Multitone Analysis

The Multitone Analysis is similar to an AC Analysis. Again the response of the system to a small perturbation signal is analysed. However, instead of multiple sinusoidal signals of different frequencies, only one multitone signal is applied. It is composed of several sinusoidal signals and therefore contains all investigated frequencies at once.

The multitone signal is computed as

\[u(t) = \sqrt{\frac{2}{N}} \sum^{N}_{k = 1} \sin \left( 2 \pi k f_b t + \frac{\pi (k - 1)^2}{N} \right) ,\]

where \(N\) is the number of tones and \(f_b\) the base frequency. In PLECS, the user can control the amplitude of the perturbation signal by a factor that is multiplied to \(u(t)\).

Algorithm

The simulation is divided into two phases. It is assumed that the system reaches its steady-state in the first phase of duration \(T_i\). In the second phase of duration \(T_b = 1 / f_b\), the response of the system is recorded for the Fourier analysis.

The Multitone Analysis performs these steps:

  1. Perform an unperturbed simulation of length \(T_i + T_b\). Record the system response during \(T_b\) in \(y_0\).

  2. Perform a simulation of the same length and perturbed by \(u\). Record the system response during \(T_b\) in \(y\).

  3. Compute the Fourier transforms \(U\) of \(u\) and \(Y\) of \(y-y_0\).

  4. Compute the transfer function as \(G = Y / U\).

Remarks

The Multitone Analysis is faster than the AC Analysis because it only needs to compute the response to one signal instead of a set of signals for each frequency. On the other hand, the analysed frequencies are restricted to multiples of the base frequency. Since the Multitone Analysis does not use the Steady-State Analysis, it still works in cases where the Steady-State Analysis fails, provided \(T_i\) is large enough.

Note that the lengths of the \(y_0\) and \(y\) vectors are different in general. To compute the difference \(y-y_0\), the missing values are linearly interpolated.

References

  • S. Boyd, “Multitone signals with low crest factor”, IEEE Transactions of Circuits and Systems, Vol. CAS-33, No. 10, 1986.

  • C. Fernández, P. Zumel, A. Fernández-Herrero, M. Sanz, A. Lázaro, A. Barrado, “Frequency response of switching DC/DC converters from a single simulation in the time domain”, Applied Power Electronics Conference and Exposition (APEC), 2011 Twenty-Sixth Annual IEEE, March 2011.