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Resistors#

The term resistor is a bit misleading because resistance is defined for every electronic component. However, it is usually used to mean an electronic component whose resistance exhibits one of the following specific behaviors.

Linear Resistors#

A linear resistor is an electronic component whose voltage scales linearly with the current flowing through and vice versa.

Theoretical Model#

Definition: Linear Resistor

A linear resistor is a one-port whose static resistance \(R\) and static conductance \(G\) are constant:

\[ V(t) = RI(t) \qquad \text{or} \qquad I(t) = GV(t) \]

Definition: Ohm's Law

The above equations are often called Ohm's law.

I-V of Linear Resistor

Notation

The following symbols are used for linear resistors:

Linear Resistor Symbols

Definition: Ohmic Resistor

An Ohmic resistor is a linear resistor such that \(R, G \gt 0\).

Theorem: Static = Dynamic

Static resistance and dynamic resistance of a linear resistor are always the same, as are static conductance and dynamic conductance.

Note

This is why we just speak of "resistance" and "conductance" when talking about linear resistors.

Proof

TODO

Theorem: Strict Linearity of Linear Resistors

All linear resistors are strictly linear.

Proof

TODO

Theorem: Linear Resistors in Series

If \(n\) linear resistors with resistances \(R_1, \dotsc, R_n\) or conductances \(G_1, \dotsc G_n\) connected in series, then they are equivalent to a single linear resistor with resistance \(R\) or conductance \(G\):

\[ R = \sum_{k=1}^n R_k \qquad \frac{1}{G} = \sum_{k = 1}^n \frac{1}{G_k} \]
Proof

Since they are connected in series, they are equivalent to a single one-port whose current \(I\) is the same, but whose voltage \(V\) is the sum of the voltages \(V_1\) and \(V_2\) across the resistors:

\[ V = V_1 + V_2 \]

We know that \(V_1 = R_1I\) and \(V_2 = R_2 I\) and so

\[ V = R_1 I + R_2 I= (R_1 + R_2)I. \]

Therefore, this equivalent one-port is an linear resistor with resistance \(R_1 + R_2\).

Similarly, since \(I = G_1 V_1\) and \(I = G_2 V_2\), we get that \(V_1 = \frac{1}{G_1}I\) and \(V_2 = \frac{1}{G_2} I\). Therefore,

\[ V = \frac{1}{G_1}I + \frac{1}{G_2}I = \left(\frac{1}{G_1} + \frac{1}{G_2}\right)I \]
\[ I = \frac{1}{\frac{1}{G_1} + \frac{1}{G_2}}V \]

Therefore, this equivalent one-port is an linear resistor with conductance \(G = \frac{1}{\frac{1}{G_1} + \frac{1}{G_2}}\). By taking the reciprocal of \(G\), we obtain the original result:

\[ \frac{1}{G} = \frac{1}{G_1} + \frac{1}{G_2} \]

Theorem: Linear resistors in Parallel

If \(n\) linear resistors with resistances \(R_1, \dotsc, R_n\) or conductances \(G_1, \dotsc G_n\) connected in parallel, then they are equivalent to a single linear resistor with resistance \(R\) or conductance \(G\):

\[ \frac{1}{R} = \sum_{k=1}^n \frac{1}{R_k} \qquad G = \sum_{k = 1}^n G_k \]
Proof

Since they are connected in parallel, they are equivalent to a single one-port whose voltage \(V\) is the same, but whose current \(I\) is the sum of the currents \(I_1\) and \(I_2\) flowing through the resistors:

\[ I = I_1 + I_2 \]

We know that \(I_1 = G_1U\) and \(I_2 = G_2 V\) and so

\[ I = G_1 V + G_2 V = (G_1 + G_2)V. \]

Therefore, this equivalent one-port is a linear resistor with conductance \(G_1 + G_2\).

Similarly, since \(V = R_1 I_1\) and \(V = R_2 I_2\), we get that \(I_1 = \frac{1}{R_1}V\) and \(I_2 = \frac{1}{R_2} V\). Therefore,

\[ I = \frac{1}{R_1}V + \frac{1}{R_2}V = \left(\frac{1}{R_1} + \frac{1}{R_2}\right)V \]
\[ V = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}I \]

Therefore, this equivalent one-port is an linear resistor with resistance \(R = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}\). By taking the reciprocal of \(R\), we obtain the original result:

\[ \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} \]

Theorem: Duality of Linear Resistors

Linear resistors are dual to linear resistors.

Proof

TODO

Implementation#

Ohmic resistors are really easy to create physically because they require nothing more than an insulating material.

Example: Non-Ohmic Linear Resistor via NIC

A linear resistor with negative resistance / conductance can be implemented by connecting an Ohmic resistor to a negative immittance converter:

Negative Linear Resistor

Piecewise Linear Resistors#

Piecewise linear resistors are electronic components which behave like linear resistors but with different resistance / conductance depending on the range of the voltage or current in which they are operated.

Theoretical Model#

Definition: Piecewise Linear Resistor

A piecewise linear resistor is a one-port whose I-V characteristic can be represented as

TODO

Concave Resistors#

Theoretical Model#

Definition: Concave Resistor

A concave resistor is a piecewise linear resistor whose I-V characteristic can be represented as

\[i = \begin{cases}0 & \text{if } v \lt V \\ Gv & \text{if } v \ge V\end{cases}\]

for some \(G, V \in \mathbb{R}\).

I-V Concave Resistor

Notation

The following symbol is used for concave resistors.

Concave Resistor Symbol

Implementation#

Example: Concave Resistor via Ideal Diode

A concave resistor can be realized using a series circuit of an ideal diode, a voltage source with constant voltage \(V\) and an Ohmic resistor with conductance \(G\):

Concave Resistor Implementation

Convex Resistors#

Theoretical Model#

Definition: Convex Resistor

A convex resistor is a piecewise linear resistor whose I-V characteristic can be represented as

\[v = \begin{cases}0 & \text{if } i \lt I \\ Rv & \text{if } i \ge I\end{cases}\]

for some \(R, V \in \mathbb{R}\).

I-V Convex Resistor

Notation

The following symbol is used for convex resistors.

Convex Resistor Symbol

Implementation#

Example: Convex Resistor via Ideal Diode

A convex resistor can be realized using a parallel circuit of an ideal diode, a current source with constant current \(I\) and an Ohmic resistor with resistance \(R\):

Convex Resistor Implementation