Shooing method in PSS analysis

[anhnha]

This is maybe a silly question but I am confused. If you can please help.

I am reading about how PSS analysis work in Cadence tutorial and encountering with Shooing method. I didn't know that method before and therefore I read up some articles about it.

Here is what I understand so far:
Shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem.
For example, with a second-order ordinary differential equation, the boundary will be:

\[y''(t) = f(t, y(t), y'(t)) , \; \; \; y( t_{0}) = y_{0}, \; y( t_{1}) = y_{1}\]

Cadence will guess the initial value:

\[ y'( t_{0}) = a\]

And then it will solve the initial problem:

\[y''(t) = f(t, y(t), y'(t)) , \; \; \; \; \; \; y'( t_{0}) = y_{0}, \; \; y'( t_{0}) = a\]

Let's say that the solution is y(t; a).

Then we compute y(t1 ; a) and compares it with y1. If they are equal => OK, the problem solved, if not then guess another initial value and try again.

Here is my questions relating to how Cadence do the algorithm.

1. How Cadence get the differential equation?
2. How it get the boundary values, y0, y1?

 

[bernhard8]

I'm also trying to understand the engines and methods behind SpectreRF during my master thesis.

ad 1: The differential equations are gathered during normal circuit analysis like DC (Kirchhoff laws). Node matrices are built like in conventional analyses in Spice, Spectre, ...
ad 2: I think it is a kind of guess like in the conventional newton algorithm the initial value x0 has to be selected. Maybe the dc solution can be used as starting point which is considered easy to calculate.

My Question:
I think I understood the harmonic balance method. It is simply a Fourier transformation of the time domain differential equations using the given beat frequency as base period (and all desired harmonics). Nonlinear behaviour is solved in time domain and converted afterwards.

What I'm unsure about is the shooting method. Can somebody answer me if I'm basically right with this?
A series of transient simulations is done using an initial guess as starting point and trying to find a solution which is periodical (period defined by beat period of PSS). The simulations are repeated for all desired harmonics. Amplitude and phase of the investigated harmonic are found basically equivalent to the direct Fourier method like in HB method. This amplitudes and phases are the complex solution for the circuit transfer function of one harmonic frequency (harmonic to the beat frequency).

General Question to PSS: can the beat frequency be seen as a kind of test signal to the circuit? The response is a series of complex transfer functions which define the frequency translations.

I read several Cadence and Kundert documents but did not come to an easy understandable explanation. Maybe somebody could answer to this topic?

 

[BigBoss]

You should have seen " Transient Solution of RLC Circuits by Differential Equations" in the circuit theory.
All differential equations are coming from circuit KCL or KVL matrices and the solution is obtained "State Variables" techniques.For more information seek internet..
Initial values are assumed for each component zero for instance initial current for coils is zero,initial voltage for capacitors is zero so on..
General form for these equations :

\[\frac{\mathrm{dX} }{\mathrm{dt} }=\left[A\right]\left[X\right]+\left[B\right]\left[W\right]\{W}^{'}+{W}^{''}+{W}^{'''}+.... \]

[X] : State Variable Vector
[A] : State Variables Matrix
:Source Vector Coefficient
[W] : Source Vector
[W]' : Source Vector's First Kind Derivative
[W]'' Source Vector's Second Kind Derivative
[W]''' Source Vector's Third Kind Derivative

etc..