Components defined in the frequency domain

The time-domain simulation of components defined in the frequency-domain can be performed using an inverse Fourier transformation of the Y-parameters of the component (giving the impulse response) and an adjacent convolution with the prior node voltages (or branch currents) of the component.

This requires a memory of the node voltages and branch currents for each component defined in the frequency-domain. During a transient simulation the time steps are not equidistant and the maximum required memory length $T_{end}$ of a component may not correspond with the time grid produced by the time step control (see section 6.2.3 on page ) of the transient simulation. That is why an interpolation of exact values (voltage or current) at a given point in time is necessary.

Components defined in the frequency-domain can be divided into two major classes.

• Components with frequency-independent (non-dispersive) delay times and with or without constant losses.
• Components with frequency-dependent (dispersive) delay times and losses.

Components with frequency-independent delay times

Components with constant delay times are a special case. The impulse response corresponds to the node voltages and/or branch currents at some prior point in time optionally multiplied with a constant loss factor.

Voltage controlled current source

With no constant delay time the MNA matrix entries of a voltage controlled current source is determined by the following equations according to the node numbering in fig. 9.8 on page .

$$I_2 = -I_3 = G\cdot \left(V_1 - V_4\right)$$ (6.108)

The equations yield the following MNA entries during the transient analysis.

$$\begin{bmatrix}0 & 0 & 0 & 0\\ +G & 0 & 0 & -G\\ -G & 0 & 0 & +G\... ...\\ V_3\\ V_4 \end{bmatrix} = \begin{bmatrix}I_1\\ I_2\\ I_3\\ I_4 \end{bmatrix}$$ (6.109)

With a constant delay time $\tau$ eq. (6.108) rewrites as

$$I_2\left(t\right) = -I_3\left(t\right) = G\cdot \left(V_1\left(t -\tau\right) - V_4\left(t -\tau\right)\right)$$ (6.110)

which yields the following MNA entries during the transient analysis.

$$\begin{bmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 ... ...-\tau\right) - V_4\left(t -\tau\right)\right)\\ I_4\left(t\right) \end{bmatrix}$$ (6.111)

Voltage controlled voltage source

The MNA matrix entries of a voltage controlled voltage source are determined by the following characteristic equation according to the node numbering in fig. 9.10 on page .

$$V_2 - V_3 = G\cdot\left(V_4 - V_1\right)$$ (6.112)

This equation yields the following augmented MNA matrix entries with a single extra branch equation.

$$\begin{bmatrix}0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0... ...\\ J_1\\ \end{bmatrix} = \begin{bmatrix}I_1\\ I_2\\ I_3\\ I_4\\ 0 \end{bmatrix}$$ (6.113)

When considering an additional constant time delay $\tau$ eq. (6.112) must be rewritten as

$$V_2\left(t\right) - V_3\left(t\right) = G\cdot\left(V_4\left(t - \tau\right) - V_1\left(t - \tau\right)\right)$$ (6.114)

This representation requires a change of the MNA matrix entries which now yield the following matrix equation.

$$\begin{bmatrix}0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0... ...t\left(V_4\left(t - \tau\right) - V_1\left(t - \tau\right)\right) \end{bmatrix}$$ (6.115)

Current controlled current source

With no time delay the MNA matrix entries of a current controlled current source are determined by the following equations according to the node numbering in fig. 9.9 on page .

$$I_2 = -I_3 = G\cdot I_1 = -G\cdot I_4$$ (6.116)

$$V_1 = V_4$$ (6.117)

These equations yield the following MNA matrix entries using a single extra branch equation.

$$\begin{bmatrix}0 & 0 & 0 & 0 & 1/G\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & ... ...\\ J_1\\ \end{bmatrix} = \begin{bmatrix}I_1\\ I_2\\ I_3\\ I_4\\ 0 \end{bmatrix}$$ (6.118)

When additional considering a constant delay time $\tau$ eq. (6.116) must be rewritten as

$$I_2\left(t\right) = -I_3\left(t\right) = G\cdot I_1\left(t -\tau\right) = -G\cdot I_4\left(t -\tau\right)\\$$ (6.119)

Thus the MNA matrix entries change as well yielding

$$\begin{bmatrix}0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 ... ...\right)\\ +G\cdot J_1\left(t -\tau\right)\\ I_4\left(t\right)\\ 0 \end{bmatrix}$$ (6.120)

Current controlled voltage source

The MNA matrix entries for a current controlled voltage source are determined by the following characteristic equations according to the node numbering in fig. 9.11 on page .

$$V_2 - V_3 = G\cdot I_2 = -G\cdot I_3$$ (6.121)

$$V_1 = V_4$$ (6.122)

These equations yield the following MNA matrix entries.

$$\begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1\\ 0... ..._2\\ \end{bmatrix} = \begin{bmatrix}I_1\\ I_2\\ I_3\\ I_4\\ 0\\ 0 \end{bmatrix}$$ (6.123)

With an additional time delay $\tau$ between the input current and the output voltage eq. (6.121) rewrites as

$$V_2\left(t\right) - V_3\left(t\right) = G\cdot I_2\left(t - \tau\right) = -G\cdot I_3\left(t - \tau\right)\\$$ (6.124)

Due to the additional time delay the MNA matrix entries must be rewritten as follows

$$\begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 ... ...\right)\\ I_4\left(t\right)\\ 0\\ G\cdot J_1\left(t - \tau\right) \end{bmatrix}$$ (6.125)

Ideal transmission line

The A-parameters of a transmission line (see eq (9.196) on page ) are defined in the frequency domain. The equation system formed by these parameters write as

$$\textrm{I.}\;\;\;\;V_1 = V_2\cdot\cosh{\left(\gamma\cdot l\right)} + I_2\cdot Z_L\cdot\sinh{\left(\gamma\cdot l\right)}$$ (6.126)

$$\textrm{II.}\;\;\;\;I_1 = V_2\cdot\dfrac{1}{Z_L}\sinh{\left(\gamma\cdot l\right)} + I_2\cdot \cosh{\left(\gamma\cdot l\right)}$$ (6.127)

Figure 6.9: ideal transmission line

Applying $\textrm{I} + Z_L\cdot \textrm{II}$ and $\textrm{I} - Z_L\cdot \textrm{II}$ to the above equation system and using the following transformations

$$\cosh{x} + \sinh{x} = \dfrac{e^x + e^{-x}}{2} + \dfrac{e^x - e^{-x}}{2} = e^x$$ (6.128)

$$\cosh{x} - \sinh{x} = \dfrac{e^x + e^{-x}}{2} - \dfrac{e^x - e^{-x}}{2} = e^{-x}$$ (6.129)

yields

$$V_1 = V_2\cdot e^{-\gamma\cdot l} + Z_L\cdot \left(I_1 + I_2\cdot e^{-\gamma\cdot l}\right)$$ (6.130)

$$V_2 = V_1\cdot e^{-\gamma\cdot l} + Z_L\cdot \left(I_2 + I_1\cdot e^{-\gamma\cdot l}\right)$$ (6.131)

whereas $\gamma$ denotes the propagation constant $\alpha + j\beta$, $l$ the length of the transmission line and $Z_L$ the line impedance.

These equations can be transformed from the frequency domain into the time domain using the inverse Fourier transformation. The frequency independent loss $\alpha \ne f\left(\omega\right)$ gives the constant factor

$$A = e^{-\alpha\cdot l}$$ (6.132)

The only remaining frequency dependent term is

$$e^{-j\beta\cdot l} = e^{-j\omega\cdot\tau} \;\;\;\; \textrm{with}... ...ta = \dfrac{\omega}{v_{ph}} = \dfrac{\omega}{c_0} = \dfrac{\omega\cdot \tau}{l}$$ (6.133)

which yields the following transformation

$$f\left(\omega\right)\cdot e^{-\gamma\cdot l} = A\cdot f\left(\ome... ...t)\cdot e^{-j\omega\cdot\tau} \Longleftrightarrow A\cdot f\left(t - \tau\right)$$ (6.134)

All the presented time-domain models with a frequency-independent delay time are based on this simple transformation. It can be applied since the phase velocity $v_{ph} \ne f\left(\omega\right)$ is not a function of the frequency. This is true for all non-dispersive transmission media, e.g. air or vacuum. The given transformation can now be applied to the eq. (6.130) and eq. (6.131) defined in the frequency-domain to obtain equations in the time-domain.

The length $T_{end}$ of the memory needed by the ideal transmission line can be easily computed by

$$T_{end} = \tau = \dfrac{l}{v_{ph}} = \dfrac{l}{c_0}$$ (6.135)

whereas $c_0$ denotes the speed of light in free space (since there is no dielectric involved during transmission) and $l$ the physical length of the transmission line.

The MNA matrix for a lossy (or lossless with $\alpha = 0$) transmission line during the transient analysis is augmented by two new rows and columns in order to consider the following branch equations.

$$V_1\left(t\right) = Z_L\cdot I_1\left(t\right) + A\cdot\left( Z_L\cdot I_2\left(t -\tau\right) + V_2\left(t -\tau\right)\right)$$ (6.136)

$$V_2\left(t\right) = Z_L\cdot I_2\left(t\right) + A\cdot\left( Z_L\cdot I_1\left(t -\tau\right) + V_1\left(t -\tau\right)\right)$$ (6.137)

Thus the MNA matrix entries can be written as

$$\begin{bmatrix}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & -Z_L & 0\\... ...1\left(t -\tau\right) + Z_L\cdot J_1\left(t -\tau\right)\right)\\ \end{bmatrix}$$ (6.138)

with $A$ denoting the loss factor derived from the constant (and frequency independent) line attenuation $\alpha$ and the transmission line length $l$.

$$A = e^{-\tfrac{\alpha}{2}\cdot l}$$ (6.139)

Ideal 4-terminal transmission line

The ideal 4-terminal transmission line is a two-port as well. It differs from the 2-terminal line as shown in fig. 6.5.1 in two new node voltages and branch currents.

Figure 6.10: ideal 4-terminal transmission line

The differential mode of the ideal 4-terminal transmission line can be modeled by modifying the branch eqs. (6.136) and (6.137) of the 2-terminal line which yields

$$V_1\left(t\right) - V_4\left(t\right) = Z_L\cdot I_1\left(t\right... ...\left(t -\tau\right) + V_2\left(t -\tau\right) - V_3\left(t -\tau\right)\right)$$ (6.140)

$$V_2\left(t\right) - V_3\left(t\right) = Z_L\cdot I_2\left(t\right... ...\left(t -\tau\right) + V_1\left(t -\tau\right) - V_4\left(t -\tau\right)\right)$$ (6.141)

Two more conventions must be indroduced

$$I_1\left(t\right) = -I_4\left(t\right)$$ (6.142)

$$I_2\left(t\right) = -I_3\left(t\right)$$ (6.143)

which is valid for the differential mode (i.e. the odd mode) of the transmission line and represents a kind of current mirror on each transmission line port.

According to these consideration the MNA matrix entries during transient analysis are

$$\begin{bmatrix}. & . & . & . & 1 & 0\\ . & . & . & . & 0 & 1\\ . ... ...4\left(t -\tau\right) + Z_L\cdot J_1\left(t -\tau\right)\right)\\ \end{bmatrix}$$ (6.144)

Logical devices

The analogue models of logical (digital) components explained in section 10.6 on page do not include delay times. With a constant delay time $\tau$ the determining equations for the logical components yield

$$u_{out}\left(t\right) = f\left(V_{in,1}\left(t - \tau\right), V_{in,2}\left(t - \tau\right), \ldots\right)$$ (6.145)

With the prior node voltages $V_{in,n}\left(t - \tau\right)$ known the MNA matrix entries in eq. (10.268) can be rewritten as

$$\begin{bmatrix}.&.&.& 1\\ .&.&.& 0\\ .&.&.& 0\\ 1 & 0 & 0 & 0 \en... ...I_{1}\left(t\right)\\ I_{2}\left(t\right)\\ u_{out}\left(t\right) \end{bmatrix}$$ (6.146)

during the transient analysis. The components now appear to be simple linear components. The derivatives are not anymore necessary for the Newton-Raphson iterations. This happens to be because the output voltage does not depend directly on the input voltage(s) at exactly the same time point.

Components with frequency-dependent delay times and losses

In the general case a component with $P$ ports which is defined in the frequency-domain can be represented by the following matrix equation.

$$\begin{bmatrix}Y_{11} & Y_{12} & \ldots & Y_{1P}\\ Y_{21} & Y_{22... ...{P} \end{bmatrix} = \begin{bmatrix}I_{1}\\ I_{2}\\ \vdots\\ I_{P} \end{bmatrix}$$ (6.147)

This matrix representation is the MNA representation during the AC analysis. With no specific time-domain model at hand the equation

$$\begin{bmatrix}Y\left(j\omega\right) \end{bmatrix} \cdot \begin{b... ...omega\right) \end{bmatrix} = \begin{bmatrix}I\left(j\omega\right) \end{bmatrix}$$ (6.148)

must be transformed into the time-domain using a Fourier transformation.

The convolution integral

The multiplication in the frequency-domain is equivalent to a convolution in the time-domain after the transformation. It yields the following matrix equation

$$\begin{bmatrix}H\left(t\right) \end{bmatrix} * \begin{bmatrix}V\left(t\right) \end{bmatrix} = \begin{bmatrix}I\left(t\right) \end{bmatrix}$$ (6.149)

whereas $H\left(t\right)$ is the impulse response based on the frequency-domain model and the $*$ operator denotes the convolution integral

$$H\left(t\right) * V\left(t\right) = \int^{+\infty}_{-\infty} H\left(\tau\right)\cdot V\left(t-\tau\right) d\tau$$ (6.150)

The lower bound of the given integral is set to zero since both the impulse response as well as the node voltages are meant to deliver no contribution to the integral. Otherwise the circuit appears to be unphysical. The upper limit should be bound to a maximum impulse response time $T_{end}$

$$H\left(t\right) * V\left(t\right) = \int^{T_{end}}_{0} H\left(\tau\right)\cdot V\left(t-\tau\right) d\tau$$ (6.151)

with

$$H\left(\tau\right) = 0 \;\;\; \forall \;\;\; \tau > T_{end}$$ (6.152)

Since there is no analytic represention for the impulse response as well as for the node voltages eq. (6.151) must be rewritten to

$$H\left(n\cdot\Delta t\right) * V\left(n\cdot\Delta t\right) = \su... ...} H\left(k\cdot\Delta t\right)\cdot V\left(\left(n-k\right)\cdot\Delta t\right)$$ (6.153)

with

$$\Delta t = \dfrac{T_{end}}{N}$$ (6.154)

whereas $N$ denotes the number of samples to be used during numerical convolution. Using the current time step $t = n\cdot\Delta t$ it is possible to express eq. (6.153) as

$$I\left(t\right) = H\left(0\right)\cdot V\left(t\right) \underbrac... ...=1} H\left(k\cdot\Delta t\right)\cdot V\left(t -k\cdot\Delta t\right)}_{I_{eq}}$$ (6.155)

With $G = H\left(0\right)$ the resulting MNA matrix equation during the transient analysis gets

$$\begin{bmatrix}G \end{bmatrix} \cdot \begin{bmatrix}V\left(t\righ... ...gin{bmatrix}I\left(t\right) \end{bmatrix} - \begin{bmatrix}I_{eq} \end{bmatrix}$$ (6.156)

This means, the component defined in the frequency-domain can be expressed with an equivalent DC admittance $G$ and additional independent current sources in the time-domain. Each independent current source at node $r$ delivers the following current

$$I_{eq_r} = \sum^{P}_{c=1}\sum^{N-1}_{k=1} H_{rc}\left(k\cdot\Delta t\right)\cdot V_c\left(t -k\cdot\Delta t\right)$$ (6.157)

whereas $V_c$ denotes the node voltage at node $c$ at some prior time and $H_{rc}$ the impulse response of the component based on the frequency-domain representation. The MNA matrix equation during transient analysis can thus be written as

$$\begin{bmatrix}G_{11} & G_{12} & \ldots & G_{1P}\\ G_{21} & G_{22... ...bmatrix} - \begin{bmatrix}I_{eq_1}\\ I_{eq_2}\\ \vdots\\ I_{eq_P} \end{bmatrix}$$ (6.158)

Frequency- to time-domain transformation

With the number of samples $N$ being a power of two it is possible to use the Inverse Fast Fourier Transformation (IFFT). The transformation to be performed is

$$Y\left(j\omega\right) \Leftrightarrow H\left(t\right)$$ (6.159)

The maximum impulse response time of the component is specified by $T_{end}$ requiring the following transformation pairs.

$$Y\left(j\omega_i\right) \Leftrightarrow H\left(t_i\right) \;\;\;\; \textrm{with} \;\;\;\; i = 0, 1, 2, \ldots, N-1$$ (6.160)

with

$$t_i = 0, \Delta t, 2\cdot\Delta t, \ldots, \left(N-1\right)\cdot\Delta t$$ (6.161)

$$\omega_i = 0, \dfrac{1}{T_{end}}, \dfrac{1}{T_{end}}, \ldots, \dfrac{N/2}{T_{end}}$$ (6.162)

The frequency samples in eq. (6.162) indicate that only half the values are required to obtain the appropriate impulse response. This is because the impulse response $H\left(t\right)$ is real valued and that is why

$$Y\left(j\omega\right) = Y^*\left(-j\omega\right)$$ (6.163)

The maximum frequency considered is determined by the maximum impulse response time $T_{end}$ and the number of time samples $N$.

$$f_{max} = \dfrac{N/2}{2\pi\cdot T_{end}} = \dfrac{1}{4\pi\cdot\Delta t}$$ (6.164)

It could prove useful to weight the Y-parameter samples in the frequency-domain by multiplying them with an appropriate windowing function (e.g. Kaiser-Bessel).

Implementation considerations

For the method presented the Y-parameters of a component must be finite for $f\rightarrow 0$ as well as for $f\rightarrow f_{max}$. To obtain $G = H\left(0\right)$ the Y-parameters at $f=0$ are required. This cannot be ensured for the general case (e.g. for an ideal inductor).