Controlled sources

The models of the controlled sources contain the transfer factor $G$. It is complex because of the delay time $T$ and frequency $f$.

$$\underline{G} = G\cdot e^{j\omega T} = G\cdot e^{j\cdot 2\pi f\cdot T}$$ (9.162)

During a DC analysis (frequency zero) it becomes real because the exponent factor is unity.

Voltage controlled current source

The voltage-dependent current source (VCCS), as shown in fig. 9.8, is determined by the following equation which introduces one more unknown in the MNA matrix.

Figure 9.8: voltage controlled current source

$$I_{out} = G\cdot\left(V_{1} - V_{4}\right) \quad \rightarrow \quad V_{1} - V_{4} - \frac{1}{G}\cdot I_{out} = 0$$ (9.163)

The new unknown variable $I_{out}$ must be considered by the four remaining simple equations.

$$I_{1} = 0 \quad I_{2} = I_{out} \quad I_{3} = -I_{out} \quad I_{4} = 0$$ (9.164)

And in matrix representation this is:

$$\begin{bmatrix}.&.&.&.& 0\\ .&.&.&.& 1\\ .&.&.&.& -1\\ .&.&.&.& 0... ...\\ I_{4}\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$$ (9.165)

As you can see the last row which has been added by the VCCS represents the determining equation (9.166). The additional right hand column in the matrix keeps the system consistent.

When pivotising the above MNA stamp (9.168) the additional row and column can be saved ensuring $G$ keeps finite (the pivot element must be non-zero). Both representations are equivalent. If $G$ is zero the below representation must be used.

$$\begin{bmatrix}0&0&0&0\\ G&0&0&-G\\ -G&0&0&G\\ 0&0&0&0 \end{bmatr... ...\\ I_{3}\\ I_{4}\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ \end{bmatrix}$$ (9.166)

The scattering matrix of the voltage controlled current source writes as follows ($\tau$ is time delay).

$$S_{11} = S_{22} = S_{33} = S_{44} = 1$$ (9.167)

$$S_{12} = S_{13} = S_{14} = S_{23} = S_{32} = S_{41} = S_{42} = S_{43} = 0$$ (9.168)

$$S_{21} = S_{34} = -2\cdot G\cdot \exp\left(-j\omega\tau\right)$$ (9.169)

$$S_{24} = S_{31} = 2\cdot G\cdot \exp\left(-j\omega\tau\right)$$ (9.170)

Current controlled current source

The current-dependent current source (CCCS), as shown in fig. 9.9, is determined by the following equation which introduces one more unknown in the MNA matrix.

Figure 9.9: current controlled current source

$$V_{1} - V_{4} = 0$$ (9.171)

The new unknown variable $I_{out}$ must be considered by the four remaining simple equations.

$$I_{1} = +\frac{1}{G}\cdot I_{out} \quad I_{2} = I_{out} \quad I_{3} = -I_{out} \quad I_{4} = -\frac{1}{G}\cdot I_{out}$$ (9.172)

And in matrix representation this is:

$$\begin{bmatrix}.&.&.&.& \frac{1}{G}\\ .&.&.&.& 1\\ .&.&.&.& -1\\ ... ...\\ I_{4}\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$$ (9.173)

The scattering matrix of the current controlled current source writes as follows ($\tau$ is time delay).

$$S_{14} = S_{22} = S_{33} = S_{41} = 1$$ (9.174)

$$S_{11} = S_{12} = S_{13} = S_{23} = S_{32} = S_{42} = S_{43} = S_{44} = 0$$ (9.175)

$$S_{21} = S_{34} = -G\cdot \exp\left(-j\omega\tau\right)$$ (9.176)

$$S_{24} = S_{31} = G\cdot \exp\left(-j\omega\tau\right)$$ (9.177)

Voltage controlled voltage source

The voltage-dependent voltage source (VCVS), as shown in fig. 9.10, is determined by the following equation which introduces one more unknown in the MNA matrix.

Figure 9.10: voltage controlled voltage source

$$V_{2} - V_{3} = G\cdot \left(V_{1} - V_{4}\right) \quad \rightarrow \quad V_{1}\cdot G - V_{2} + V_{3} - V_{4}\cdot G = 0$$ (9.178)

The new unknown variable $I_{out}$ must be considered by the four remaining simple equations.

$$I_{1} = 0 \quad I_{2} = -I_{out} \quad I_{3} = I_{out} \quad I_{4} = 0$$ (9.179)

And in matrix representation this is:

$$\begin{bmatrix}.&.&.&.& 0\\ .&.&.&.& -1\\ .&.&.&.& 1\\ .&.&.&.& 0... ...\\ I_{4}\\ 0\\ \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}$$ (9.180)

The scattering matrix of the voltage controlled voltage source writes as follows ($\tau$ is time delay).

$$S_{11} = S_{23} = S_{32} = S_{44} = 1$$ (9.181)

$$S_{12} = S_{13} = S_{14} = S_{22} = S_{33} = S_{41} = S_{42} = S_{43} = 0$$ (9.182)

$$S_{21} = S_{34} = G\cdot \exp\left(-j\omega\tau\right)$$ (9.183)

$$S_{24} = S_{31} = -G\cdot \exp\left(-j\omega\tau\right)$$ (9.184)

Current controlled voltage source

The current-dependent voltage source (CCVS), as shown in fig. 9.11, is determined by the following equations which introduce two more unknowns in the MNA matrix.

Figure 9.11: current controlled voltage source

$$V_{1} - V_{4} = 0$$ (9.185)

$$V_{2} - V_{3} = G\cdot I_{in} \quad \rightarrow \quad V_{2} - V_{3} - I_{in}\cdot G = 0$$ (9.186)

The new unknown variables $I_{out}$ and $I_{in}$ must be considered by the four remaining simple equations.

$$I_{1} = I_{in} \quad I_{2} = -I_{out} \quad I_{3} = I_{out} \quad I_{4} = -I_{in}$$ (9.187)

The matrix representation needs to be augmented by two more new rows (for the new unknown variables) and their corresponding columns.

$$\begin{bmatrix}.&.&.&.& 1 & 0\\ .&.&.&.& 0 & -1\\ .&.&.&.& 0 & 1\... ..._{4}\\ 0\\ 0 \end{bmatrix} = \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}$$ (9.188)

The scattering matrix of the current controlled voltage source writes as follows ($\tau$ is time delay).

$$S_{14} = S_{23} = S_{32} = S_{41} = 1$$ (9.189)

$$S_{11} = S_{12} = S_{13} = S_{22} = S_{33} = S_{42} = S_{43} = S_{44} = 0$$ (9.190)

$$S_{21} = S_{34} = \frac{G}{2}\cdot \exp\left(-j\omega\tau\right)$$ (9.191)

$$S_{24} = S_{31} = -\frac{G}{2}\cdot \exp\left(-j\omega\tau\right)$$ (9.192)