2.1 Resistor

A resistor is a two-terminal passive electronic component that resists the flow of electric current. Simply put, it limits how much current can pass through a circuit. It’s one of the fundamental building blocks in electronics.

The following circuit shows a resistor connected to a voltage source. The xschem simulation file can be found here.

Resistor V-I curve
Figure 2.1.1: Simple circuit of a resistor.

Characteristics

Resistor behavior is governed by Ohm’s Law, which states:

\[V=IR\]


Where:

  • \(V\) is the voltage across the resistor,
  • \(I\) is the current through the resistor,
  • \(R\) is the resistance in ohms (\(Ω\)).

Characteristics curve

The ideal resistor has a linear V-I characteristic, meaning voltage and current are directly proportional. When plotted on a graph:

Resistor V-I curve
Figure 2.1.2: V-I characteristics of a resistor.

Resistance

Resistance (\(R\)) can be modeled by the mathematical equation,

\[R = \frac{\rho A}{L}\]

where:

  • \(R\) is the resistance in ohms (Ω),
  • \(\rho\) is the resistivity of the material in ohm-meters (Ω·m),
  • \(L\) is the length of the conductor in meters (m),
  • \(A\) is the cross-sectional area of the conductor in square meters (m²).

In manufacturing process this equation is the fundamental model to make a resistor.

IC level Implimentation

At the integrated circuit (IC) level, resistors are implemented directly on the silicon chip using materials such as poly-silicon or diffused regions within the silicon wafer. These resistive elements are carefully structured as thin, straight lines or serpentine (zig-zag) shapes to control the resistance value while fitting into a compact area. The fabrication of these resistors is done using standard CMOS (Complementary Metal-Oxide-Semiconductor) processes, which often overlap with the steps used for creating transistors.

Resistor V-I curve
Figure 2.1.3: IC level Implementation of a resistor. [https://www.circuitstoday.com/monolithic-ic]

The sheet Resistance

Sheet resistance is defined by resistance per squre sheet in IC designing.

since,

\[R=\frac{\rho l}{A}\]

making the squre shape \(L=W\) will result with, sheet resistance;

\[R_s=\frac{\rho}{D}\]

where \(D\) is the depth of the resistor. so the sheet resistance will independent of the size of the squre.

then any resistance can be implemented as,

\[R=R_s.\frac{L}{W}\]
Resistor V-I curve
Figure 2.1.4: IC level layout of a resistor. [https://ihp-open-pdk-docs.readthedocs.io/en/latest/]