02 · Introductory

Resistance and conductance

Resistance measures opposition to current; conductance is its reciprocal and often makes parallel reasoning easier.

75–100 minIntroductory

In this lesson

Use ohms and siemens correctly.
Convert R to G and back.
Interpret large/small resistance physically.
Perform order-of-magnitude checks.

Starting question

What does it physically mean for one resistor to be “large” and another to be “small”?

Resistance measures how much terminal voltage is required per unit current in the linear model. Conductance expresses the same behaviour from the reciprocal viewpoint: how much current results per unit voltage.

02.1Resistance and conductance are reciprocal descriptions

R = V/I

G = 1/R

I = GV

Resistance is measured in ohms; conductance in siemens:

1 Ω = 1 V/A

1 S = 1 A/V = 1/Ω

A large R means a given voltage produces little current. A large G means a given voltage produces a large current. They are not two independent component properties.

02.2Build an order-of-magnitude map

RGI at 1 V

1 Ω1 S1 A

1 kΩ1 mS1 mA

1 MΩ1 µS1 µA

This table is worth internalizing. At a fixed 1 V, multiplying R by 1000 divides I and G by 1000.

Example. For R=4.7 kΩ,

G = 1/4700 S ≈ 213 µS

I(1 V) ≈ 213 µA

02.3Physical bridge: geometry and material

For a uniform conductor of length L and constant cross-sectional area A made from a material described by resistivity ρ under the relevant conditions, the familiar macroscopic relation is

R = ρ L/A

The formula captures three intuitions: a longer path gives larger resistance; a larger cross-section gives smaller resistance; the material enters through ρ.

This is a physical bridge, not a new rule needed for every circuit problem. Packaged resistors achieve their specified R through material, geometry and fabrication details that the lumped symbol intentionally hides.

02.4Real resistance has tolerance and temperature dependence

A component marked 10 kΩ is nominally 10 kΩ; manufacturing tolerance defines a permitted range around that value. A ±1% resistor, for example, may lie between

9.90 kΩ ≤ R ≤ 10.10 kΩ

Over a limited temperature range, a common first-order model is

R(T) ≈ R(T₀)[1 + α(T−T₀)]

where α is a component/material temperature coefficient valid around the reference condition. The approximation and α are not universal; real datasheets specify how a particular resistor family behaves.

02.5Open and short are ideal limiting models

R → ∞ ⇒ G → 0

R → 0 ⇒ G → ∞

At fixed finite voltage, the first limit drives current toward zero and models an open circuit. The second limit is the ideal short-circuit model: zero voltage can coexist with finite current determined by the rest of the circuit.

02.6Practice for mastery

Find G for 2.2 kΩ.
A resistor has G=5 mS. Find R and the current at 2 V.
A uniform wire doubles in length with the same material and area. What happens to R?
The same wire keeps its length but doubles its cross-sectional area. What happens to R?
Give the resistance range of a 47 kΩ ±5% resistor.

Solutions

1. G≈0.455 mS.

2. R=200 Ω and I=10 mA.

3. R doubles.

4. R halves.

5. 44.65 kΩ to 49.35 kΩ.