**5. Conclusions**

By considering this, we can easily deduce that the size of the **X**

**^** (*tE*,*<sup>i</sup>*

**4. Efficiency of the partitioned time-frequency technique**

(35), as well as for the components of **H**(**X**

66 Modeling and Simulation in Engineering Sciences

value *x* ^ *<sup>v</sup>*(*tE*,*<sup>i</sup>*

Considerable Jacobian **J**(**X**

With the state variable **X**

the resulting Jacobian **J**(**X**

contain the dc sensitivity of **H**(**X**

savings when computing the solution.

used by SPICE-like commercial simulators).

in the *tC* fast carrier time scale.

Given that the *k* = 0 order Fourier coefficient *Xv*,0(*tE*,*<sup>i</sup>*

**^** (*tE*,*<sup>i</sup>*

**^** (*tE*,*<sup>i</sup>*

**^** (*tE*,*<sup>i</sup>* **^** (*tE*,*<sup>i</sup>*

)) matrix size reductions will also be achieved with this technique.

)) matrix size reduction, it is possible to avoid dealing with large

**^** (*tE*,*<sup>i</sup>* ).

)) corresponding to latent blocks of the circuit.

) is exactly the same as the constant *tC* time

)) vector size reductions, as also

be significantly decreased, as well as the total number of equations in the HB system of (23). Furthermore, another crucial aspect is that we are no longer forced to carry out the conversion between time and frequency domains for the latent state variables expressed in the form of

), components of the HB system of (23) that have no dependence on active state

)) to the latent components of **X**

**^** (*tE*,*<sup>i</sup>*

**^** (*tE*,*<sup>i</sup>*

variables will not be required for any direct or inverse Fourier transformation operations.

) and the error function **H**(**X**

Indeed, by considering the latency of state variables in some parts of the circuit, some blocks of the Jacobian matrix (26) are simply reduced to 1×1 scalar elements. These scalar elements

linear systems in the iterations of (25). Thus, a less computationally expensive Newton-Raphson iterative solver is required to solve (23). In conclusion, partitioning the circuit into active and latent sub-circuits (blocks) can lead us to significant computation and memory

The effectiveness of the multitime ETHB technique is nowadays widely recognized by the RF and microwave community. The efficiency of the partitioned time-frequency simulation technique described in the previous section was also already established, as a consequence of the considerable reductions in the computational effort required to obtain the numerical solution of several RF circuits with distinct topologies and levels of complexity [16]. Even so, a brief comparison between this method, the previous state-of-the-art multitime ETHB and a conventional univariate time-step integration scheme (SPICE-like simulation), is included in this section. This will help the reader to get a perception of the potential of the partitioned hybrid technique. For that, we considered the RF mixer (frequency translation device) depicted in **Figure 3** as the illustrative application example. The circuit was simulated in MATLAB with three different techniques: (i) the partitioned time-frequency simulation technique, (ii) the multitime ETHB, and (iii) the Gear-2 multistep method [5] (a time-step integrator commonly

Numerical computation times for simulations in the [0, 1.0 *μ*s] and [0, 10.0 *μ*s] intervals are presented in **Table 1**. For simplicity, in the hybrid time-frequency techniques we assumed a uniform grid in the *tE* slow time scale (we have chosen *hE* = 10 ns as the step size in that dimension) and we considered *K* = 11 as the maximum harmonic order for the HB evaluations

) vector defined by (21) will

In this chapter, we have presented a partitioned time-frequency numerical technique espe‐ cially designed for the efficient simulation of RF circuits operating in a periodic fast time scale and an aperiodic slow time scale. This technique can be viewed as a wise combination of multitime ETHB based on a multivariate formulation, with a conventional univariate timestep integration scheme. With this technique fast changing (active) state variables are com‐ puted in a bivariate mixed time-frequency domain, whereas slowly varying (latent) state variables are evaluated in the natural one-dimensional time domain. By partitioning the circuits into active and latent parts and exploiting the fact there is no obligation to perform conversion between time and frequency for latent blocks, considerable reductions in the computational effort can be achieved without compromising the accuracy of the results. Although the speedups obtained with the simulation of the illustrative application example presented in Section 4 are already notable, it must be noted that higher efficiency gains should be expected when simulating RF networks containing a number of latent blocks larger than the active ones.
