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Multiports#

Definition: Port

A port in a lumped circuit is any pair of terminals such that the current flowing from one terminal is equal to the current flowing into the other.

Many electronic components can be modeled as ports. When a device has multiple terminals which can be arranged into pairs which satisfy the port condition, we call it a multiport or an \(n\)-port, where \(n\) is the number of ports.

Multiports

Independence#

Definition: Independence

An \(n\)-port is independent if all currents or all voltages (or both) across its terminals are independent of all other electrical elements connected to the same circuit as the \(n\)-port.

We can think of an independent \(n\)-port as a magical box which can force an entire circuit to behave in a way which conforms with the \(n\)-port. Naturally, no existing electronic component can be truly independent. However, under the right conditions, many can be approximated as such within appropriate bounds.

I-V Characteristic#

We analyze every \(n\)-port as a collection of \(n\) one-ports.

Definition: Current Vector

The current vector of an \(n\)-port at time \(t\) is the column vector whose components are the currents through each of its one-ports at \(t\):

\[ \begin{bmatrix}i_1(t) \\ \vdots \\ i_n(t) \end{bmatrix} \]

Currents through Multiport

Notation

\[ \mathbf{i}(t) \qquad \boldsymbol{i}(t) \]

Definition: Voltage Vector

The voltage vector of an \(n\)-port at time \(t\) is the column vector whose components are the voltages across its one-ports at \(t\):

\[ \begin{bmatrix}v_1(t) \\ \vdots \\ v_n(t)\end{bmatrix} \]

Voltages across Multiport

Notation

\[ \mathbf{v}(t) \qquad \boldsymbol{v}(t) \]

When analyzing a port, we are interested in what currents and voltages it can be operated at.

Definition: Current-Voltage Characteristic

The current-voltage characteristic of an \(n\)-port is the set of all admissible voltage and current vector pairs:

\[ \mathcal{F} \subseteq \mathbb{R}^{2n} \]

where any \(\begin{bmatrix} \boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\) represents a state in which the multiport can physically exist.

Example

For example, a one-port could have the following current-voltage characteristic:

\[ \mathcal{F} = \left\{(9 \mathop{\mathrm{V}}, 0.011 \mathop{\mathrm{A}}), (12 \mathop{\mathrm{V}}, 0.024 \mathop{\mathrm{A}}), (9 \mathop{\mathrm{V}}, 0.013 \mathop{\mathrm{A}}), (12 \mathop{\mathrm{V}}, 0.018 \mathop{\mathrm{A}}) \right\} \]

This means that the one-port can function only under one of the following conditions:

  • a voltage of \(9 \mathop{\mathrm{V}}\) and a current of \(0.011 \mathop{\mathrm{A}}\);
  • a voltage of \(12 \mathop{\mathrm{V}}\) and a current of \(0.024 \mathop{\mathrm{A}}\);
  • a voltage of \(9 \mathop{\mathrm{V}}\) and a current of \(0.013 \mathop{\mathrm{A}}\);
  • a voltage of \(12 \mathop{\mathrm{V}}\) and a current of \(0.018 \mathop{\mathrm{A}}\).

Important: Sign Convention

When dealing with I-V characteristics, the standard is to assume the passive sign convention for each of the constituent one-ports. If you are using the active sign convention for whatever reason, always make this clear. Failure to do so can lead to a lot of confusion both for you and for other people.

Important: Time Evolution

In general, the I-V characteristic of a port may change over time, i.e. the set \(\mathcal{F}\) itself may be given by some function of time:

\[ \mathcal{F} = F(t) \]

The most trivial example is that of a twenty year old electronic component. Due to wear and tear, we certainly can't expect it to be operable at exactly the same currents and voltages as it was back when it was first manufactured.

However, there are devices whose I-V characteristic can change much faster such as due to changes in the intensity of the light shining on the component or other factors.

Definition: Time Variance

A multiport is time-variant if its I-V characteristic is different at different times.

A multiport is time-invariant if its I-V characteristic is the same at all times.

We assume that multiports are time-invariant unless explicitly stated otherwise.

Representations#

Since I-V characteristics can be very complicated and dealing with sets directly is tedious, we are interested in finding mathematical formulas which describe the relationships between voltage and current.

Definition: Implicit Representation

An implicit representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is any function \(f: \mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}\) such that

\[ f \left(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix}\right) = \boldsymbol{0} \]

if and only if \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\).

Definition: Parametric Representation

A parametric representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is a functions \(f: \mathbb{R}^n \to \mathbb{R}^{2n}\) such that

\[ \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F} \]

if and only if there exists some \(\boldsymbol{\lambda} \in \mathbb{R}^{n}\) such that

\[ \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} = f(\boldsymbol{\lambda}) \]

There are also the following explicit representations:

Definition: Admittance Representation

An admittance representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is any function \(g: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n\) such that

\[ \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F} \]

if and only if

\[ \boldsymbol{i} = g(\boldsymbol{v}). \]

Definition: Voltage-Controlled Multiport

If the I-V characteristic of a multiport has an admittance representation, then the multiport is said to be voltage-controlled.

Definition: Impedance Representation

An Impedance representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is any function \(r: \mathcal{D} \subseteq \mathbb{R}^n \to \mathbb{R}^n\) such that

\[ \begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F} \]

if and only if

\[ \boldsymbol{v} = r(\boldsymbol{i}). \]

Definition: Current-Controlled Multiport

If the I-V characteristic of a multiport has an impedance representation, then the multiport is said to be current-controlled.

Definition: Hybrid Representation

A hybrid representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is any function \(H: \mathbb{R}^n \to \mathbb{R}^n\) such that

\[\begin{bmatrix}\boldsymbol{v}_a \\ \boldsymbol{i}_b\end{bmatrix} = H\left( \begin{bmatrix}\boldsymbol{i}_a \\ \boldsymbol{v}_b\end{bmatrix} \right),\]

where

\[\boldsymbol{i}_a = \begin{bmatrix} i_1 \\ \vdots \\ i_m\end{bmatrix} \qquad \boldsymbol{v}_a = \begin{bmatrix} v_1 \\ \vdots \\ v_m\end{bmatrix} \qquad \boldsymbol{i}_b = \begin{bmatrix} i_{m+1} \\ \vdots \\ i_n \end{bmatrix} \qquad \boldsymbol{v}_b = \begin{bmatrix} v_{m+1} \\ \vdots \\ v_n\end{bmatrix}\]

for some \(m \in \{1, \dotsc, n\}\).

Definition: Inverse Hybrid Representation

An inverse hybrid representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port is any function \(H': \mathbb{R}^n \to \mathbb{R}^n\) such that

\[\begin{bmatrix}\boldsymbol{i}_a \\ \boldsymbol{v}_b\end{bmatrix} = H'\left(\begin{bmatrix}\boldsymbol{v}_a \\ \boldsymbol{i}_b\end{bmatrix}\right), \]

where

\[\boldsymbol{i}_a = \begin{bmatrix} i_1 \\ \vdots \\ i_m\end{bmatrix} \qquad \boldsymbol{v}_a = \begin{bmatrix} v_1 \\ \vdots \\ v_m\end{bmatrix} \qquad \boldsymbol{i}_b = \begin{bmatrix} i_{m+1} \\ \vdots \\ i_n \end{bmatrix} \qquad \boldsymbol{v}_b = \begin{bmatrix} v_{m+1} \\ \vdots \\ v_n\end{bmatrix}\]

for some \(m \in \{1, \dotsc, n\}\).

Definition: Forwards Transmission Representation

A forwards transmission representation of the I-V characteristic \(\mathcal{F}\) of an \(n\)-port with \(n = 2m\) is any function \(T: \mathbb{R}^n \to \mathbb{R}^n\) such that

\[\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i}\end{bmatrix} \in \mathcal{F} \qquad \iff \qquad \begin{bmatrix}\boldsymbol{v}_1 \\ \boldsymbol{i}_1\end{bmatrix} = T \left( \begin{bmatrix}\boldsymbol{v}_2 \\ -\boldsymbol{i}_2\end{bmatrix} \right),\]

where

\[\begin{bmatrix} \boldsymbol{v}_1 \\ \boldsymbol{i}_1 \end{bmatrix} = \begin{bmatrix}v_1 \\ \vdots \\ v_m \\ i_1 \\ \vdots \\ i_m\end{bmatrix} \qquad \begin{bmatrix} \boldsymbol{v}_2 \\ -\boldsymbol{i}_2 \end{bmatrix} = \begin{bmatrix}v_{m+1} \\ \vdots \\ v_n \\ -i_{m+1} \\ \vdots \\ -i_n\end{bmatrix}.\]

Power#

Definition: Lossless Multiport

A multiport with I-V characteristic \(\mathcal{F}\) is lossless if

\[ \boldsymbol{v} \cdot \boldsymbol{i} = 0 \]

for all \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\).

Definition: Lossy Multiport

A multiport with I-V characteristic \(\mathcal{F}\) is lossy if

\[ \boldsymbol{v} \cdot \boldsymbol{i} \ne 0 \]

for some \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\).

Definition: Passive Multiport

A multiport with I-V characteristic \(\mathcal{F}\) is passive if

\[ \boldsymbol{v} \cdot \boldsymbol{i} \ge 0 \]

for all \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\).

Definition: Active Multiport

A multiport with I-V characteristic \(\mathcal{F}\) is active if

\[ \boldsymbol{v} \cdot \boldsymbol{i} \lt 0 \]

for some \(\begin{bmatrix}\boldsymbol{v} \\ \boldsymbol{i} \end{bmatrix} \in \mathcal{F}\).

Duality#

Definition: Duality

Two \(n\)-ports \(\mathcal{F}\) and \(\mathcal{F}^d\) are dual if there exists a constant \(R_d \in \mathbb{R}\) such that

\[ (\boldsymbol{v}, \boldsymbol{i}) \in \mathcal{F} \iff (\boldsymbol{v}^d, \boldsymbol{i}^d) \in \mathcal{F}^d, \]

where

\[ \boldsymbol{v}^d = R_d \boldsymbol{i} \qquad \boldsymbol{i}^d = \frac{1}{R_d}\boldsymbol{v}. \]