Concept

Short-circuit time constants — where it appears

The time constant each capacitor of a high-pass network sees with every other capacitor shorted. Their reciprocals sum exactly to the network's lowest-order denominator coefficient, which gives the low corner and a pulse's sag without solving for a single pole.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The lower corner, estimated from short-circuit time constants. computed by solving, not by drawing, on three coupling capacitors and three shunt resistors at 28 spreads of the capacitor values. Each capacitor's short-circuit time constant is its own value times the resistance between its terminals with the other two shorted; the sum of the RECIPROCALS is 60000 s⁻¹ here, and it equals the ratio of the denominator's two highest coefficients to 1.4e-12 and the negated sum of the poles to 1.4e-12. That much is the same theorem as the other end. What is an estimate is the corner: 9549 Hz against a measured 8192 Hz, high by 16.6%. It is high at every spread drawn — the error reverses direction with the construction, so both ends of a band are estimated inwards.

Shorted instead of opened, and the error changes sign

The same construction with the other capacitors shorted rather than removed sums the reciprocals of the products, and that sum is the ratio of the denominator's two HIGHEST coefficients — the negated sum of the poles, exact to a part in 10¹². Divided by 2π it estimates the lower corner of a band, and it is 16.6 per cent HIGH with three coupling capacitors and never once low. Two settings of the slider give the same three time constants in a different order, the same sum, and corners two per cent apart.

transients · Open circuit time constants

Without the buffer, a second stage of 1 times the first's impedance: the top sags to 0.175 instead of 0.303, and the tail crosses at 2.00 s and overshoots to +0.010. computed by solving, not by drawing, marched and checked against the closed form to 4.5e-4: a half-second pulse through two one-second coupling stages, buffered (dashed) and with the second hung directly on the first (solid), its resistor 1 times the first's and its capacitor 1/1 of it, so that each stage alone keeps its one-second time constant. Buffered, the top sags to 0.3033 and the tail crosses zero 2.27 s after the pulse began. Unbuffered, the denominator gains the cross term R1·C2 and its poles move to −0.382 and −2.618 per second; the top sags to 0.1751, undershoots to −0.8245, crosses zero at 2.002 s and overshoots to +0.0103. The slider is the second stage's impedance relative to the first's.

Two coupling stages without a buffer

Hang a second coupling stage directly on the first, with equal parts, and the pair's denominator gains a third time constant — the first resistor carrying the second capacitor's current. The poles spread from a double pole to −0.382 and −2.618 per time constant, a half-second pulse's top sags to 0.175 instead of the buffered pair's 0.303, and the sag rate is exactly 2 + 1/k per time constant for a second stage k times the first's impedance — the sum of the two short-circuit time constants. The tail still crosses zero, because the two zeros at the origin are untouched, and a step's lowest point is still at exactly twice its zero crossing. Then a reversal: specified by the overall corner rather than by the parts, the loaded pair sags only 1.12 times as fast as one stage and undershoots 7.6% rather than 13.5% — the buffer the cascade essay assumed makes the specified input worse.

instruments · AC coupling

Named alongside it

The objects these essays reach for when they reach for this one.

AC coupling Poles Bandwidth Baseline shift Charging exponential Corner frequency Loading Model range Open circuit time constants Step response Verification

All concepts