Circuit Response
every model has an edge, and the figure carries it
Concept

Ladder filter — where it appears

A filter realised as alternating series and shunt reactances between a source and a load resistance. Doubly terminated at maximum power transfer, its passband response is stationary with respect to every reactive element, which is the property a cascade of active sections cannot have.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Where one pole goes when each component is 5% high. computed by solving, not by drawing. A series R–L–C, its poles recovered by rooting the determinant, and the derivative of the upper one with respect to each element taken exactly from the two null vectors at the pole. The dashed lines are the first-order prediction for a 5 per cent change; the filled circles are where the root actually goes when the element is changed and the determinant re-rooted. The three directions are the argument: the resistance moves the pole along a circle of constant radius, because the natural frequency does not contain it — its normalised sensitivity has a real part of 3.1e-16. The inductance and the capacitance each carry exactly −½ of the radius, and imaginary parts that are exact negatives. At 5 per cent the prediction is out by 0.122 per cent of the pole's own magnitude.

The derivative of a root

An earlier essay turned one transposed solve into the derivative of a response with respect to every element, and found a doubly terminated ladder filter stationary at its ripple peaks to a part in ten to the eighth. A pole is a different object — a value of s at which the matrix loses rank — and its derivative comes from two null vectors and a division. Pointed at the same two realisations, the ladder filter's advantage is a factor of 2.17, not eight orders of magnitude: what is stationary is the magnitude at one frequency, and it says nothing about where the poles are.

networks · Sensitivity At a stationary point a tolerance has a mean, not a spread. computed by solving, not by drawing. Six hundred ladder filters with every reactance drawn independently from ±1 per cent, measured at the ripple peak and at a frequency between the peaks. Away from the peak the distribution is centred on nominal and 325 of 600 are above it. At the peak none is: the whole distribution lies below, with a mean of -0.0030 per cent and a worst case of -0.0140. A yield calculation that assumes a symmetric spread at a frequency that has a stationary point is wrong in both directions at once — it allows parts above a limit that cannot exist, and it misses that the whole batch has moved.

The tolerance that can only take away

At a doubly terminated ladder filter's ripple peak the first derivative of the magnitude with respect to every reactance is zero to ten digits, which an earlier essay measured and which says nothing about how much the response moves. This says how: every second derivative is negative, so of six hundred ladder filters built from one per cent components not one is above nominal, the mean has shifted rather than the spread having grown, and doubling the tolerance quadruples the damage instead of doubling it.

networks · Sensitivity Inside the passband the matrix is worst at the edge, and at no ripple peak. computed by solving, not by drawing. The condition number of a 5th-order Chebyshev filter's nodal matrix at every frequency from a hundredth of its cutoff to ten times it. It is 4.004e+3 at direct current, rises to 1.5868e+7 at 964.0 Hz — located by golden section rather than read off the sweep — falls to 4.970e+6 at 1520 Hz and rises again through the stopband. The maximum is above every ripple peak (the highest is at 897.9 Hz) and within 3.6 per cent of the half-power frequency: the worst-conditioned place in the passband is where the filter stops passing and starts blocking, which is a place the response curve has no feature at. It is a local maximum: above the band κ climbs again, reaching 2.702e+7 at ten times the cutoff, because the susceptances in the matrix grow with the frequency and κ counts them. Nothing about any of these numbers says whether a digit is actually lost anywhere.

Where the matrix is worst, and where the answer is not

When a solve stops being exact has been answered with two networks at direct current. Swept along the frequency axis, a resistive chain's nodal matrix has a condition number of 4.00×10³ at direct current and 1.59×10⁷ at 964 Hz — a maximum at the band edge, at no ripple peak and at no feature the response has. It predicts nothing. The solution vector is right to three units of round-off everywhere, the response taken out of it loses four decades into the stopband, and the same filter written at 400 kΩ instead of 10 Ω has a condition number 1.4×10⁹ times larger and returns the same twelve digits.

networks · Conditioning What the accuracy costs: dynamic range against the resistance scale. computed by solving, not by drawing. The earlier essay found the active realisation's response converging on the passive one's as the resistance scale rises. This is the price. The floor rises as the square root of the scale — fitted at 0.500 — because the resistors are the noise. The largest internal swing rises as the scale itself — fitted at 1.012 — because each gyrator forces the inductor's own current through its own resistors, so an amplifier inside it carries that current times R. Dynamic range on a ±15 V supply therefore falls as the three-halves power: 78 dB at 100 kΩ and 18 dB at 10 MΩ. The passive ladder realising the same response has 141 dB, and its worst internal node carries 1.10 times the input.

Eight amplifiers, and what they add

The rung below realised a Chebyshev ladder out of floating gyrators and found the response converging on the passive one's as the resistance scale rises — twenty-two decibels out at ten kilohms, a twentieth of a decibel at ten megohms. It closed by naming two quantities it had not measured. They are the same quantity: the resistors that buy the accuracy are the noise, and the amplifiers inside the gyrators carry the inductor's own current through them, so the floor rises as the square root of the scale and the ceiling falls as the scale.

filters · Gyrator A fifth-order Chebyshev ladder built from parts of Q = 100: its passband sags by 0.94 dB near the edge and 0.38 dB at low frequency — 8.686·δ·τg, the group delay times the dissipation. computed by solving, not by drawing. A doubly terminated 0.5 dB Chebyshev ladder, fifth order, 1 kHz edge, 1 kΩ terminations, lossless (dashed) and with every inductor given a series resistance and every capacitor a shunt conductance for a uniform dissipation δ = ωc/Q = 62.8 s⁻¹ (solid), and the lossless response plus 8.686·δ·τg(f), with τg read off the lossless filter's phase (dots). The prediction holds to 3.6% of the extra loss across the passband. The extra is smallest where the delay is smallest, 0.383 dB at 38 Hz, and largest at 953 Hz near the edge, 0.943 dB, where the delay peaks at 1.79 ms: the ripple the design specified is rounded off where it was steepest.

The derivative that is a delay

A doubly terminated ladder filter is stationary in every element's value at its ripple peaks, and that insensitivity says nothing about the parts' losses. Give every reactance the same dissipation δ — a series resistance δL in each inductor, a shunt conductance δC across each capacitor — and every reactance has the impedance it had at s + δ, so the lossy ladder is exactly H(s + δ). By the Cauchy–Riemann equations the loss then rises by 8.686 times δ times the group delay, in decibels: the derivative of a ladder filter's response with respect to its parts' dissipation is its group delay. Solved with the resistors in, a fifth-order 0.5 dB Chebyshev of Q = 100 parts sags 0.383 dB at low frequency and 0.943 dB near its edge, where the delay peaks at 1.79 ms, the law holding to 3.6%. The sag falls as one over Q, grows with the order — 0.37 dB at third order, 2.74 at ninth — and is set by the mean of the inductors' and capacitors' dissipations.

networks · Sensitivity One step of refinement in the same double precision returns the stopband's lost digits: 1.1e-12 at 10 kHz before, under 2.4 units of round-off after. computed by solving, not by drawing. The 5th-order Chebyshev ladder solved at every frequency in double precision (solid), and again after one step of iterative refinement — the residual b − A·x formed in double, the correction solved with the same elimination and added (dashed) — each against the same network eliminated in a pair of doubles. Plain, the response's error tracks the round-off divided by the response (dotted), from 1.1e-16 at 10 Hz to 1.1e-12 at 10 kHz, where the output is 6.78e-7 of the input. One step brings every frequency to within 2.4 units of round-off of the truth. No more digits were used: the loss belonged to the elimination's errors in the small component, and the residual, computed against the original matrix, sees them.

The step that returns the digits

A fifth-order ladder filter solved in double precision loses four decades in its stopband — 1.1 × 10⁻¹² of relative error at ten times the cutoff — on a solution vector that is right to round-off. One step of iterative refinement, in the same double precision, returns all of it: form the residual b − A·x, solve for a correction with the same elimination, add it, and every frequency is within 2.4 units of round-off of a 31-digit solve. The reason is that the plain solve is stable by the norm and not by the row — its componentwise backward error is 5.3 × 10⁻¹³ in the stopband rows — and one step makes it componentwise stable. The bridge walked to balance cannot be helped at all. Its node voltages are already right to the last bit; its digits are lost storing two numbers whose difference is the answer, and again storing its conductances, which moves the network itself by round-off over δ before anything is solved.

networks · Conditioning

Named alongside it

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

Closed formComponent sensitivityAdjoint networkComponent toleranceConditioningNumerical errorQuality factorVerificationBiquad cascadeChebyshevDesign tradeoffDevice mismatch

All concepts