4.3 Resistance

Definition of resistance

Resistance quantifies how strongly a material opposes the flow of electric current. It's measured in ohms (Ω) and plays a central role in determining how current and voltage behave throughout a circuit. Whenever current flows through a resistor, electrical energy gets converted into heat, which is why resistance also matters for understanding energy dissipation.

Ohm's law

Ohm's law is the most important relationship in basic circuit analysis. It states that the voltage across a conductor is directly proportional to the current flowing through it:

$V = IR$

where $V$ is voltage (in volts), $I$ is current (in amps), and $R$ is resistance (in ohms). If you know any two of these quantities, you can solve for the third. This law holds for ohmic materials, meaning materials whose resistance stays constant regardless of the applied voltage. Most metals at constant temperature behave this way, but not all devices do (more on that below).

Resistance vs. conductance

Conductance (G) is simply the flip side of resistance. It measures how easily current flows rather than how much the material opposes it. The unit is the siemen (S), and the relationship is:

$G = \frac{1}{R}$

Metals like copper have high conductance (low resistance), while insulators like rubber have extremely low conductance (high resistance). Conductance becomes especially handy when analyzing parallel circuits, since conductances in parallel just add together directly.

Factors affecting resistance

Three main factors determine a component's resistance: what it's made of, its temperature, and its physical dimensions. Knowing how each factor contributes lets you predict and control resistance in circuit design.

Material properties

A material's resistivity ($\rho$) describes its intrinsic tendency to resist current, independent of shape or size.

• Metals like copper ($\rho \approx 1.7 \times 10^{-8}$ Ω·m) and silver have very low resistivity because they contain abundant free electrons that move easily.
• Insulators like rubber and glass have extremely high resistivity because their electrons are tightly bound to atoms.
• Semiconductors like silicon and germanium fall in between. Their resistivity can be tuned over a wide range through doping (adding impurity atoms).

Temperature dependence

For most metals, resistance increases with temperature. As the material heats up, atoms vibrate more vigorously, and these vibrations scatter the flowing electrons more frequently. This is called a positive temperature coefficient of resistance (TCR), and for metals the relationship is roughly linear over moderate temperature ranges.

Semiconductors often behave the opposite way: heating them generates more free charge carriers, so resistance decreases with temperature (negative TCR). Superconductors are the extreme case, dropping to exactly zero resistance below a critical temperature.

Geometry and dimensions

Even for the same material, a longer or thinner conductor has more resistance. The relationship is:

$R = \frac{\rho L}{A}$

• $\rho$ = resistivity of the material (Ω·m)
• $L$ = length of the conductor
• $A$ = cross-sectional area

Think of it like water flowing through a pipe: a longer pipe offers more resistance to flow, and a wider pipe offers less. Double the length, double the resistance. Double the cross-sectional area, halve the resistance.

Types of resistors

Resistors are passive components designed to provide a specific amount of resistance in a circuit. They come in several varieties depending on the application.

Fixed vs. variable resistors

• Fixed resistors have a set resistance value that doesn't change. Common types include carbon film, metal film, and wirewound resistors.
• Variable resistors let you adjust resistance. A potentiometer has three terminals and is used for things like volume knobs. A rheostat uses two terminals to control current.
• Photoresistors (light-dependent resistors) change resistance based on how much light hits them, with resistance dropping as light intensity increases.

Linear vs. non-linear resistors

A linear resistor obeys Ohm's law: its resistance stays constant no matter what voltage you apply, so its $V$-vs-$I$ graph is a straight line.

A non-linear resistor has resistance that changes with voltage, current, or some other condition:

• Varistors have high resistance at normal voltages but drop to low resistance during voltage spikes, making them useful for surge protection.
• Thermistors change resistance with temperature and are widely used in temperature-sensing circuits.

Series and parallel resistance

Most real circuits combine resistors in series and parallel configurations. Knowing how to simplify these combinations into a single equivalent resistance is essential for circuit analysis.

Equivalent resistance calculations

Series: Resistances simply add up. Current must pass through each resistor one after another, so every resistor adds to the total opposition.

$R_{eq} = R_1 + R_2 + \cdots + R_n$

Parallel: The reciprocals add. Current has multiple paths, so adding resistors in parallel actually decreases the total resistance.

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$

For the common case of just two resistors in parallel, there's a shortcut:

$R_{eq} = \frac{R_1 \cdot R_2}{R_1 + R_2}$

For more complex networks, simplify step by step: identify groups that are purely in series or purely in parallel, reduce them, and repeat.

Voltage and current distribution

In a series circuit, the same current flows through every resistor, but voltage divides proportionally to each resistance. The voltage divider formula gives the voltage across $R_2$:

$V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2}$

In a parallel circuit, every resistor sees the same voltage, but current splits inversely proportional to resistance (more current takes the path of least resistance). The current divider formula for two parallel resistors gives the current through $R_1$:

$I_1 = I_{total} \cdot \frac{R_2}{R_1 + R_2}$

Notice the subscripts carefully: in the current divider, the current through $R_1$ depends on $R_2$, not $R_1$. That trips people up on exams.

Power dissipation in resistors

When current flows through a resistor, electrical energy is converted into thermal energy (heat). This process is called Joule heating (also called resistive or ohmic heating).

Joule heating

The power dissipated by a resistor can be calculated three equivalent ways, depending on which quantities you know:

$P = IV = I^2R = \frac{V^2}{R}$

All three forms come from combining the power equation $P = IV$ with Ohm's law. Pick whichever version uses the two quantities you already have.

Joule heating is the operating principle behind electric heaters, incandescent light bulbs, and fuses (which melt and break the circuit when current gets too high).

Power rating of resistors

Every resistor has a power rating, the maximum power it can safely dissipate without overheating and failing. Common ratings for small resistors are 1/4 W, 1/2 W, 1 W, and 5 W.

If your circuit requires a resistor to dissipate more power than its rating allows, you need a higher-rated resistor or a heat sink. In high-temperature environments or tightly packed circuit boards, you may need to derate (use a resistor rated for more power than your calculation suggests) to maintain reliability.

Resistivity and conductivity

While resistance depends on a specific object's shape and size, resistivity ($\rho$) and conductivity ($\sigma$) are intrinsic material properties. They describe how well a material conducts regardless of its geometry.

• Resistivity is measured in Ω·m. A low value means the material conducts well.
• Conductivity is the reciprocal: $\sigma = \frac{1}{\rho}$, measured in S/m.

Microscopic origins

At the atomic level, resistivity comes from electrons scattering off obstacles as they move through the material. These obstacles include:

• Lattice vibrations (phonons), which increase with temperature
• Impurities and foreign atoms in the crystal structure
• Crystal defects like vacancies and dislocations

Matthiessen's rule says these contributions are roughly additive: $\rho_{total} \approx \rho_{thermal} + \rho_{impurity} + \rho_{defect}$. The mean free path is the average distance an electron travels between scattering events; longer mean free paths correspond to lower resistivity.

Relationship to resistance

The bridge between the intrinsic property (resistivity) and the measurable quantity (resistance) is the geometry equation:

$R = \frac{\rho L}{A}$

This is the same formula from the geometry section above. It tells you that if you know a material's resistivity and the dimensions of your conductor, you can calculate its resistance directly.

Applications of resistance

Circuit design

• Voltage dividers produce a specific fraction of an input voltage, useful for signal conditioning and creating reference voltages.
• Current-limiting resistors protect sensitive components (like LEDs) from drawing too much current. For example, placing a 330 Ω resistor in series with an LED limits current to a safe level.
• Pull-up and pull-down resistors hold digital logic inputs at a defined HIGH or LOW state when no other signal is driving them.
• Feedback resistors in op-amp circuits set the gain and bandwidth of the amplifier.

Electrical measurements

• Shunt resistors are low-value, precision resistors placed in series with a circuit. You measure the voltage drop across them and use Ohm's law to calculate the current.
• Voltmeters use very high internal resistance to minimize the current they draw, avoiding disturbance to the circuit being measured.
• Strain gauges are thin resistive elements bonded to a surface. When the surface deforms, the gauge stretches, changing its resistance. That resistance change is proportional to the mechanical strain.

Temperature sensors

• Resistance Temperature Detectors (RTDs) use the predictable resistance-vs-temperature behavior of metals (often platinum) to measure temperature with high accuracy and linearity.
• Thermistors offer much higher sensitivity than RTDs, making them ideal for precise measurements over narrower temperature ranges.

Both types can suffer from self-heating: the measurement current itself warms the sensor, introducing error. Circuit designers keep the sensing current small to minimize this effect.

Superconductors vs. normal conductors

Zero resistance phenomenon

Below a material-specific critical temperature ($T_c$), superconductors lose all DC electrical resistance. This happens because electrons form Cooper pairs that move through the lattice without scattering. A current started in a superconducting loop will flow indefinitely with no energy input.

Superconductors also exhibit the Meissner effect: they expel magnetic fields from their interior, behaving as perfect diamagnets.

Critical temperature

$T_c$ varies widely. Conventional superconductors like niobium have $T_c$ values around 9 K, requiring liquid helium for cooling. High-temperature superconductors (certain copper-oxide ceramics called cuprates) have $T_c$ values above 77 K, meaning they can be cooled with liquid nitrogen, which is far cheaper and more practical. The search for a room-temperature superconductor remains one of the most active areas in materials science.

Quantum effects in resistance

At very small scales (nanometers), classical models of resistance break down and quantum mechanics takes over.

Quantum tunneling

Quantum mechanics allows electrons to pass through thin potential barriers that classical physics says they shouldn't be able to cross. This is quantum tunneling. In modern transistors with insulating layers only a few atoms thick, tunneling causes unwanted leakage currents. On the other hand, tunneling is deliberately exploited in devices like tunnel diodes and scanning tunneling microscopes.

Ballistic transport

When a conductor is so small that the electron mean free path exceeds the device dimensions, electrons travel through without scattering. This is ballistic transport. In this regime, resistance no longer depends on the conductor's length; instead, it's determined by the quantum of conductance at the contacts. Ballistic transport has been observed in carbon nanotubes, graphene, and semiconductor nanowires.

Measurement techniques

Four-point probe method

The standard two-probe measurement includes the resistance of the leads and contacts themselves, which can be a problem when measuring low resistances. The four-point probe method solves this by using separate pairs of probes for current injection and voltage measurement. Since negligible current flows through the voltage probes, contact and lead resistance don't affect the reading.

This technique is especially useful for characterizing thin films and semiconductor wafers. The Van der Pauw method extends the four-point approach to samples of arbitrary shape.

Wheatstone bridge

A Wheatstone bridge measures an unknown resistance by balancing it against known resistances. Four resistors are arranged in a diamond (bridge) configuration with a galvanometer across the middle.

When the bridge is balanced, no current flows through the galvanometer, and the unknown resistance can be calculated from the known values. This null method is very precise because it doesn't depend on the accuracy of the galvanometer itself. Variations include the Kelvin bridge (for very low resistances) and various AC bridges (for measuring impedance).

Resistance in semiconductors

Doping effects

Pure (intrinsic) semiconductors have relatively few free charge carriers at room temperature, giving them high resistance. Doping introduces impurity atoms to dramatically change this:

• N-type doping (e.g., adding phosphorus to silicon) provides extra electrons, increasing conductivity.
• P-type doping (e.g., adding boron to silicon) creates "holes" (missing electrons that act as positive charge carriers), also increasing conductivity.

The higher the doping concentration, the lower the resistance. This tunability is what makes semiconductors so useful for building transistors, diodes, and integrated circuits.

Temperature dependence

Semiconductors typically show a negative temperature coefficient: resistance decreases as temperature rises, because thermal energy frees more charge carriers. This is the opposite of metals.

At very low temperatures, carriers can "freeze out" onto their donor or acceptor atoms, causing resistance to spike. At high temperatures, intrinsic carrier generation dominates and resistance drops. Heavily doped semiconductors behave more like metals at moderate temperatures, since they already have so many carriers that the thermal contribution is relatively small.

Resistance in AC circuits

Impedance concept

In AC circuits, opposition to current flow is described by impedance (Z), not just resistance. Impedance has two parts:

$Z = R + jX$

• $R$ is the resistance (real part), which dissipates energy as heat.
• $X$ is the reactance (imaginary part), which stores and releases energy. Reactance comes from capacitors ($X_C$) and inductors ($X_L$).

The magnitude of impedance is $|Z| = \sqrt{R^2 + X^2}$, and the phase angle between voltage and current is $\theta = \arctan(X/R)$.

Frequency dependence

At higher frequencies, AC current tends to flow near the surface of a conductor rather than through its full cross-section. This is the skin effect, and it effectively reduces the usable cross-sectional area, increasing resistance. The higher the frequency, the thinner the conducting layer and the greater the effective resistance.

The proximity effect further redistributes current when conductors are close together. Both effects matter in power transmission and high-frequency circuit design, where engineers may use stranded wire (Litz wire) to mitigate these losses.