One solve, read four ways
A resistor and a capacitor in series, with the output taken across the capacitor. It is the second circuit anybody meets and the most-drawn response in the subject, and the interesting thing about it is not the shape of the curve. It is that the curve requires no new theory at all: it is the same nodal solve that answers questions about batteries, evaluated at a different kind of number.
The one substitution
Take the direct-current analysis of a network and change one thing: allow the quantity multiplying each element to be complex. A capacitance C contributes an admittance sC; an inductance L contributes an impedance sL; a resistance contributes 1/R as before, with no s in it at all. Then set s = j2πf and solve.
What comes out at each node is a complex number, and the two things it carries are exactly the two things a sinusoidal steady state has: a magnitude, which is how large the sinusoid is at that node, and an angle, which is how far it lags or leads the source. Nothing else is needed. There is no separate theory of alternating current, no separate set of laws, and no separate solver — the matrix, the elimination, the checks on the answer and the refusals are all the ones from the direct-current case, running on complex arithmetic.
This is worth insisting on because the subject is usually taught the other way round, with phasors introduced as a new formalism after resistive circuits are finished. Presented as a substitution instead, several apparently separate results collapse into one:
- Direct current is the case s = 0. A capacitor’s admittance is then zero, which is an open circuit, and an inductor’s impedance is zero, which is a short. Neither of those has to be remembered as a rule; both follow.
- A frequency response is the case s on the imaginary axis.
- A natural frequency — a mode the circuit rings at with nothing driving it — is a value of s at which the matrix has no inverse, so a non-zero answer is possible with a zero source.
- A transient is a sum over those natural frequencies.
Four topics, one solve, three kinds of argument.
Reactance is not resistance, and the difference is the angle
The magnitude of a capacitor’s impedance is 1/(2πfC), which has the units of resistance and falls with frequency. That much is usually the first thing said about it, and taken alone it produces a persistent error: treating reactances and resistances as quantities that add.
They do not, because the capacitor’s impedance is not merely 1/(2πfC) — it is that magnitude at an angle of −90°. A resistance of 500 ohms and a reactance of 500 ohms in series do not make 1000 ohms; they make 707 ohms, at an angle of −45°. The Pythagorean answer is not an approximation or a convention. It is what adding two complex numbers at right angles gives, and the entire reason complex arithmetic is used here rather than being an elegant option.
The consequence for the low-pass above is that at the corner frequency — where the reactance of the capacitor happens to equal the resistance — the output is not half the input, which is what adding the magnitudes would give. It is 1/√2 of the input, which is 0.707, which is −3.01 decibels. The famous number is the diagonal of a square, and it appears in this subject about as often as any number does.
The sketch, and where it costs something
Every engineer draws the asymptotic version of that response, and it is a genuinely good tool: horizontal at unity below the corner, falling at twenty decibels per decade above it, meeting at the corner in a sharp angle. It can be drawn on the back of anything, it composes — a response with four corners is four straight segments — and it gets the shape right everywhere.
It is also a model, and this site’s rule applies to it as much as to anything else. Its error is exactly computable and is largest precisely where the response is usually read:
- at the corner, the sketch is 3.01 dB high
- half a decade either side, about 1 dB
- it is within a tenth of a decibel only below about a seventh of the corner and above about seven times it
Those numbers are measured in the figure above against the solved response, not quoted. The one-tenth-of-a-decibel boundary is found by bisection on the difference between the two curves, and it moves with the slider — which is to say it does not move at all as a ratio, which is the point: the error of the sketch depends only on the distance from the corner in decades, never on where the corner is.
The practical reading is that the sketch is excellent for the shape and unusable for the value near a corner, which is a shame, because near a corner is where circuits are usually operated. A filter specified to pass a band up to its corner passes the top of that band at 0.707 of its intended amplitude, and the sketch says 1.00.
The phase is the part that gets forgotten
The magnitude of a first-order response is a curve that a reader will recognise. The phase is a curve that is repeatedly ignored, and it decays far more slowly than intuition suggests.
At the corner, the output lags the input by 45°, which is unsurprising. What is surprising is how far the influence reaches: at a tenth of the corner frequency, a decade below anything that could be called filtering, the phase is already −5.7°. At ten times the corner it is −84°, not −90°. The phase of a single pole is essentially never at either of its asymptotes anywhere a reader is looking.
That matters much more than it appears to, and the reason is the feedback field later in this collection. A feedback loop becomes unstable when the phase around it reaches −180° while the gain is still above unity, and the phase contributed by each pole in the loop reaches deep into the decades below it. A pole a full decade above the frequency where a loop crosses unity gain still takes six degrees of margin away. Poles that “are not in the way” are in the way.
Three routes to the same number
The figure at the top of this essay has its magnitude computed three separate ways, and all three are required to agree before it is drawn.
By nodal analysis. Assemble the matrix at s = j2πf, solve, read the output node.
By the pole–zero factorisation. The determinant of that matrix is a polynomial in s; sampling it around a circle in the complex plane and transforming recovers the polynomial, whose roots are the poles. Evaluating the resulting rational function is then a completely different computation from inverting a matrix. The two agree to about a part in 1015.
By the chain-matrix product. A ladder network — a series element, then a shunt element, then a series element, and so on — is the product of a sequence of two-by-two matrices, one shape for series and another for shunt, and the voltage ratio falls out of the product’s first entry. Nothing in that method touches nodal analysis, node numbering or elimination. It agrees with the other two to twelve decimal places at the corner frequency.
Three routes is one more than this collection usually manages, and the reason for going to the trouble here is that this is the figure everything else is checked against. A first-order low-pass is the calibration standard of the whole subject: if the machinery is wrong about this, it is wrong about everything, and it would be wrong quietly.
Where the schematic goes
This is the point at which the site’s drawing convention starts to earn itself, so it is worth stating why the schematic in the figure above is small and in a corner.
A resistor and a capacitor can be drawn in a hundred arrangements and they are all the same circuit. Nothing about the placement, the wire lengths or the orientation carries information; the netlist is the content. Give a schematic the middle of the canvas and the canvas has been spent on the one part of the figure that could not have been wrong.
The response, on the other hand, can be wrong in a dozen ways — the wrong corner, the wrong slope, the wrong phase, an axis mislabelled, a curve that does not reach its asymptote — and every one of those is a thing a reader could catch. So the response gets the canvas. This is not a stylistic preference; it is the same reasoning that decides what a figure asserts.
What the corner frequency is a property of
A small thing that is worth being exact about, because it is the source of a common confusion when a second component is added.
The corner of the low-pass in this essay is at 1/(2πRC) — a property of the pair, not of either element. Doubling the resistance and halving the capacitance leaves it where it was, and changes something else entirely: the impedance the circuit presents to whatever drives it, and the impedance it presents to whatever it drives.
So a first-order filter has two independent quantities in it, and only one of them appears in the response. The other is the loading behaviour from the first field of this collection, and it decides whether the filter drawn is the filter obtained. A one-megohm and one-picofarad low-pass has the same corner as a one-kilohm and one-microfarad one, and the first is destroyed by any load below fifty megohms while the second drives almost anything.
That is the same observation the divider essay makes, arriving here through a different door: the quantity a response plot shows is a ratio, and the quantity that decides whether the plot is true of the assembled circuit is a magnitude.
Where the second corner comes from
Cascading two of these does not give a response with two corners at the same frequency, and the reason is worth a paragraph because the mistake is easy and the symptom is subtle.
Connect the output of one resistor-capacitor pair to the input of another, and the second pair loads the first: it draws current that the first’s capacitor would otherwise have received. The result has two poles, as expected, but they are not both at 1/(2πRC) — they are split, one lower and one higher, by an amount that depends on the ratio of the two sections’ impedances.
For two identical sections the poles land at about 0.38 and 2.6 times the nominal corner, which is a factor of seven apart. The response is therefore much gentler around the corner than two coincident poles would give, and a designer expecting a sharp second-order roll-off finds a lazy one.
The remedies are the two from the first field, and they are the two used everywhere: make the second section’s impedance much larger than the first’s, so the loading is small, or put a follower between them, so there is none. The filters elsewhere in this collection use the second, and say so.
Decades, octaves and the shape of the axis
One convention worth stating plainly, because it is assumed everywhere afterwards. Frequency axes on this site are logarithmic, always, and gains are in decibels.
The reason is not that the numbers are large. It is that both of the operations the subject cares about become straight lines under that transformation. Cascading two circuits multiplies their responses, which adds their decibels; a response falling as a power of frequency becomes a straight line whose slope is that power. A first-order response falls as 1/f far above its corner, which is twenty decibels per decade, which is one straight line — and a fifth-order filter falls at a hundred decibels per decade, which is another. On a linear axis neither is a line and neither can be read.
The unit deserves a note as well. Twenty decibels per decade and six decibels per octave are the same slope, since an octave is a factor of two and 20·log₁₀2 is 6.02. The second form is more common in audio, the first in almost everything else, and the two are used interchangeably in the literature in a way that has confused generations of readers into thinking they describe different things.
The other three responses of the same pair
A resistor and a capacitor make four two-terminal-pair networks depending on where the source and the output go, and they are worth listing because they exhaust the first-order possibilities.
Output across the capacitor gives the low-pass in the figure. Output across the resistor gives a high-pass with the same corner: flat above it, falling at twenty decibels a decade below, and with a phase that runs from +90° to zero. The two are complementary in a precise sense — their magnitudes squared add to one at every frequency — which is the reason a first-order crossover splits a signal into two halves that recombine exactly.
Swapping which element is in series and which is across gives the same two responses again, so there are two distinct behaviours rather than four. And there is no first-order band-pass and no first-order notch, because a single pole cannot produce a peak: peaks require a pole pair, which requires two reactive elements, which is the subject of the resonance essay in this field.
That exhaustiveness is worth having because it bounds what can be expected from one capacitor. A requirement that cannot be met by a monotone twenty-decibel-per-decade slope needs a second reactive element, and no cleverness with a single one will substitute.
Reading a Bode plot backwards
A last practical note, since the figures in this field are usually read in the other direction.
Given a measured response, the poles and zeros can be read off it approximately by fitting straight lines: every corner where the slope changes by −20 dB/decade is a pole, every corner where it changes by +20 is a zero, and the frequency of the corner is the frequency of the pole or zero. That is the asymptotic sketch used as a measuring instrument rather than as a drawing aid.
It works, it is how a great deal of practical characterisation is done, and its accuracy is exactly the accuracy this essay measured: about three decibels at each corner, better than a tenth of a decibel a factor of seven away from one. So a corner frequency read off a measured plot by extending asymptotes is good to a few per cent, and a level read off the same plot near a corner is not good to three decibels — which is a factor of 1.41 and rather more than most readers expect from a technique they have been taught to trust.
What comes next
The rest of this field takes the same solve and asks harder questions of it. The next essay draws the complex numbers as arrows rather than as curves, which makes Kirchhoff’s voltage law visible as a closed polygon and puts a number on the assumption underneath the whole frequency axis — that the circuit has already settled. Then resonance, where two reactances cancel and the bandwidth that results can be measured and compared against what the components predict. And then the figure that breaks the symbol: what a capacitor’s impedance actually does, which follows 1/(2πfC) for three or four decades and then turns round and climbs.