One-Port Linearity#

Definition: One-Port Linearity

Theorem: Linear One-Port

A one-port is linear if and only if its V-I characteristic can be written as

\[ av + bi + c = 0 \]

Proof

TODO

Graphically, the I-V characteristic of a linear one-port is a straight line:

I-V Linear One-Port

Theorem: Thévenin Equivalent

The Thévenin equivalent of a linear one-port with I-V characteristic \(v = Ri + V\) is a series circuit of an ideal voltage source and a linear resistor with resistance \(R\):

Thévenin Equivalent of Linear One-Port

Proof

TODO

Theorem: Norton Equivalent

The Norton equivalent of a linear one-port with I-V characteristic \(i = Gv + I\) is a parallel circuit of an ideal current source and a linear resistor with conductance \(G\):

Norton Equivalent of Linear One-Port

Proof

TODO

Linearization#

Non-linear one-ports can have very complicated I-V characteristics, which often makes their analysis in the context of a particular circuit difficult. However, if we know that the one-port will only be operated within a small region around a specific point on its I-V characteristic, then it can be approximated very well as a linear one-port.

Definition: Linearization

Suppose we have a non-linear one-port with I-V characteristic \(\mathcal{F}_{\text{non-linear}}\) and a point \((v, i) \in \mathcal{F}_{\text{non-linear}}\).

Linearization is the process of finding a linear one-port whose I-V characteristic \(\mathcal{F}_{\text{linear}}\) resembles \(\mathcal{F}_{\text{non-linear}}\) as much as possible around the point \((v, i)\).

Theorem: Linearization via Implicit Representation

Suppose we have a non-linear one-port with I-V characteristic \(\mathcal{F}_{\text{non-linear}}\) and let \((v, i) \in \mathcal{F}_{\text{non-linear}}\).

If \(\mathcal{F}_{\text{non-linear}}\) has an implicit representation \(f(V, I) = 0\), then the I-V characteristic \(\mathcal{F}_{\text{linear}}\) of the linear one-port which best approximates \(\mathcal{F}_{\text{non-linear}}\) around \((v, i)\) has an implicit representation which can be obtained using \(f\)'s partial derivatives:

\[ \frac{\partial f}{\partial V}(v, i) \cdot (V - v) + \frac{\partial f}{\partial I}(v, i) \cdot (I - i) = 0 \]

Proof

TODO

Theorem: Linearization via Explicit Representation

Suppose we have a non-linear one-port with I-V characteristic \(\mathcal{F}_{\text{non-linear}}\) and let \((v, i) \in \mathcal{F}_{\text{non-linear}}\).

If \(\mathcal{F}_{\text{non-linear}}\) has an explicit representation \(I = G(V)\), then the I-V characteristic \(\mathcal{F}_{\text{linear}}\) of the linear one-port which best approximates \(\mathcal{F}_{\text{non-linear}}\) around \((v, i)\) has an explicit representation which can be obtained using \(G\)'s derivative:

\[ I = G'(v) \cdot (V - v) + i \]

If \(\mathcal{F}_{\text{non-linear}}\) has an explicit representation \(V = R(I)\), then the I-V characteristic \(\mathcal{F}_{\text{linear}}\) of the linear one-port which best approximates \(\mathcal{F}_{\text{non-linear}}\) around \((v, i)\) has an explicit representation which can be obtained using \(R\)'s derivative:

\[ V = R'(i) \cdot (I - i) + v \]

Proof

TODO