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

Bode plot — where it appears

A response drawn as magnitude in decibels and phase in degrees against a logarithmic frequency axis. Both halves come from one solve of the same matrix, and the straight-line sketch drawn over them is three decibels wrong at the corner.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

A single-pole low-pass with its corner at 995 Hz. Solved at 209 frequencies. The straight-line sketch, drawn faintly, is 3.01 dB wrong at the corner and within a tenth of a decibel only below 152 Hz. The phase is already −5.7° a decade before the corner and −84° a decade after it.

One solve, read four ways

Reactance, phase, the corner frequency and the roll-off are not four ideas. They are four readings of one complex number, obtained from the same matrix that answers direct-current questions — and the straight-line sketch every engineer draws of them is itself a model, three decibels wrong exactly where it is read.

frequency · Frequency response One magnitude curve, two phase curves, and the one of them the magnitude decides. computed by solving, not by drawing. A passive lead network — a resistor with a capacitor across it, over a second resistor, its zero at 1.00 kHz — and the same network followed by a first-order all-pass. The two magnitudes agree to the last bits of a double at every one of the 1601 frequencies sampled, and the phases differ by as much as 180°. Bode's gain–phase integral, fed the magnitudes alone with their phases discarded, returns 39.29° at the corner against a solved 39.29°, and tracks the minimum-phase curve to 0.02° across the band — while being wrong about the second network by the all-pass's own phase, which is what excess phase means. The edge here is the span of the sweep rather than a frequency of the circuit: ±3 decades of magnitude carries 99.92% of the integral's weight, and what is left out is the tail of a logarithm.

The phase the magnitude already knows

For one class of network the phase is not an independent measurement: it is fixed everywhere by the magnitude, through an integral Bode wrote down in 1945. Fed nothing but the magnitudes of a solved lead network, that integral returns 39.289 degrees at the corner against a solved 39.289, and tracks the whole curve to two hundredths of a degree. Cascade an all-pass and the magnitudes agree to the last bits of a double while the phases part by 180 degrees — so the recovery is exact for one of the two and cannot see the other at all.

frequency · Minimum-phase A pair at Q = 0.7071: the sketch is -3.01 dB out at the corner. computed by solving, not by drawing. The solved magnitude against the two straight lines that stand in for it. At a conjugate pair the error at the corner is 20 log Q = -3.010 dB, which is unbounded in both directions, and the worst error anywhere is 3.010 dB at 1.00× the corner. No damping brings it inside 0.770 dB: that is the minimax, at Q = 0.9152, where the corner error and an interior maximum are equal.

The straight lines, and where they are not the curve

Two straight lines through a corner is the most-used approximation in this subject and almost the only one with no number attached. It has one, and it is exact: at a single real pole the sketch is 3.0103 decibels high at the corner and nowhere worse, and its error is the same a factor above the corner as the same factor below — a symmetry the construction does not suggest. A pole pair has no such bound at all, and the best any two-slope sketch can do on one is 0.770 decibels, at a quality factor of 0.9152.

frequency · Asymptotic approximation 40.0 dB of gain change and 180.0 degree-decades of phase, peaking at 109.8°. computed by solving, not by drawing. The gain and the phase of two leads of 10:1, together, each section buffered from the next. The gain changes by 40.000 dB from one end of the sweep to the other. The area under the phase, integrated on the solve over seven decades beyond the outermost corner, is 180.000 degree-decades, against 180.000 for ninety per twenty decibels of gain change. The peak is 109.806° at 3.16 kHz, below the 131.93° no arrangement of real leads sharing that rise can pass.

The phase a decibel buys

The area under a minimum-phase network's phase, counted in degrees across decades of frequency, is fixed by how far its gain moves from one end of the spectrum to the other: ninety degree-decades for every twenty decibels, however the network is arranged. One 100:1 lead peaks at 78.579 degrees and two 10:1 leads stacked together at 109.806, and both enclose 180.000. The peak is a design decision and the area is a bill — and a network that gives its gain back gives its phase back with it.

frequency · Minimum-phase The straight lines report 45.00° of margin, and the solved loop has 51.83°. computed by solving, not by drawing. The gain and phase of a loop made of an integrator and one pole, the corner at 1.00 kHz, with the integrator set so that the straight-line asymptotes cross unity at 1.00 kHz. The solved loop crosses at 786 Hz instead. Read at the lines' crossover, with the phase taken from the solve, the margin is 45.00°; at the loop's own crossover it is 51.83°. Closed, the loop is stable: the largest real part among its closed-loop poles is -5.00e-1 of the corner's angular frequency. The solved gain never rises above its asymptotes, so the lines cannot report more margin than the loop has.

The margin the straight lines report

A phase margin read off a sketch is read where the straight lines cross unity, and that is not where the loop does. For an integrator and one real pole the lines report 45.00 degrees on a loop that has 51.83 — short, and always short, because a real pole's response never rises above its asymptotes. A pair that peaks reverses the sign and removes the bound: at a quality factor of two the lines report 71.57 degrees on a loop that has no margin at all, and at five they report 82.41 on a loop that is unstable.

frequency · Asymptotic approximation The straight lines report 6.02 dB of gain margin on a loop that has none. computed by solving, not by drawing. The gain and phase of a loop made of an integrator and a pair at Q = 2, the corner at 1.00 kHz, the integrator set so that the straight-line asymptotes cross unity at 500 Hz. The loop's phase passes −180° at 1.00 kHz. There the lines put the loop gain at −6.02 dB and the solve at 0.00 dB, so the gain margin they report is 6.02 dB against 0.00 dB. The difference is 6.0206 dB, which is 20 log Q exactly with Q = 2, at any integrator gain. Closed, the loop is on the edge: the largest real part among its poles is -5.87e-17 of the corner's angular frequency.

The gain margin the straight lines get exactly wrong

A phase margin read off the straight lines is wrong by an amount that depends on where the loop crosses unity. A gain margin read off them is not: for an integrator and a pole pair the phase passes −180° at the pair's own frequency whatever the gain, and the lines are out there by exactly 20 log Q — 6.0206 dB for two real poles, 13.98 dB at a quality factor of five, at every integrator gain drawn. The stability condition for the loop turns out to be the same inequality: it is stable exactly when the gain margin the lines report exceeds that error.

frequency · Asymptotic approximation An order-8 Butterworth: the whole sketch is 3.0103 dB out at the corner, and its 4 sections −5.85 to +8.17 dB. computed by solving, not by drawing. The error of the straight-line sketch against the solved response — for each buffered section of an order-8 Butterworth lowpass at 1.00 kHz, and for the whole cascade. At the corner the sections are out by −5.852 dB (Q = 0.5098), −4.418 dB (Q = 0.6013), −0.915 dB (Q = 0.9000), +8.175 dB (Q = 2.5629), which add to −3.0103 dB: the whole filter's error, the same 10 log 2 as a single pole. The whole sketch is never further out than that anywhere; the section with the highest quality factor is +8.343 dB out at 1.04 kHz. The quality factors multiply to 1/√2.

The corner error a filter hides in its sections

The straight lines of an eighth-order Butterworth filter are 3.0103 dB out at the corner and nowhere worse — the same as one pole, at every order. The lines of the four sections it is built from are out by −5.85, −4.42, −0.92 and +8.17 dB there, which must add to the whole because the sections multiply. The whole sketch never gets worse with order and the worst section's error grows as 20 log(n/π). And the section the sketch misrepresents most is the one whose frequency error moves the filter most: 0.312 dB for one per cent, against the 0.173 dB any section's stopband shift gives.

frequency · Asymptotic approximation Lead zero at 500 Hz: the lines over-report by at most 3.12° with the loop's phase, and short by up to 20.6°. computed by solving, not by drawing. The error in the phase margin read off the straight lines of an integrator, two poles at 1.00 kHz and a lead section with its zero at 500 Hz and its pole 10 times higher, against where the lines cross unity, measured in decades from the zero. Gaps are placements where the lines are flat at unity or the loop crosses more than once. Read with the loop's own phase the reading is over by at most +3.12°, with the lines crossing −0.01 decades from the zero on a loop with 73.4° of margin, and never over on any loop with 60° or less; it is short by as much as 20.65°. Read with the phase off its straight lines too, on loops with 60° or less, it is never over.

The zero that lifts the lines

A phase margin read off the straight lines of an all-pole loop can only be short, and the proof takes three steps. A lead section breaks two of them: a zero's response lies above its lines, and between a zero and its pole the phase rises. The two breaks pull opposite ways, and measured across every placement of the crossing, in every sweep drawn, they leave an over-report of at most 6.58°, on a loop with 87.5°; on loops with sixty degrees or less it never exceeds 1.52°. The phase sketch is another matter: with the lines crossing at a lead zero above the plant's corner, it reports 17.91° on a loop with 1.17°.

frequency · Asymptotic approximation Four readings of one distance: |H| is |p| over the length of this arrow. The complex plane of a single-pole low-pass with its corner at 995 Hz. The pole sits at −995 Hz on the real axis and the faint circles are contours of constant magnitude — circles centred on the pole, because |H| = |p|/|s − p| and a constant magnitude is a constant distance. The imaginary axis crosses them, and where it crosses is the frequency response: 1.00 corners up the distance is 1.4142 times the pole's own, so the magnitude is 0.70711 and the phase is minus the 45.00° the separation subtends. Checked against the nodal solve at sixty frequencies over five decades, the two agree to 6.7e-16. So the four readings are one geometric fact: the corner is where the distance is √2 times its smallest, the roll-off is the distance becoming ω, the flat passband is the distance not yet changing, and the phase is the angle.

Four readings of one distance

Reactance, phase, the corner frequency and the roll-off are four readings of one complex number, and the number has a geometry. A single pole's magnitude is the pole's distance from the origin divided by its distance from the point on the imaginary axis, and the phase is minus the angle that separation subtends — agreeing with the nodal solve to a part in 10¹² over five decades. So the corner is where the distance is root two times its smallest, the roll-off is the distance becoming omega, the flat passband is the distance not yet changing, and the phase is the angle. One distance, four readings.

frequency · Frequency response The gain recovered from the phase alone, right in shape, with its level supplied by the gain at direct current. computed by solving, not by drawing. The lead, 1.00 kHz to 10.0 kHz, solved at 1601 frequencies over sixteen decades, and its gain rebuilt by Bode's companion integral from the phase alone over ±3 decades of 1.00 kHz. The integral returns a difference of log-magnitudes and nothing else, so the curve it gives has no level: drawn as if the direct-current gain were one, it is the dashed curve. Supplying the direct-current gain, −20.000 dB, places it. The placed curve lies within 0.055 dB of the solved gain across the drawn band, worst at 794 kHz.

The level the phase cannot know

Bode's gain–phase relation runs both ways. Given only the phase of a lead network, its companion integral returns the gain's shape across six decades to 0.055 dB. It cannot return the gain's level, because multiplying a network by a constant changes no phase anywhere. The level is one number, −20.000 dB here, and it has to come from outside the phase: one gain reading at any frequency supplies it. The same integral fed an all-pass reads a flat gain as a double pole, 6.021 dB down at the all-pass's frequency and 40.086 dB down a decade above it.

frequency · Minimum-phase A zero on the right at 1.00 kHz: the straight lines report 46.47° of margin, and the loop has 18.19°. computed by solving, not by drawing, the closed form held to the netlist — an inverting differentiator added back to its input makes the zero — at three frequencies and to the closed loop's poles. An integrator with a zero in the right half-plane at 1.00 kHz, the integrator set so that the straight lines cross unity at 950 Hz. The magnitude, and so the lines, are those of a zero on the left; the phase falls by the zero's angle where the left zero's would rise (dashed). The loop crosses at 3.04 kHz. Read at the lines' crossing with the loop's own phase the margin is 46.47°, and 133.53° with the phase the magnitude implies; at the loop's own crossing it is 18.19°.

The zero the lines put on the wrong side

A zero in the right half-plane has exactly the magnitude of its mirror on the left, so it has exactly the same straight lines, and nothing drawn from a magnitude can say which side it is on. The phase can: it falls where the mirror's rises. An integrator and such a zero, read off its lines, reports 90° − atan x of margin on a loop that has acos x, where x is where the lines cross as a fraction of the zero — so the lines say 45° of a loop with none. Behind one real pole the over-report reaches 37.6° on a loop of 2.1° when the zero sits below the pole, and vanishes when it sits above it: the reading is unsafe exactly when the zero comes first.

frequency · Asymptotic approximation

Named alongside it

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

Asymptotic approximationMinimum-phaseModel rangePhase marginCorner frequencyLoop gainThe quality factorAll-passDamping ratioExcess phaseGain phase relationComplex frequency

All concepts