Resistors - Detailed Study Notes

Chapter 1: Resistors

1. Introduction

• In circuit models, passive electronic components are treated as perfect components.
• Three types of perfect components: resistors, capacitors, and inductors.
• Their characteristics depend on how they handle energy.
• Understanding the technology and models of passive electronic components is essential for their effective use.

2. Resistance and Resistivity

2.1. General Information:

• Resistance is a component's ability to oppose the passage of current (electron flow).
• This opposition results in energy dissipation within the component through the Joule effect.
• The unit of resistance is the Ohm, symbolized by the Greek letter Omega ($\Omega$).
• Resistors are non-polarized passive components; their function is independent of current direction.
• Resistance values range from tens of milli Ohms ($m\Omega = 10^{-3} \Omega$) to tens of Megaohms ($M\Omega = 10^6 \Omega$).
• Values are standardized unless a special order is placed for a large quantity.
• Resistance values are specified at 25°C with a tolerance (precision of 10%, 5%, 2%, 1%, 0.5%, 0.1%).
• Values are indicated either by direct marking or by a color code.
• A temperature coefficient (ppm/°C) indicates the resistance value's variation with temperature.
• A resistor in a circuit diagram can be represented in two forms (Figure 1).

2.2. Resistivity:

• Resistivity is a material's inherent ability to oppose electrical current.
• Low for conductors, very high for insulators, and moderate for semiconductors (germanium, silicon).
• Varies with temperature:
  • Increases for conductors.
  • Decreases for insulators.
  • Decreases significantly for semiconductors.
• Symbol for resistivity is $\rho$, expressed in Ohm-meters ($\Omega "." m$).
• It's the resistance of a 1m long and 1 $m^2$ cross-section material.
• The formula for the resistance of a wire with resistivity $\rho$, length $l$, and cross-sectional area $A$ is: $\R = \rho \frac{l}{A}$. (Figure 1 shows resistor symbols)

Example: What is the resistance of a 100m copper wire with a 2.5mm diameter, given the resistivity of hardened copper is $\rho = 1.8 \times 10^{-8} \Omega "." m$? Resistivity varies with temperature. Table 1 provides resistivity and temperature coefficient values for some materials. As a first approximation, the formula giving the resistance at temperature $T$ ($R_0$ is the resistance at 0°C) is:

3. Basic Formulas

3.1. Ohm's Law

• Relates voltage and current (V-I relationship) characterizing a resistor's behavior.
• Ohm's Law: $V = R \cdot I$

3.2. Power Dissipated

• Electron collisions in the conductor release energy, transforming into heat.
• This irreversible transformation is analogous to mechanical friction.
• The power dissipated as heat is: $P = R \cdot I^2 = \frac{V^2}{R} = V \cdot I$

3.3. Joule Effect

• The Joule effect is the transformation of electrical energy into heat, or heat and light.
  • Purely thermal receivers (resistors) convert all received electrical energy into heat.
  • Lamps convert electrical energy into heat and light.
  • Motors convert most electrical energy into work, with the remainder lost as heat.

3.4. Resistor Combinations

• Resistors can be grouped to achieve non-standard resistance values or for space/heat dissipation reasons.
• Two types of groupings: series and parallel.
  • Resistors in Series
  • Resistors in Parallel

3.5. Voltage Divider

4. Technological Variety

• Selecting the correct resistor type is crucial based on circuit requirements.
• Considerations include:
  • Current type: DC, AC (frequency range).
  • Electrical parameters: value, precision, maximum power dissipation (steady-state or transient, with/without forced ventilation), maximum supporting voltage, temperature stability.
  • Physical parameters: dimensions, mechanical stress resistance (vibration, etc.).
  • Cost: Generally low but important for high precision/power dissipation requirements.

4.1. Agglomerated Resistors

• The oldest type, made from carbon powder mixed with an insulator and binder, surrounded by a coating. (See cross-section in Figure 2).
• Tolerance: 20% (no tolerance ring) or 10% (silver ring).
• Non-inductive but noisy.
• Available in various power ratings.

4.2. Carbon Film Resistors

• Consist of a thin carbon layer deposited on a ceramic insulating rod and covered with varnish.
• The value is adjusted by carving a helical groove in the carbon layer (Figure 3).
• Available with or without connecting rings fixed to the ends.
• Has some self-induction (coil).
• Less noisy and more stable than agglomerated resistors.
• Most common and least expensive type.

4.3. Metal Film Resistors

• Usually consists of a metal film deposited on a ceramic insulating rod.
• The value is adjusted by carving a helical groove in the metal layer.
• Available with or without connecting rings (more reliable but fragile).
• Has some self-induction (coil).
• Less noisy than agglomerated and carbon film resistors.
• Generally more precise with a lower temperature coefficient but more expensive.

4.4. Wirewound Power Resistors

• Typically made of a resistive wire wound in non-contiguous turns on a cylindrical ceramic support.
• Possesses significant inductive characteristics, limiting use to low frequencies.
  • Wirewound in heat-dissipating housing type RH (Dale, Vishay, Sfernice, ….)
  • Vitrified wound type RB (Manufacturers: Vishay, Sfernice, Welwyn,….)
• Nickel-Chrome wire is wound on a ceramic cylinder, covered with a glass layer.

4.5. Precision Resistors

• Used for precise measurements or polarization.
• Environment must be considered to preserve precision.
• Available with precision of 1%, 0.1%, and 0.01%; stability in temperature is usually paired (e.g., ± 5 ppm / °).
• Cost is proportional to precision.

4.6. Surface Mount Resistors – CMS (SMD)

• Part of surface-mount components (CMS).
• Miniature components without connecting wires, directly soldered onto a printed circuit board.
• Allow for high integration (space-saving) and are recommended for very high frequencies due to low parasitic inductance.
• Difficult to handle and solder for amateurs without industrial equipment.
• Value is indicated by a marking code, except for the smallest ones.

5. Color Code

• The value of a component is not always clearly indicated; the color code is used to identify the value of most resistors, some capacitors, inductors, thermistors, and the reference of some diodes and the gain of certain transistors.
• To determine a resistor’s value:
  1. Count the number of color rings:
     • 3 rings: 2 significant digits, 1 multiplier, no tolerance ring = ±20%
     • 4 rings (one wider): 2 significant digits, 1 multiplier, 1 tolerance ring
     • 5 rings (one wider): 3 significant digits, 1 multiplier, 1 tolerance ring
     • 6 rings (two wider): 3 significant digits, 1 multiplier, 1 tolerance ring, 1 temperature coefficient ring
  2. The first ring is closest to the edge. The first 2 or 3 rings are significant digits. The next ring is the multiplier, then the wider ring indicates tolerance (brown, 1% for the E96 series). Sometimes a 2nd wider ring indicates the temperature stability coefficient (for precision resistors).

Example: A resistor with 5 rings (green = 5, blue = 6, red = 2 → 562), 1 multiplier (brown = 1 → x 10), 1 tolerance (red = 2%) is 5620,$R = 5.62k\Omega$ value given at ±2%.

6. Resistor Characteristics

• Characteristics specified by the manufacturer:

6.1. Nominal Resistance Value

• The nominal resistance in Ohms ($\Omega$) at an ambient temperature of 25°C. Higher values are expressed using multiples of the Ohm.

6.2. Tolerance

• A percentage, plus or minus around the nominal value, that the supplier guarantees for all delivered parts. It applies to new parts before use because variations can be larger after prolonged operation.
• For example: A 120 $\Omega$ resistor with a tolerance of ±10% has a deviation of ±12 $\Omega$ ($120 \times ±10 / 100 = ±12$). The actual resistance value is between 108 $\Omega$ (120 - 12= 108) and 132 $\Omega$ (120 + 12 = 132). If the tolerance is ±2%, the deviation is ±2.4 $\Omega$ ($120 \times ±2 / 100 = ±2.4$); the actual value is between 117.6 $\Omega$ (120 - 2.4 = 117.6) and 122.4 $\Omega$ (120 + 2.4 = 122.4).

6.3. Nominal Power

• The power a resistor can dissipate in still air at normal atmospheric pressure and an ambient temperature of 20 or 25°C. It is determined so that no point in the resistor exceeds the specified temperature limit for that type of construction under these conditions. If the ambient temperature is higher, the allowable power must be reduced to stay within the resistor's temperature limits. Derating curves are established for each model.

6.4. Temperature Coefficient

• The quotient of the relative change in resistance per degree change in temperature. Expressed in parts per million per degree centigrade (ppm / °C).
• For common resistors, the temperature coefficient is between 50 ppm/°C and 100 ppm/°C.
• Example: A 2 000 $\Omega$ resistor at 20°C with a temperature coefficient of 80 ppm/°C increases its value by 0.16 $\Omega$ for each 1°C increase in temperature: 2 000 x 80 / 1 000 000 = 0.16 $\Omega$ /°C.
• In a 60°C environment, the resistance becomes 2006.4 $\Omega$ (0.16 $\Omega$ /°C x 40° C = 6.4 $\Omega$).
• For negative temperature coefficients, resistance decreases as temperature increases.

6.5. Maximum Voltage Across Terminals

• Voltage across a resistor is given by the formula: $U = \sqrt{P \cdot R}$, where U is voltage in volts, P is power in watts (nominal power of the resistor), and R is resistance in ohms.
• Without overloading the resistor in power, very high voltages are obtained at the terminals for high ohmic values.

6.6. Stability

• A stable resistor maintains a value close to the original after prolonged use. Value variation depends on the type and manufacturing technology. Wirewound or metal film resistors are very stable; agglomerated resistors are less so.

6.7. Voltage Coefficient

• The measure of the change in resistance value as a function of the voltage across terminals, expressed as a percentage of variation per volt.
• Negligible for wirewound and metal film resistors; appreciable for agglomerated resistors (0.02 % V).

6.8. Noise Voltage

• All resistors produce parasitic voltage due to thermal agitation of molecules. This voltage is very low for wirewound and metal film resistors but more significant for agglomerated resistors due to current flow through heterogeneous material.
• Measured in microvolts per volt applied to terminals (µV / V). Limits the possibility of amplification because it becomes bothersome at a certain level. This noise mainly consists of low frequencies below 10 kHz (audible) and increases with the ohmic value of the resistance.

7. Uses and Applications

7.1. Adjustable Resistor – Adjustable Potentiometer

• A component whose resistance value can be varied by moving a contact on a resistive track (carbon or metal) using an external adjustment device (adjusting screw).
• Used in the final phase of manufacturing to adjust the device in order to compensate for the tolerance of the components used (component precision).
• Symbols of adjustable resistance and potentiometer (European standard).

7.2. Rheostat

• An adjustable resistance used to modify the current in a circuit by connecting it in series.
• Generally consists of a variable power resistance.
• The maximum current that can pass through it is indicated; a momentary overload is often mentioned.
• Adjustment is made using a linear slider or a crank.
  Example: 100$\Omega$ rheostat with a maximum current of 1.8 A and 2.5A (max 15 min).

7.3. Non-Radiating Load (Power Resistance)

• A non-inductive power resistance used as a load when adjusting the emission part of a transmitter station (in place of the antenna) to avoid disturbing surrounding radio communications. It is part of the mandatory equipment of a Radioamateur transmitting station.

7.4. Shunt

• A low-value resistor placed in parallel to divert most of the current. Used in ammeters to measure current larger than what the measuring part of the device can support (e.g., a galvanometer).

7.5. Strap - 0 $\Omega$ Resistor

• Has a resistance value of 0 $\Omega$. Used in printed circuit board design to simplify the circuit layout by avoiding overlapping traces.

7.6. Thermistor – NTC – PTC

• A special resistor whose resistance value varies with temperature.
• Classified into two categories: negative temperature coefficient (NTC, value decreases as temperature increases) and positive temperature coefficient (PTC, value increases as temperature increases).

7.7. Varistor – VDR (Voltage Dependent Resistor)

• A special resistor, based on metal oxide, whose resistance value varies with the voltage applied to its terminals. The resistance decreases as the voltage increases above a threshold voltage, protecting the circuit from overvoltage. It can absorb large currents (100A to 1000A) for a short duration (8 to 20 µS).

8. Testing and Failures

8.1. Testing with an Ohmmeter

• An ohmmeter is a device that measures the resistance of a component. It can be analog (with a needle galvanometer) or digital (with an LED or LCD display) and can be part of a universal controller, an RLC measurement bridge, or dedicated to this type of measurement. The measurement is based on Ohm's law.

8.2. Practical Tips for Measuring Resistance:

• For high resistance values, do not touch the contacts of the resistor and/or measuring leads, as this adds the resistance of your body (≈ 100k$\Omega$) in parallel with the component under test, falsifying the measurement.
• For low resistance values, use measuring leads as short as possible and account for their resistance by subtracting it from the displayed value. For those with a (milli) ohmmeter, do a