Op-amps and comparators as building blocks

The operational amplifier, or op-amp, is the most useful analog building block a digital designer meets. With two or three resistors it becomes a buffer that isolates a soft source from its load, an amplifier that scales a tiny sensor voltage up to fill the ADC's range, or a circuit that subtracts one voltage from another. Its close relative, the comparator, turns an analog voltage into a clean logic signal. You do not need to know the transistor circuit inside an op-amp to use one well; a simple model and two rules are enough for most designs, and the circuit analysis methods from lessons 4 to 6 do the rest.

The NS-4 uses three op-amp circuits. A buffer drives the ADC from the battery divider, an amplifier measures the solar charging current from the 50 mΩ sense resistor, and a comparator watches the battery voltage and cuts power to the radar if the battery runs too low, protecting it from deep discharge. Jonah is asked to check all three for the NS-5.

The op-amp model

An op-amp has two inputs, the non-inverting input (+) and the inverting input (−), one output and two supply pins. It amplifies the difference between its inputs by a very large open-loop gain A, typically 100,000 or more at low frequencies. A useful circuit model is a voltage-controlled voltage source: the output equals A times (v₊ − v₋), drawn behind a small output resistance, with a very large input resistance between the inputs. The output can never go beyond the supply rails; a rail-to-rail output op-amp gets within a few tens of millivolts of each rail, and older designs stay a volt or more away.

Because A is so large, an op-amp used on its own saturates at one rail or the other for almost any input difference: with A = 100,000, a difference of just 33 µV drives the output across a 3.3 V range. Op-amps become useful when negative feedback connects part of the output back to the inverting input. The feedback adjusts the output until the inputs are almost equal, and the circuit's behavior is then set by the external resistors rather than by the poorly controlled value of A.

The two golden rules

For an ideal op-amp with negative feedback and an output that is not saturated, two rules give the answer to almost every circuit. First, no current flows into either input, because the input resistance is effectively infinite. Second, the output does whatever is needed to make the voltage difference between the inputs zero, so v₋ = v₊. The second rule is sometimes called the virtual short: the inputs sit at the same voltage, but no current flows between them. Apply the rules with KCL at the inverting input and almost any op-amp circuit can be solved in a few lines.

The voltage follower

The simplest feedback circuit connects the output directly to the inverting input and the signal to the non-inverting input. By the second rule the output equals the input. The circuit has a gain of exactly one, so it seems to do nothing, but it has an input resistance of many megohms and an output resistance of a fraction of an ohm. It copies a voltage from a soft source to a load without loading the source, which is why it is called a buffer.

Non-inverting and inverting amplifiers

Adding a resistive divider in the feedback path gives gain. In the non-inverting amplifier, the signal drives the + input and a divider of R_f and R_g feeds a fraction of the output back to the − input. The output settles where that fraction equals the input, so the gain is one plus the resistor ratio. The input resistance is still very high.

In the inverting amplifier, the + input is grounded and the signal drives the − input through R_in, with R_f from output to − input. By the second rule the − input sits at 0 V, a virtual ground, so the input current is v_in/R_in. By the first rule that current cannot enter the op-amp, so it all flows through R_f, making the output −R_f/R_in times the input. The inverting amplifier's input resistance is only R_in, but its virtual ground makes it ideal for summing several signals, each through its own resistor.

The difference amplifier and current sensing

The NS-4 charger's 50 mΩ sense resistor drops only 75 mV at 1.5 A, and one end of it sits at the battery voltage rather than at ground. A difference amplifier measures the voltage across the resistor while rejecting the common voltage that both ends share. With four resistors matched in the ratio R₂/R₁, its output is (R₂/R₁)(v₂ − v₁). Choosing R₂/R₁ = 20 turns 75 mV into 1.5 V. In practice, high-side current measurement is usually done with a dedicated current-sense amplifier, a difference amplifier with precision-matched resistors built in, because ordinary 1 percent resistors limit how well the common voltage is rejected.

Comparators

A comparator compares two voltages and outputs a logic level: high if v₊ > v₋ and low otherwise. It is designed to be used without negative feedback, to switch quickly between rails, and often to have an open-drain output that can drive a logic input or an LED directly. You can use an op-amp as a comparator in a pinch, but a true comparator switches faster, recovers from saturation cleanly and is specified for this job.

A plain comparator has a problem with slowly changing or noisy inputs. As the battery voltage drifts down past the threshold, small noise from the radar's current pulses makes it cross back and forth, and the output chatters on and off many times. If that output switches a load, the load's own current changes the battery voltage and makes the chatter worse. The cure is hysteresis: positive feedback from the output to the + input, through a resistor, so that the threshold moves away from the input once the output switches. The circuit is called a Schmitt trigger, and it has two thresholds, a rising one and a falling one.

The derivation above shows a common design pattern: the width of a Schmitt trigger's hysteresis band depends only on the resistor ratio and the output swing, and its center depends on the reference. Choose the ratio for the width first, then adjust the reference to put the band in the right place. The code below checks the final values.

# Low-battery Schmitt trigger: thresholds from the resistor ratio and reference
def thresholds(v_ref, r1, r2, v_out_high=3.3, v_out_low=0.0):
    k = r1 / (r1 + r2)                       # fraction of the output fed back
    t_hi = v_ref * (1 - k) + v_out_high * k  # threshold while output is high
    t_lo = v_ref * (1 - k) + v_out_low * k   # threshold while output is low
    return t_lo, t_hi

ratio = 0.2                                  # battery divider ratio
for v_ref in (2.50, 2.44):
    lo, hi = thresholds(v_ref, 10e3, 200e3)
    print("Vref %.2f V: comparator thresholds %.3f / %.3f V -> battery %.2f / %.2f V"
          % (v_ref, lo, hi, lo / ratio, hi / ratio))

# Finite open-loop gain: how close is a non-inverting amplifier to its ideal gain?
def closed_loop(A, Rf, Rg):
    beta = Rg / (Rf + Rg)
    return A / (1 + A * beta)

for A in (1e3, 1e5, 1e6):
    g = closed_loop(A, 100e3, 3e3)
    print("A = %8.0f: gain %.3f (ideal %.3f), error %.3f%%" % (A, g, 1 + 100 / 3, (1 - g / (1 + 100 / 3)) * 100))

Vref 2.50 V: comparator thresholds 2.381 / 2.538 V -> battery 11.90 / 12.69 V
Vref 2.44 V: comparator thresholds 2.324 / 2.481 V -> battery 11.62 / 12.40 V
A =     1000: gain 33.194 (ideal 34.333), error 3.319%
A =   100000: gain 34.322 (ideal 34.333), error 0.034%
A =  1000000: gain 34.332 (ideal 34.333), error 0.003%

Finite gain and the limits of the ideal model

The golden rules assume infinite gain. With finite gain A and feedback fraction β = R_g/(R_f + R_g), the non-inverting amplifier's gain is A/(1 + Aβ). When Aβ is large this approaches 1/β, the ideal gain, with a fractional error of about 1/(Aβ). The code shows that A = 100,000 gives a gain error of 0.03 percent for a gain of 33, far smaller than the 1 percent resistors that set the gain. Real op-amps also have an input offset voltage of microvolts to millivolts, input bias currents, limited output current, a maximum rate of change of output called the slew rate, and a gain that falls with frequency, roughly in inverse proportion above a few hertz. A datasheet's gain-bandwidth product tells you how much gain is left at a given frequency, and for the slow sensor signals on the NS-4 it is never the limit.