Comparators, Schmitt trigger and timers

Why it matters

Comparators turn analog measurements into decisions: a level switch trips an alarm, a zero-crossing detector times a thyristor firing angle, a flash ADC is a bank of comparators. Real signals are noisy, so a plain comparator chatters near its threshold; the Schmitt trigger cures this with hysteresis. The 555 timer packages two comparators and a flip-flop into the most widely used timing chip, giving one-shot delays and clock oscillators from one resistor-capacitor pair.

Key ideas

Comparator. An op-amp (or a dedicated comparator IC) used without negative feedback. If v₊ > v₋ the output goes to its high level, otherwise to its low level. A non-inverting comparator has the signal on v₊ and the reference on v₋; an inverting comparator swaps them. With the reference at 0 V it is a zero-crossing detector. Dedicated comparators (often with open-collector outputs) switch much faster than op-amps, which are compensated for linear use and recover slowly from saturation.

Chatter and hysteresis. Noise on a slowly varying input makes a comparator switch many times as the signal crosses the threshold. A Schmitt trigger adds positive feedback, so the threshold depends on the present output state: an upper threshold V_UT for switching one way and a lower threshold V_LT for switching back. Noise smaller than the hysteresis width V_H = V_UT − V_LT cannot cause false switching.

Inverting Schmitt trigger. Input to v₋; a divider R₁ (to ground or V_ref) and R₂ (from the output) sets v₊. With output ±V_sat the thresholds are ±β·V_sat (plus a shift from V_ref), where β = R₁/(R₁ + R₂). The output goes low when the input rises above V_UT and high when it falls below V_LT.

Non-inverting Schmitt trigger. Input through R₁ to v₊, R₂ from output to v₊, v₋ at reference. Thresholds ±(R₁/R₂)·V_sat for V_ref = 0.

Op-amp astable (square-wave generator). An inverting Schmitt trigger whose input is a capacitor charged from its own output through R. The capacitor swings between the two thresholds, giving a square wave at the output and an exponential near-triangle across C.

555 timer. Inside: a resistor divider of three equal resistors sets 2/3·V_CC and 1/3·V_CC; the threshold comparator resets a flip-flop when its pin exceeds 2/3·V_CC; the trigger comparator sets it when its pin drops below 1/3·V_CC; a discharge transistor shorts the timing capacitor when the output is low.

• Monostable: a trigger pulse sets the output high and releases C, which charges through R from 0 to 2/3·V_CC: pulse width 1.1·RC, independent of V_CC.
• Astable: C charges through R_A + R_B from 1/3 to 2/3·V_CC (output high) and discharges through R_B alone back to 1/3·V_CC (output low). Since ln 2 = 0.693, each interval is 0.693 × resistance × C. The duty cycle is above 50 % in the basic circuit; a diode across R_B allows below 50 %.

Formulas

β = R₁ / (R₁ + R₂) — feedback fraction of the Schmitt trigger.

V_UT = β·V_sat + (1 − β)·V_ref, V_LT = −β·V_sat + (1 − β)·V_ref — inverting Schmitt with V_ref applied to the bottom of R₁ and symmetric output ±V_sat.

V_H = V_UT − V_LT = 2β·V_sat — hysteresis width (V).

V_UT,LT = ±(R₁/R₂)·V_sat — non-inverting Schmitt with v₋ = 0.

T = 2·R·C·ln((1 + β)/(1 − β)) — op-amp astable period (s).

T = 1.1·R·C — 555 monostable pulse width (s).

t_H = 0.693·(R_A + R_B)·C, t_L = 0.693·R_B·C, f = 1.44 / ((R_A + 2R_B)·C) — 555 astable.

D = (R_A + R_B) / (R_A + 2R_B) — 555 astable duty cycle (fraction of period output is high).

Worked examples

Example 1 (standard: Schmitt trigger). An inverting Schmitt trigger uses an op-amp saturating at ±12 V, R₁ = 10 kΩ (to ground) and R₂ = 40 kΩ (from output). Find the thresholds and hysteresis. Will 1 V peak-to-peak noise on a slowly rising input cause chatter?

1. β = R₁/(R₁ + R₂) = 10/50 = 0.2.
2. V_UT = β·V_sat = 0.2 × 12 = +2.4 V; V_LT = −2.4 V.
3. V_H = 2β·V_sat = 4.8 V.
4. 1 V of noise is far smaller than 4.8 V, so once the output has switched the input cannot get back across the other threshold: no chatter.

Answer: V_UT = +2.4 V, V_LT = −2.4 V, V_H = 4.8 V; no chatter.

Example 2 (GATE level: 555 astable). A 555 astable uses R_A = 4.7 kΩ, R_B = 10 kΩ and C = 10 nF. Find t_H, t_L, the frequency and the duty cycle.

1. t_H = 0.693(R_A + R_B)C = 0.693 × 14.7×10³ × 10×10⁻⁹ = 101.9 µs.
2. t_L = 0.693·R_B·C = 0.693 × 10×10³ × 10×10⁻⁹ = 69.3 µs.
3. T = t_H + t_L = 171.2 µs, so f = 1/T = 5.84 kHz (the 1.44 formula gives 5.83 kHz, the difference being rounding of ln 2).
4. D = t_H/T = 101.9/171.2 = 59.5 %.

Answer: t_H ≈ 102 µs, t_L ≈ 69 µs, f ≈ 5.84 kHz, D ≈ 59.5 %.

Common mistakes

• Using negative-feedback analysis (virtual short) on a comparator or Schmitt trigger; with positive feedback the inputs are not equal except at the switching instant.
• Mixing up which threshold applies: the inverting Schmitt switches low at V_UT and high at V_LT.
• Calling the 555 astable duty cycle R_B/(R_A + 2R_B); that is the low-time fraction.
• Using 0.693·RC for the 555 monostable (it is 1.1·RC because C charges from 0 to 2/3·V_CC, and ln 3 = 1.1).
• Assuming a sine wave compared with a non-zero reference gives a 50 % duty cycle; it does not.
• Using a slow general-purpose op-amp as a high-speed comparator.

For GATE IN

Expect threshold and hysteresis calculations for inverting and non-inverting Schmitt triggers (including with a reference), duty cycle of a comparator driven by a sine, output waveforms for a given input, and 555 monostable and astable timing. Practise drawing the transfer characteristic (v_o against v_i) with arrows showing the direction of the hysteresis loop.

Quick check

1. A 555 monostable has R = 100 kΩ and C = 10 µF. Pulse width?
2. A Schmitt trigger has V_UT = 3 V and V_LT = 1 V. Hysteresis?
3. A zero-crossing detector is driven by a 50 Hz sine. Output frequency and duty cycle?
4. In a 555 astable, R_A = R_B. Duty cycle?

Answers: 1. 1.1 s. 2. 2 V. 3. 50 Hz square wave, 50 %. 4. 2/3, about 66.7 %.