Resistor

Electrical resistance is a characteristic of a given object, measured in ohms ($\Omega =\frac{V}{A}$). We determine the resistance between any two points by applying a potential difference and measuring the resulting current, given by Ohm’s law. $R=\frac{V}{i}$ In electrical circuits, we have components called resistors.

Circuit analysis

Some tips and tricks for resistor identification:

• Resistors that have been shorted are connected to the same node at both ends.
  • We eliminate shorted resistors from our analysis.
• For resistors in parallel, they share both nodes in common.
• For resistors in series:
  • They have one node $x$ in common that cannot be output nodes.
  • Only two resistors connected to $x$.
• We can safely eliminate dangling resistors from our analysis. Usually this happens when we try to calculate the open circuit voltage at a port.

Equivalence

Resistors in series have the same current. The equivalent resistance is: $R_{eq}={\sum }_{i=1}^n{R}_i$ Resistors in parallel have the same voltage. $\frac{1}{R_{eq}}={\sum }_{i=1}^n\frac{1}{R_i}$ Observe that if $R_2$ is far greater than $R_1$, we can roughly have $R_{eq}\approx R_2$.

For circuits with controlled sources, we can’t find equivalent resistance the same way (by turning off sources). We can instead assume an external $v=1V$ and measure $i$, then $R_{eq}=\frac{v}{i}$.

Input and output resistance

Input resistance is the specific resistance that a specific circuit might present to an input signal. Figuring out what it might be is an important task, because it gives some idea of the circuit’s performance. Amplifiers should have a high input resistance.

Sometimes this is an obvious thing to find (could literally be the first resistor at the beginning of the circuit). Sometimes this requires extra work. The output resistance could literally be the last resistor over which $v_o$ is defined. A formula we can use is:

$$R_i=\frac{v_i}{i_i}$$

Whether this actually holds is not clear. There are answers in Sedra/Smith that literally do not follow this or skip steps to the point that the input resistance comes out of nowhere. We should turn off DC sources.

I find that for amplifier circuits, if a resistor network is ultimately connected to ground it is included in the input resistance. This holds for op amps. For MOSFET amplifiers, we know $R_i=\infty$ because $i_G=0$.

For BJT amplifiers, it’s worth also knowing the resistance reflection rule, where the input looking into the base is:

$$R_B=(\beta +1)\sum R_E+r_{\pi }$$

Looking into each individual node of the BJT, the small-signal input resistance is given by the formulas described in the small signal page.

Electromagnetism

Resistance depends only on the physical properties of the material (resistivity), the length, and the cross-sectional area. A derivation comes from the idea of a linear resistor:

$$i={\iint }_S(-\hat{z})\frac{\sigma V_0}{h}\cdot (-\hat{z})dxdy$$

where the $\hat{z}$ unit vector is perpendicular to the ends, $h$ is the distance between the ends, and $V_0$ is the applied voltage. Then $\vec{E}=V_0/h$:

$$i=\frac{\sigma V_0}{h}\iint dxdy=\frac{\sigma A}{h}{V}_0$$

We define the resistance as:

$$R=\frac{h}{\sigma A}=\frac{V_0}{i}$$ $$R=\frac{{\int }_{+}^{-}\vec{E}\cdot d\vec{l}}{{\iint }_S\sigma \vec{E}\cdot d\vec{s}}$$

In problems, we start with the potential and current integrals (no derivation above!).

See also

• Conductance, the reciprocal of resistance