Series, parallel and divider circuits in digital hardware

Most of the resistors on a microcontroller board belong to a handful of standard circuits: voltage dividers that scale a signal, pull-up and pull-down resistors that give an input a defined state, series resistors that limit current, and current-sharing paths. Each is a series or parallel combination, and each can be designed with nothing more than Ohm's law and Kirchhoff's laws. This lesson develops the combination rules properly and then works through the circuits that appear on every digital board, using the NS-4's service button, its I²C sensor bus and its 5 V radar interface as examples.

Resistors in series

Resistors are in series when the same current must flow through each, because they share nodes that nothing else connects to. KVL says the total voltage across the chain is the sum of the individual drops, and Ohm's law gives each drop as iR. The chain therefore behaves like one resistor whose value is the sum of the parts. Series resistance is always larger than the largest individual resistor.

The voltage divider follows directly. In a series chain the current is the total voltage divided by the total resistance, so the voltage across any one resistor is the total voltage multiplied by that resistor's share of the total. This voltage divider rule is used so often that it is worth memorizing: a resistor's voltage is its fraction of the series resistance times the applied voltage.

A divider gives the right voltage only when nothing draws significant current from its output. Any load connected across R2 is in parallel with it, which lowers the effective lower resistance and the output voltage, as the leakage example in lesson 2 showed. A good rule of thumb is that the current drawn by the load should be less than about one percent of the current through the divider. Lesson 6 replaces the rule of thumb with an exact method.

Resistors in parallel

Resistors are in parallel when they connect between the same two nodes and therefore share the same voltage. KCL says the total current is the sum of the branch currents, and each branch current is v/R. The combination behaves like a single resistor whose conductance is the sum of the conductances. Parallel resistance is always smaller than the smallest individual resistor.

For two resistors the formula simplifies to the product over the sum, written with the parallel operator R1 ∥ R2. Two equal resistors in parallel give half the value, and a resistor in parallel with one ten times larger is reduced by about 9 percent. These shortcuts let you estimate quickly whether a parallel path matters.

The current divider is the parallel counterpart of the voltage divider. In two parallel branches the current splits in inverse proportion to resistance: the smaller resistance takes the larger share. Expressed with conductances the rule has the same form as the voltage divider, each branch taking its share of the total conductance.

Simplifying networks

Ladder networks and other combinations can be reduced step by step: find a pair of elements that are clearly in series or in parallel, replace them with their equivalent, redraw, and repeat. The key is to check the definitions rather than the appearance of the drawing. Two resistors drawn side by side are in parallel only if both ends connect to the same two nodes; two resistors drawn in a line are in series only if no other branch joins the node between them. Some networks, such as a bridge, cannot be reduced this way at all and need the systematic methods of lessons 4 and 5.

Pull-up and pull-down resistors

A digital input pin has a very high input resistance and draws almost no current. If it is connected to nothing, its voltage floats: it drifts with static charge, picks up noise from nearby traces and may sit between the logic thresholds, where the input buffer wastes current and reads randomly. A pull-up resistor connects the input to the supply rail so that it reads 1 when nothing else drives it. A pull-down resistor connects it to ground so that it reads 0. A switch or an open-drain output then pulls the line the other way when activated.

The NS-4's service button connects the microcontroller pin to ground when pressed, with a 47 kΩ pull-up to 3.3 V. With the button open, the only current is the pin's leakage, at most 1 µA by the datasheet, so the pin sits at 3.3 − (1 µA × 47 kΩ) = 3.25 V, well above the input-high threshold. With the button pressed, the pin is at ground and the resistor draws 3.3 V / 47 kΩ = 70 µA, but only while a technician is holding the button. Choosing the pull-up value is a trade between that pressed current and the noise immunity of the released state: a smaller resistor holds the line more firmly against interference, and a larger one wastes less power.

Open-drain outputs and the I²C bus

Many digital buses use open-drain (or open-collector) outputs, which can pull a line low but cannot drive it high. A single pull-up resistor per line restores the high level. Any device on the bus can pull the line low, and the line is high only when every device releases it, an arrangement called wired-AND. The NS-4's temperature and humidity sensor and its real-time clock share a two-wire I²C bus, with pull-up resistors on the clock line (SCL) and the data line (SDA).

The pull-up value on an I²C bus is bounded on both sides. It cannot be too small, because when a device pulls the line low it must sink the pull-up current while keeping the line below the low threshold; the I²C specification limits that sink current to 3 mA in standard and fast mode with the low level at most 0.4 V. It cannot be too large, because the pull-up must charge the bus capacitance quickly enough to meet the required rise time, a constraint that lesson 9 calculates. For now the lower bound follows from Ohm's law.

Two NS-4 boards from an early build had I²C errors that appeared only in cold weather. The cause was two modules on the bus each fitted with their own 2.2 kΩ pull-ups, in parallel with the main board's 2.2 kΩ pull-ups. Three 2.2 kΩ resistors in parallel make 733 Ω, below the 967 Ω minimum, so the weakest device could not pull the line low enough, and its output transistor weakened further as the temperature dropped. Removing the extra pull-ups from the modules fixed it. Parallel combinations hide in places like this, spread across boards and connectors.

Dividers for level shifting

The NS-4's radar module was designed for 5 V systems, and its serial transmit line swings from 0 to 5 V. Feeding 5 V into a 3.3 V microcontroller input would exceed its absolute maximum rating and could damage it. A simple voltage divider scales the 5 V signal down: with R1 = 10 kΩ from the radar output to the pin and R2 = 20 kΩ from the pin to ground, the pin sees 5 × 20/30 = 3.33 V when the radar output is high. Lesson 9 shows that the divider also slows the signal's edges, which limits how fast the serial link can run.

# Worst-case analysis of the radar level-shifting divider
from itertools import product

def divider(vin, r1, r2):
    return vin * r2 / (r1 + r2)

def worst_case(tol):
    R1, R2 = 10e3, 20e3
    results = []
    for vin, s1, s2 in product([4.6, 5.25], [-1, 1], [-1, 1]):
        r1 = R1 * (1 + s1 * tol)
        r2 = R2 * (1 + s2 * tol)
        results.append(divider(vin, r1, r2))
    return min(results), max(results)

for tol in (0.05, 0.01):
    lo, hi = worst_case(tol)
    ok = (lo >= 2.31) and (hi <= 3.6)
    print("tolerance %2.0f%%: pin from %.3f V to %.3f V -> %s" % (tol * 100, lo, hi, "pass" if ok else "fail"))

# Parallel pull-ups on the I2C bus
def par(*rs):
    return 1 / sum(1 / r for r in rs)
for n in (1, 2, 3):
    rp = par(*([2.2e3] * n))
    print("%d x 2.2k in parallel = %4.0f ohm, sink current %.2f mA" % (n, rp, (3.3 - 0.4) / rp * 1e3))

tolerance  5%: pin from 2.963 V to 3.615 V -> fail
tolerance  1%: pin from 3.046 V to 3.523 V -> pass
1 x 2.2k in parallel = 2200 ohm, sink current 1.32 mA
2 x 2.2k in parallel = 1100 ohm, sink current 2.64 mA
3 x 2.2k in parallel =  733 ohm, sink current 3.95 mA

Current-sense resistors

Not every resistor exists to set a voltage. The NS-4's solar charger measures its output current with a 50 mΩ resistor in series with the battery lead; at 1.5 A it drops 75 mV, which an amplifier scales up for the microcontroller's ADC. A small resistance keeps the wasted power low, here 1.5² × 0.05 = 0.11 W, but means the measured voltage is small and the resistor's tolerance and the trace resistance of its connections matter. Designers connect the amplifier to the resistor's pads with separate traces, a Kelvin connection, so that the voltage dropped in the high-current traces is not added to the reading.