The heat a delay predicts

What a filter does not pass, it sends back found that a passive LC ladder filter built from lossless reactances has two destinations for power and no third: what it does not transmit, it reflects, and the two sum to what the source makes available to two parts in $10^{15}$. Giving each inductor a finite Q opened a third destination — heat — and the budget still closed, to $10^{-10}$, with the dissipated power computed separately from the resistors’ own currents. At Q 50 the filter absorbed up to 19.5 per cent of the available power, and in every case the absorption peaked near the band edge.

That essay offered a reason — the band edge is where a Chebyshev ladder filter stores the most energy for the power it passes, and where its group delay peaks — and left the reason as a question. Is the absorbed fraction, to first order, simply the group delay times the loss rate? If it is, every lossy filter’s droop is predicted by one curve the lossless design already has, and a designer can read the heat off a delay plot.

The answer is yes, with two findings that turn out to matter more than the rule. The group delay predicts the heat in the stopband as well as the passband, weighted by the power the source makes available rather than the power the filter passes; and with losses in only one kind of element, the rule misses by a factor that says where the filter keeps its energy.

Energy, delay, and the identity that joins them #

An inductor carrying a current stores magnetic energy, a capacitor holding a voltage stores electric energy, and a resistance in series with the inductor, of value R, dissipates a power that is exactly $2(R/L)$ times the inductor’s time-averaged stored energy. A conductance across a capacitor does the same with $2(G/C)$. So a filter whose every element loses energy at the same fractional rate σ — every inductor with $R/L = \sigma$, every capacitor with $G/C = \sigma$ — dissipates $2\sigma$ times its total stored energy.

The other half is an identity about lossless networks. For a lossless two-port driven from a matched source, the time-averaged energy it stores, divided by the power the source makes available, is

$$\frac{W}{P_{\text{avail}}} = |t|^2\,\tau_{21} + |\rho|^2\,\tau_{11},$$

the group delay of the transmission weighted by the fraction transmitted, plus the group delay of the reflection weighted by the fraction reflected. It is the circuit form of what physicists call the Wigner–Smith delay: energy that enters a network and comes out later was stored inside it in between, and how much later is how much was stored. In the passband, where almost nothing is reflected, it is the familiar group delay; in the stopband, where almost everything is, it is the delay of the reflection.

Put the two together and, to first order in the loss, the absorbed fraction is $2\sigma$ times that delay sum.

The hero figure tests it in the case a real filter is in: inductors lossy, capacitors not. Each inductor has a series resistance of $\omega_c L/Q$, so each loses energy at $\sigma = 2\pi f_c/Q$, and with the magnetic energy roughly half the total the prediction is $\sigma$ times the delay sum rather than $2\sigma$. At Q 100 the absorption peaks at 10.53 per cent at 0.957 of the corner, where the prediction is 11.24 per cent; across the ripple band from 0.3 to 0.95 of the corner the two agree to seven per cent. That seven per cent does not shrink as the Q rises — it is still seven at Q 400 — so it is not an error of order in the loss. It is the energy split, which the next figures separate.

The reflection essay’s own figure, drawn at Q 50, shows what is being predicted: an absorption rising slowly through the passband, peaking hard at the band edge at 19.5 per cent, and falling through the stopband without reaching zero. The prediction there is 22.5 per cent at the peak. A lossy Chebyshev’s familiar rounded corner, the passband sagging towards the edge rather than rippling evenly, is this curve seen from the load: the power the delay says will be stored near the edge is the power that is not delivered.

Why the stopband still stores energy #

The identity has a consequence that is easy to state and hard to believe until it is measured: for this ladder filter the reflection’s delay equals the transmission’s, at every frequency, to a part in ten thousand. A Chebyshev ladder filter’s reflection vanishes only at isolated frequencies on the frequency axis — the passband maxima — and a reflection whose zeros lie on the axis has a phase that only jumps there, so between the jumps it turns at exactly the rate the transmission’s does. The weighted sum then collapses: $|t|^2\tau_{21} + |\rho|^2\tau_{11} = \tau_{21}$, and the stored energy over the available power is the group delay itself, whatever fraction is transmitted.

That is what makes the stopband absorption at the right of the hero figure non-zero. At 1.3 times the corner the filter transmits under one per cent of the available power, yet its group delay is still a quarter of a period of the corner, and so it stores a quarter of a period’s worth of the available power. The stored energy belongs to the wave that enters and is reflected, not to the wave that passes; a filter rejecting a harmonic is holding that harmonic’s energy for a quarter of a period on its way back out, and a lossy coil takes its share during the stay. A designer who estimated the stopband heat from the transmitted power — the obvious guess, and the one that gives $\sigma\tau_{21}|t|^2$ — would predict almost nothing there and be wrong by a factor of a hundred.

The same is true of the fourth-order ladder filter, which is not symmetric and is terminated in 1.984 times its source resistance: its reflection’s delay equals its transmission’s too. A fifth-order Butterworth ladder filter, whose reflection zeros all sit at direct current, shows the same equality. The property belongs to where the reflection zeros sit, not to the symmetry: a reflection polynomial whose roots lie on the frequency axis adds phase only in jumps. A filter designed for flat delay rather than flat magnitude has reflection zeros off the axis, by the same argument has a reflection delay of its own, and for it the weighted sum is the quantity to use rather than the group delay alone.

Exact at small loss, and what the split adds #

The seven per cent is two different errors, and they separate cleanly when the loss is made small and put into each kind of element in turn.

At Q 10,000, with the loss rate put first into the inductors alone and then into the capacitors alone, the average of the two absorptions is the delay sum times the loss rate to 0.087 per cent at every frequency from direct current to 1.6 times the corner. That is the identity confirmed to first order, with nothing fitted: the lossy filters are solved as netlists, the delays are central differences of the lossless filter’s phases, and the two are compared.

Each alone departs from the average by the share of the energy it can reach. The inductors dissipate in proportion to the magnetic energy and the capacitors to the electric, so their ratio at equal loss rates is the ratio of magnetic to electric stored energy. Near direct current it is 2.37 for this ladder filter: the series inductors carry the full load current while the shunt capacitors see only a small voltage swing, so the energy sits in the inductors. Through the ripple band it is near one — 0.95 at half the corner, 1.02 at 0.95 — because at the frequencies a Chebyshev ladder filter is designed for, its sections are close to resonance and resonance shares energy equally. In the stopband it climbs again, to 4.63 at 1.6 times the corner.

That is the first amendment. A designer with lossy coils and good capacitors, which is nearly every real LC filter, can use the delay sum for the passband with confidence and must not use it in the stopband without the split, because there the coils hold most of the energy and absorb most of the power.

The dual stores the other kind #

A fifth-order ladder filter can be built two ways from the same prototype: starting with a series inductor, as here, or with a shunt capacitor, swapping every element for its dual. The two transmit identically — to $10^{-10}$ in the figure below — and so have identical delays. They do not store their energy in the same places.

The two curves are mirror images about one. Where the series-first ladder filter stores 4.63 times as much magnetic energy as electric, at 1.6 times the corner, the shunt-first ladder filter stores 4.63 times as much electric; near direct current 2.37 against 0.42. The product of the two ratios is one at every frequency, to a fiftieth, because duality exchanges every current with a voltage and every inductor with a capacitor while keeping the network’s response.

The practical reading is unexpected and useful. The stopband absorption of a filter with lossy coils depends on which way round it was built: the series-first ladder filter, whose stopband energy is in its coils, absorbs 1.64 times what the delay sum predicts at the coils’ loss rate at 1.6 times the corner, while the shunt-first ladder filter with the same coils absorbs 0.36 times it — four and a half times less. A designer who must absorb stopband power rather than reflect it — protecting a source that cannot take its own harmonics back — can choose the realisation for that, at no cost to the passband, which is the same in both.

The higher the order, the hotter the edge #

The delay at a Chebyshev’s band edge grows with the order, and the absorption follows it. The orders are odd because an even-order Chebyshev ladder filter cannot be built between equal terminations at all — the termination an even order cannot have is 1.984 times the source at this ripple — and equal terminations are what every figure here assumes. With inductors of Q 100 the peak absorption is 4.55 per cent at order three, 10.53 at five, 18.58 at seven and 27.68 at nine, against predictions of 4.33, 11.26, 21.68 and 35.55 from the peak delay sum. Making the capacitors lossy at the same Q nearly doubles each: 8.09, 19.09, 32.02 and 44.55 per cent.

The second amendment shows here. At order three the prediction is slightly low, by 5 per cent, and from order five upward it is increasingly high, by 7, 17 and 28 per cent. The prediction is a first-order statement, true as the loss goes to zero, and a ninth-order Chebyshev with inductors of Q 100 is not at small loss: it absorbs over a quarter of the power at its edge. A filter that absorbs a quarter of its power is no longer storing what the lossless filter stores, because the lossy filter’s resonances are broadened and its delay peak is lower, and the first-order law cannot know that.

A first-order law, and the size of its error #

With every element at the same Q, so that the split drops out entirely, the law’s overstatement of the band-edge peak is a clean measure of its order. It is 1.8 per cent at Q 1,000, 4.5 at 400, 18 at 100 and 78 at 25, and its product with Q tends to a constant, 17.5. The error is second order in the loss, as the law’s derivation says it must be: a correction of relative size one over Q on a quantity that is itself proportional to one over Q.

For design that gives a usable rule with its own error bar. The delay sum times the loss rate predicts a lossy filter’s absorption, and the prediction is high by roughly $17.5/Q$ at the band edge of a fifth-order 0.5 dB Chebyshev — under 5 per cent above Q 400, under 20 above Q 100. A filter whose components are worse than that is losing enough power that its response has to be computed rather than predicted; the Q the components allow is where that regime is drawn as a ceiling rather than a perturbation.

The droop, read in decibels #

Turned into the units a filter’s response is plotted in, the law says the transmission loss added by the components is

$$\Delta A \approx 4.343\cdot 2\sigma\cdot\left(\tau_{21} + \frac{|\rho|^2}{|t|^2}\tau_{11}\right)\ \text{dB},$$

and in the passband, where $|\rho|^2$ is a few per cent at most, that is $8.686\,\sigma\tau_{21}$ decibels — the familiar rule that a filter’s dissipative loss follows its group delay, which this essay has measured rather than assumed. For the fifth-order ladder filter at Q 100, with both kinds of element lossy, the edge delay of 1.79 periods of the corner gives a predicted loss of a decibel there, and the passband’s 0.79 periods at half the corner gives 0.43 dB. A delay plot of the lossless design is therefore a plot of where the lossy filter will sag, and the sag’s depth in decibels is the delay in periods times $54.6/Q$.

The stopband version of the rule is worth stating separately because it is not about the response at all. There the transmission is already tens of decibels down and a fraction of a decibel more changes nothing anyone measures; what the loss changes is where the rejected power goes. The absorbed fraction is $2\sigma\tau_{21}$ there too, so a filter rejecting a harmonic whose delay is a tenth of a period at that frequency, built from components of Q 100, turns 1.3 per cent of that harmonic’s available power into heat and reflects the rest. For a signal filter that is negligible; for the harmonic filter on a power amplifier’s output it is the share of the harmonic power the filter’s own components have to be rated to dissipate, and the delay plot says it before any part is chosen.

How the numbers were obtained #

Every filter here is a netlist solved at each frequency. The lossy versions add a series resistance of $\omega_c L/Q$ to each inductor or a shunt conductance of $\omega_c C/Q$ to each capacitor, so every element loses energy at the same rate $2\pi f_c/Q$; the absorbed power is the sum of the loss resistors’ dissipations from their own currents, as in the reflection essay. The delays are central differences of the lossless ladder filter’s transmission phase and reflection phase, with a step of $10^{-5}$ of the frequency. The dual ladder filter is built from the same prototype values with every series inductor replaced by a shunt capacitor and every shunt capacitor by a series inductor, at the same impedance and frequency scaling.

What it leaves out #

Loss that depends on frequency. A real coil’s resistance rises with frequency from skin and proximity effect and a real capacitor’s dissipation factor is nearly constant rather than its conductance, so neither has a constant loss rate. The law is local — it uses the loss rate at each frequency — and survives that; the constant 17.5 does not.

The sources and loads. The terminations are exact resistances, the two the ladder filter was designed between. A filter driven from an amplifier’s output impedance stores energy differently, because what it reflects is reflected again, and the delay identity holds for the network the source sees rather than for the filter alone.

Still open: the stopband’s impedance, the matched filter, and the delay a designer can buy back #

What the source sees in the stopband. The split showed the series-first ladder filter storing its stopband energy in its coils and the shunt-first one in its capacitors, which is to say that one looks inductive from its input and the other capacitive. For a source that is an amplifier rather than a resistor, that reactance is the load it sees at its own harmonics, and whether it can make an amplifier misbehave depends on which way round the filter was built. The load a filter is in its stopband measures it from both ends.

A filter designed for equal ripple in its reflection. Since reflected and transmitted power are complementary, a filter equiripple in one is equiripple in the other, and the reflection delay $\tau_{11}$ then has its own band-edge peak. A diplexer whose two filters share an input stores energy in both, and the delay identity summed over both filters would say where its heat goes.

Delay as a design variable. The law says that a filter which stores less at its edge absorbs less there, and the same transition band can be met with less edge delay by giving up some stopband steepness. How much heat a designer buys back per decibel of rejection surrendered — between a Chebyshev and a filter with a gentler edge — is the trade the law makes computable without building either.