4. BLA-based DCA with S-parameters

The previous expressions can be used in a DCA on system-level simulations, where every sub-circuit is represented by a SISO system. In actual electronic circuits however, a port-based representation of the sub-blocks has to be used to represent the terminal impedances and to include the forward and reverse gain of each sub-circuit in the circuit. In the remainder of this paper, S-parameters will be used to represent the behaviour of the different circuit blocks. Similar expressions can be obtained for the Y and Z parameters, but this is considered to be outside of the scope of this paper.

The reasoning in this section is very similar to the one detailed in previous section but, instead of working with SISO BLA, each of the sub-circuits is described by a MIMO BLA.

Figure 4.1 shows the general circuit under test for the DCA. There are $N$ non-linear sub-circuits embedded in a package. The whole circuit is excited by different-phase multisines $R$ and the output of the circuit is terminated in a load impedance.

Figure 4.1 The circuit under test will consist of $N$ non-linear sub-circuits embedded in a linear package. The whole circuit is excited by a reference signal $R$. The goal of the DCA is split the distortion in the output wave $B_{t}$ into its contributions.

The steady-state port voltages and currents of the sub-circuits are measured and transformed into waves using the classical expression [23]:

where $V_{i}$ is the port voltage and $I_{i}$ is the port current flowing into the sub-circuit port. $Z_{0}$ is a user-chosen reference impedance. The $A$ and $B$ waves at the $p_{n}$ ports of the $n^{\mathrm{th}}$ sub-circuit are gathered in vectors, giving:

The relation between $\mathbf{A}_{\left[n\right]}$ and $\mathbf{B}_{\left[n\right]}$ is given by the MIMO BLAs $\mathbf{S}_{\mathbf{A}_{\left[n\right]}\rightarrow\mathbf{B}_{\left[n\right]}}^{\mathrm{BLA}}$:

in which $\mathbf{D}_{\left[n\right]}$ is the vector of distortion sources. Determining the MIMO BLAs is more complex than what has been done for the SIMO procedure described in Section 2. The algorithm needed is described in the following section. For now, assume the MIMO BLAs to be known. All the different BLAs are gathered in a block diagonal matrix, similarly to what was done in (3.4):

The total number of ports of all the sub-circuits is denoted by $P$. The vector of distortion sources $\mathbf{D}\in\mathbb{C}^{P\times1}$ is again noise-like, so the covariance matrix $\mathbf{C}_{\mathbf{D}}\!=\!\mathbb{E}\left\{ \mathbf{D}\mathbf{D}^{\mathsf{H}}\right\}$ is used to describe it. Determining $\mathbf{C}_{\mathbf{D}}$ is done in the same way as explained in Section 3 and equation (3.7).

The distortion at the output of the system is defined by considering the BLA from the reference multisine to the output wave $B_{t}$:

The goal of the DCA is to split the power in $D_{t}$ as a sum of contributions from $\mathbf{C}_{\mathbf{D}}$. Classical papers on wave-based circuit and noise analysis have dealt with this problem already [24] [25], and describe how to determine a row vector $\mathbf{T}_{\mathrm{out}}$ that is used to refer $\mathbf{C}_{\mathbf{D}}$ to the output wave (See Appendix 3). This results in the following formula:

This expression can be re-written in a similar way as in (3.8) to obtain a list of direct distortion contributions and correlation distortion contributions.

In circuits, each port of each sub-circuit will create several distortion contributions: a single direct contribution and some correlation contributions. The number of contributions can therefore rise quickly, especially in fully differential, complex, circuits. In a circuit with $P$ ports, there are $\frac{1}{2}P\left(P-1\right)$ distortion contributions to the output.

If the amount of contributions is too large to be easily tractable and interpretable, the different contributions of a single sub-circuit can be combined into a single contribution by simply summing the contributions of each of the ports of one sub-circuit. The covariances can also be combined in the same way. Combining the contributions of each stage reduces the amount of contributions for $N$ sub-circuits to $\frac{1}{2}N\left(N-1\right)$. If this amount of contributions is still too large for an easy interpretation of the results, the contributions of several sub-circuits can be combined into one contribution of a larger sub-circuit. It is a clear advantage that the BLA-based DCA can easily be applied hierarchically as this allows to zoom in selectively on the most contributing parts of the circuit, while leaving the other sub-circuits aggregated at a higher level of abstraction.