term Transient* method allows you to specify the period of the converter as well as choosing the number of periods to simulate before convergence is assumed.

Convergence of the first PSS algorithm requires finding the set of initial states which satisfies the PSS requirement. While convergence is straighforward and very fast with simple converters such as buck, boost and other non-isolated topologies, flyback and other isolated topologies can often cause trouble. Leakage inductances and high frequency ringing can cause large state variable changes in a short amount of time. This ringing creates a system sensitivity that may prevent convergence with the first PSS method. The next section discusses how to mitigate the effect of this ringing. If the ringing cannot be mitigated, the second PSS method can be used.

The second method uses a transient response to find the PSS conditions. While the second method requires a stable controller and assumes convergence in a certain number of periods, it does not require the high-frequency states to converge. The second method is best when the first method does not converge. Because AC-DC and DC-AC circuits have two independent periods, corresponding to the line and switching frequencies, the first method will not work on them, and the second method must be used. Controllers which utilize jitter or introduce jumps in control (e.g. valley-mode switching controllers) will also require the second method.

The second PSS method, as it relies on the transient response, will usually not achieve the same level of accuracy as the first PSS method. Without that level of PSS accuracy, simulations such as the control loop analyses may be inaccurate.

There are several methods to improve the convergence of the standard (first) PSS method. Transformerless topologies will typically converge just fine, but transformer-based isolated converters will run into challenges. Let's examine these challenges. First, let's examine the converter design itself. We'll run a *Transient Simulation* to understand the topology, including examining output stability and the appearance of the switching nodes.

Figure 3 shows waveforms from a flyback converter, including the switch-node voltage and the primary- and secondary-side current. When the primary-side transistor is off, the leakage inductance rings with the primary-side transistor's output capacitance, which appears throughout the circuit. In addition, when the primary-side transistor is on, the primary-side current rings at a very high frequency. The PSS algorithm has challenges including this lightly-damped, high-frequency ringing in its solution, and will often fail.

Real power converters will have much better damped waveforms due to unmodeled parasitics. If there is still unnecessary ringing in the real circuit, R-C snubbers will often be added to damp or eliminate this ringing. Adding similar snubbers to the simulation will likewise improve the ringing and help convergence.

Figure 5 shows the improved switching waveforms from the added snubbers. As the ringing subsides before the end of the switching period, the PSS solution will be less impacted by these high-frequency modes. Without these high-frequency modes, the PSS algorithms can converge faster and more reliably.

The converter's control scheme also impacts the PSS convergence. In all but the simplest of converters, a stable controller is necessary to reach convergence. Besides global stability, including oscillation or exponential runaway, the controller must prevent subharmonic oscillation in current-mode controllers. Finally, certain controller types, like valley switch-mode controllers, may introduce quantized switching periods or other types of jitter into the system. In these cases, the PSS simulation will not be able to find a stationary steady-state solution.

Examine the controller stability by running a long transient response and sweeping the input and output parameters to the extremes to understand the stability over corners. If any loss of regulation, jitter, or subharmonic oscillation occurs, modify the controller to fix these issues before continuing. If a controller cannot achieve a steady-state solution due to jitter or quantized stepping, a simpler controller can be substituted to study the converters efficiency.

PSS convergence is important for many of the features in Eta Designer. For example, efficiency and control simulations rely on finding the steady-state operation of the converter. However, finding the PSS solution of many converters is difficult, due to several causes. By understanding methods of improving PSS convergence, you can modify your Eta Designer projects to simulate more reliably.