Negative Immittance Converters#

Negative immittance converters (NICs) are circuits capable of transferring the voltage and current of an electronic component, whilst reversing the direction of one or the other and potentially scaling them:

A negative immittance converter which reverses the direction of current, whilst retaining the direction of voltage, is known as a current inversion negative immittance converter (INIC);
A negative immittance converter which reverses the direction of voltage, whilst retaining the direction of current, is known as a voltage inversion negative immittance converter (VNIC).

Negative immittance converters are extremely useful because they can be used for many purposes:

Reversing the current or voltage direction results in the reversal of the sign of the resistance / conductance (both static and dynamic) of the electronic component. This allows us to physically realize resistors with negative resistance.

TODO

Theoretical Model#

Definition: Negative Immittance Converter

A negative immittance converter is a two-port whose I-V characteristic is

\[\left\vert\begin{aligned}v_1 &= -k v_2 \\ i_1 &= -\frac{1}{k} i_2 \end{aligned}\right.\]

for some constant $k \in \mathbb{R}_{\ne 0}$ known as the conversion ratio.

A negative immittance converter is:

an INIC when $k \lt 0$;
a VNIC when $k \gt 0$.

Theorem: Strict Linearity

Every negative immittance converter is strictly linear.

Proof

TODO

Theorem: Implicit Representations

If a negative immittance converter has a conversion ratio $k$, then

\[\begin{bmatrix}1 & k \\ 0 & 0\end{bmatrix}\boldsymbol{v} + \begin{bmatrix}0 & 0 \\ k & 1\end{bmatrix}\boldsymbol{i} = \boldsymbol{0}\]

is an implicit representation of it.

Proof

TODO

Theorem: Explicit Representations

The negative immittance converter has neither an admittance representation nor an impedance representation.

It has the following hybrid representation and inverse hybrid representation:

\[\begin{bmatrix}v_1 \\ i_2\end{bmatrix} = \boldsymbol{H} \begin{bmatrix}i_1 \\ v_2\end{bmatrix} \qquad \boldsymbol{H} = \begin{bmatrix}0 & -k \\ -k & 0\end{bmatrix}\]

\[\begin{bmatrix}i_1 \\ v_2\end{bmatrix} = \boldsymbol{H}' \begin{bmatrix}v_1 \\ i_2\end{bmatrix} \qquad \boldsymbol{H}' = \begin{bmatrix}0 & -\frac{1}{k} \\ -\frac{1}{k} & 0\end{bmatrix}\]

It has the following forwards transmission representation and backwards transmission representation:

\[\begin{bmatrix}v_1 \\ i_1\end{bmatrix} = \boldsymbol{T} \begin{bmatrix} v_2 \\ -i_2 \end{bmatrix} \qquad \boldsymbol{T} = \begin{bmatrix}-k & 0 \\ 0 & \frac{1}{k}\end{bmatrix}\]

\[\begin{bmatrix}v_2 \\ i_2\end{bmatrix} = \boldsymbol{T}'\begin{bmatrix}v_1 \\ -i_1\end{bmatrix} \qquad \boldsymbol{T}' = \begin{bmatrix}-\frac{1}{k} & 0 \\ 0 & k\end{bmatrix}\]

Proof

TODO

Theorem: Activity of NICs

Negative immittance converters are active.

Proof

TODO

Theorem: Antireciprocity of NICs

Negative immittance converter are always antireciprocal.

Proof

TODO

Theorem: Symmetry Condition for NICs

A negative immittance converter is symmetrical if and only if $|k| = 1$.

Proof

TODO

Implementation#

Example: INIC from Nullor

A INIC can be constructed using a nullor and two identical Ohmic resistors:

INIC from Nullor

The voltage across the nullator is zero and so Kirchhoff's voltage law gives us the following:

\[v_1 = v_2\]

Similarly, applying Kirchhoff's voltage law to the nullator and the two Ohmic resistors, we get that $R i_1 = R i_2$. We therefore have:

\[i_1 = i_2\]

We have obtained the equations of a negative immittance converter

\[\left\vert\begin{aligned}v_1 &= -k v_2 \\ i_1 &= -\frac{1}{k} i_2 \end{aligned}\right.\]

with $k = -1$.

TODO FIX

Example: VNIC from Nullor

A VNIC be constructed using a nullor and two identical Ohmic resistors:

VNIC from Nullor

The voltage across the nullator is zero and so Kirchhoff's voltage law gives us the following:

\[v_1 = -v_2\]

Similarly, there is no current flowing through the nullator and so the two Ohmic resistors and the norator form a parallel circuit. This immediately makes them a one-port and we get the following:

\[i_2 = -i_1\]

We therefore obtain the equations of a negative immittance converter
$$
\left\vert\begin{aligned}v_1 &= -k v_2 \ i_1 &= -\frac{1}{k} i_2 \end{aligned}\right.$$

with $k = +1$.

TODO FIX

Example: NIC from Op-Amp

A negative immittance converter can be constructed using an ideal operational amplifier and Ohmic resistors:

NIC from Ideal Op-Amp

As long as the ideal op-amp is operated in its linear region, the above circuit behaves like a negative immittance converter with $k = -1$.

To ensure that the ideal op-amp is indeed operated in its linear region, we need to have $v_1 - R i_1 \in [-V_{\text{sat}}; +V_{\text{sat}}]$.

We can see this by analyzing the network.

Linear region:

When the ideal op-amp is operated in its linear region, we know that it is equivalent to a nullor. In this case, it would be connected in such a way so as to act as a NIC with $k = -1$

Saturation regions:

When the ideal op-amp is operated outside its linear region, we know that $v_d \ne 0$. The differential voltage is given by $v_d = v_+ - v_- = v_2 - v_1$.

According to Kirchhoff's voltage law, the relationships between the port voltages and currents still hold:

\[\left\vert\begin{aligned}v_1 &= v_{\text{out}} + R i_1 \\ v_2 &= v_{\text{out}} + R i_2\end{aligned}\right.\]

By substituting these into the expression for $v_d$, we can determine the saturation conditions based on the currents:

\[v_d = (v_{\text{out}} + R i_2) - (v_{\text{out}} + R i_1) = R(i_2 - i_1)\]

When the ideal op-amp is operated in its negative saturation region, we have $v_d \lt 0$ and $v_{\text{out}} = -V_{\text{sat}}$:

\[R(i_2 - i_1) \lt 0 \implies i_2 \lt i_1\]

In this state, the voltage at port 1 is clamped relative to the current:

\[v_1 = -V_{\text{sat}} + R i_1\]

By contrast, when the ideal op-amp is operated in its positive saturation region, we have $v_d \gt 0$ and $v_{\text{out}} = +V_{\text{sat}}$:

\[R(i_2 - i_1) \gt 0 \implies i_2 \gt i_1\]

In this state, the voltage at port 1 is clamped as follows:

\[v_1 = +V_{\text{sat}} + R i_1\]