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In electronics, a steady state (S.S.) occurs in a circuit or network when all transients have died away. It is an equilibrium condition that occurs as the effects of transients are no longer important. DC steady state is that state, if it exists, of a circuit described by a differential equation where the transient(s) have decayed away--typically the solution as time goes to infinity. Steady state determination is an important topic today, because many design specifications in a power electronic system are given in terms of the system’s steady-state characteristics. Periodic steady-state solution is also a prerequisite for small signal dynamic modeling. Steady-state analysis is therefore an indispensable component of the design process. In some types of circuits, such as lightly damped systems this integration could extend over many periods making the computation costly. The same happens for the so-called "stiff" circuits, and for circuits were a high frequency carrier is modulated by a much slower signal (e.g. Cellphone mixers): such circuits, for example, require a computational-costly integration over a wide span of time, with much smaller integration step. Faster numerical methods than the classical brute force integration are available to find the steady state (periodic, quasiperiodic) of non-autonomous and autonomous circuits(such as, the period T is not known a priori). Such methods are often referred as fast steady state algorithms.
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Types
Steady state algorithms can be sorted into:
- TD Time domain algorithms (Time domain sensitivities,Shooting)
- FD Frequency domain algorithms (Harmonic Balance)
Harmonic balance methods, are the best choice for most microwave circuits excited with sinusoidal signals (e.g. mixers, power amplifiers).
Time domain methods
Time domain methods can be further divided into:
- One step methods (Time domain Sensitivities)
- Iterative methods (Shooting methods).
One step methods require Derivatives to compute the S.S; whenever those are not readily available at hand or at the output of the simulator involved, iterative methods come into focus. SPICE, for example, doesn't output derivatives, and it's not readily suitable to be the simulator of choice to compute SS thru time domain sensitivities. There's the slight option to rebuild derivatives numerically, but iterative methods are often preferred.
Steady-state response
A steady-state response is the electrical response of a system at equilibrium. The steady-state response does not necessarily mean the response is a fixed value. An AC power supply has no fixed voltage on the output but the output is steady (a voltage of a fixed frequency and amplitude). The steady-state response follows the transient response. It is also sometimes referred to as the forced response in systems involving damping, though this is not an entirely accurate description. The forced response of a system has both transient and steady-state components. In aerospace engineering the steady-state response is used in conjunction with the theory governing aircraft flight controls as well as for topics such as aeroacoustics, vibrations, and nonlinear dynamics. In flight control theory, for example, the steady-state response of a system can be thought of in terms of deflections of the aerodynamic control surfaces such as ailerons, elevators, and rudders. For example, if a pilot were to pull back on the yoke to pitch the aircraft up (by deflecting the elevators up) the response would be a plot of the aircraft's pitch as a function of time. This plot typically looks sinusoidal damping to a constant value. This constant value is the steady-state response -- it is the response of the aircraft after a long period of time has passed (mathematically at infinity) such that the transient response (the initial oscillations after the input) no longer has an effect.
Examples
Many commercial and non-commercial simulators embed a fast steady state algorithm: For example:
See also
- Sinusoidal response
- Stiff circuits
- Harmonic Balance
- Time domain sensitivities
- Shooting methods
- Transient response
Further Reading
- Jan Ogrodsky - Circuit Simulation and Algorithms. CRC Press
