Resistive networks in practice

Lessons 1 and 2 treated resistors as ideal: exact values, unlimited power, no effect from the instruments used to measure them. Real circuits are built from parts with tolerances, temperature coefficients and power limits, and they are measured with instruments that draw current and add resistance of their own. This lesson applies Kirchhoff's laws to the questions an engineer meets when a resistive circuit leaves the page: how far can the output of a divider stray, how hot will a resistor run, how much does a meter disturb the reading, how are tiny resistance changes detected with a bridge, and how are large currents measured with a shunt?

Real resistors

A resistor's marked value is nominal. Three specifications describe how the real value differs:

• Tolerance: the permitted deviation at manufacture, typically ±5% or ±1% for general-purpose parts and ±0.1% or better for precision parts.
• Temperature coefficient of resistance (TCR), in parts per million per degree Celsius (ppm/°C): about ±100 ppm/°C for thick-film chips, ±25 ppm/°C or better for thin-film and metal-film parts, and a few ppm/°C for precision foil resistors.
• Long-term stability: drift with age, load and humidity, often a few tenths of a percent over the life of a general-purpose part.

Standard values follow the IEC 60063 E-series, in which values are spaced evenly on a logarithmic scale. The E12 series (12 values per decade: 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82) suits ±10% parts; E24 suits ±5%; E96 suits ±1%. A designer who calculates a needed resistance of 13.7 kΩ picks the nearest standard value, or a combination, and checks that the circuit still meets its specification.

Because a divider's output depends on a ratio, matched resistors help: resistor networks made on one substrate track each other closely in temperature and age, so the ratio stays stable even when absolute values drift.

Power ratings and derating

Every resistor has a rated power, the maximum it can dissipate continuously at a stated ambient temperature, usually 70 °C. Above that, the allowed power is reduced linearly, typically to zero at the maximum element temperature (often 125 °C or 155 °C). Resistors also have a maximum working voltage, which limits high-value parts before their power rating does.

Measurement and loading

Every instrument connected to a circuit becomes part of it. An ideal voltmeter has infinite resistance and an ideal ammeter zero; real ones do not, and the difference is called loading.

Voltmeter loading

A digital multimeter typically has an input resistance of 10 MΩ on its DC voltage ranges. Connected across a low-resistance part of a circuit, it draws negligible current; connected across a high-resistance node, it forms a parallel combination that pulls the voltage down.

Ammeter burden

An ammeter is inserted in series and adds its own resistance, causing a burden voltage. A multimeter's current ranges use internal shunt resistors, often adding a few hundred millivolts at full scale. In a low-voltage circuit this can change the current being measured. Clamp meters, which sense the magnetic field around a conductor, avoid the problem entirely for larger currents.

Current shunts

Large currents are measured with a shunt: a precise, low-value resistor placed in series, across which the voltage is measured. Shunts are rated by current and full-scale voltage, such as 400 A at 50 mV. They are made of alloys such as manganin, whose resistance hardly changes with temperature, and have separate sense terminals so that the voltage is measured across the resistive element alone, not the high-current connections.

Four-wire (Kelvin) measurement

When the resistance being measured is small, or the leads are long, the resistance of the leads themselves adds to the reading. A four-wire (Kelvin) connection solves this: one pair of wires carries a known test current, and a separate pair, carrying almost no current, senses the voltage directly at the resistor. Because negligible current flows in the sense leads, their resistance causes no error.

Bridge circuits

The Wheatstone bridge (Lesson 2) is two dividers fed from the same source, with the output taken between their midpoints. When the ratios match, the output is zero regardless of the supply voltage, which makes the bridge ideal for detecting small changes in one resistance.

Strain gauges, load cells and pressure sensors use this principle. A strain gauge is a fine metal foil pattern bonded to a surface; stretching it lengthens and narrows the conductor, raising its resistance in proportion to the strain \(\varepsilon\): \(\Delta R/R = GF \cdot \varepsilon\), where the gauge factor \(GF\) is about 2 for metal foil.

Ladder networks

A ladder network alternates series and shunt elements. It is solved from the far end back toward the source, combining as you go, or by working forward from an assumed output voltage (the "unit output" method): assume 1 V at the output, compute currents and voltages back to the source, then scale everything so that the source has its real value.

An infinite ladder of identical series resistors \(R\) and shunt resistors \(R\) has an input resistance that is unchanged by adding one more section, so \(R_{in} = R + R \parallel R_{in}\). Solving the resulting quadratic gives \(R_{in} = R(1 + \sqrt{5})/2 \approx 1.618R\), the golden ratio, a neat result and a useful model of long transmission lines with leakage.

The R-2R ladder

The R-2R ladder uses only two resistor values and is the basis of many digital-to-analog converters (DACs). Looking toward the terminated end from any node, the resistance is exactly R, so at each node the current splits equally. Each bit therefore contributes a current half as large as the bit before, and the output is proportional to the binary number applied.