Frequency Response and Bode Plots

Decibels and decades, first- and second-order factors, straight-line asymptotes, resonance and Q

Bell Labs — the 1930s

Hendrik Bode joins Bell Labs straight from graduate school to design electrical filters and equalizers for the telephone network, and in 1929 moves to its Mathematical Research Group. Amplifiers for long-distance lines need dozens of stages and feedback around them, and their frequency responses are products of many factors — tedious to compute and hard to see. In the 1930s Bode devises a simple but accurate way to draw them: plot the gain in decibels and the phase against frequency on a logarithmic axis, and each factor of the response becomes, approximately, a pair of straight lines. Add the lines and you have the whole system, sketched with a ruler (Bode plot; Hendrik Wade Bode). Nearly a century later, every control engineer, circuit designer and audio engineer still draws frequency responses that way. This lesson is how.

A frequency response

H(j\omega)

is a complex number at each frequency. Bode plots show its magnitude in decibels,

20\log_{10}|H(j\omega)|

, and its phase

\angle H(j\omega)

, both against

\log\omega

. Two facts make the axes worth it:

logarithms turn products into sums, so a response that is a product of factors becomes a sum of curves;
a logarithmic frequency axis gives each decade (factor of 10) the same width, so behaviour from 1 Hz to 1 MHz fits on one page.

Remember the anchors: a factor 10 in amplitude is

+20

dB, a factor 2 about

+6

dB,

1/\sqrt2

-3

dB.

💡A first-order pole

H(j\omega) = \dfrac{1}{1 + j\omega/\omega_p}

— the RC low-pass, with corner frequency

\omega_p

Magnitude: flat at 0 dB below $\omega_p$ ; above it, a straight line falling at −20 dB per decade; at the corner itself, −3 dB.
Phase: 0° well below the corner, −90° well above, −45° at the corner, changing over about two decades centred on it.

A first-order zero,

1 + j\omega/\omega_z

, is the mirror image: +20 dB/decade and +90° above

\omega_z

Derivation

Where the straight lines come from

For the pole,

|H| = 1/\sqrt{1 + (\omega/\omega_p)^2}

, so

20\log_{10}|H| = -10\log_{10}\left(1 + (\omega/\omega_p)^2\right).

For $\omega \ll \omega_p$ the bracket is about 1: the magnitude is about 0 dB.
For $\omega \gg \omega_p$ the bracket is about $(\omega/\omega_p)^2$ : the magnitude is about $-20\log_{10}(\omega/\omega_p)$ dB — a straight line in $\log\omega$ , falling 20 dB per decade, that passes through 0 dB at $\omega_p$ .
At $\omega = \omega_p$ exactly, $-10\log_{10}2 = -3.01$ dB: the true curve is 3 dB below the corner of the asymptotes, and only about 1 dB below one octave either side.

The phase is

-\arctan(\omega/\omega_p)

-45°

at the corner, within

6°

of its asymptotes a decade away.

💡A second-order factor

H(j\omega) = \dfrac{\omega_n^2}{(j\omega)^2 + 2\zeta\omega_n(j\omega) + \omega_n^2}

flat below $\omega_n$ , then −40 dB per decade above it;
phase from 0° to −180°, passing −90° at $\omega_n$ ;
for small damping a resonant peak near $\omega_n$ of height about $\dfrac{1}{2\zeta}$ , i.e. $Q = \dfrac{1}{2\zeta}$ — a damping ratio of 0.1 gives a peak of 5, or +14 dB. The asymptotes miss the peak entirely, so it must be added by hand.

📐Sketching a response

Sketch

H(j\omega) = \dfrac{10\,(1 + j\omega/10)}{(1 + j\omega)(1 + j\omega/100)}

Low frequencies: $|H| \to 10$ , so start at +20 dB.
Corner at 1 rad/s (pole): bend down at −20 dB/decade. By 10 rad/s the asymptote has fallen to 0 dB.
Corner at 10 (zero): +20 dB/decade cancels the slope: flat at 0 dB.
Corner at 100 (pole): bend down again, −20 dB/decade.

The exact curve misses each corner by 3 dB — below at the poles, above at the zero: 17.0 dB at 1 rad/s instead of 20, 2.9 dB at 10 instead of 0, −3.0 dB at 100. The phase dips to about −57° near 3 rad/s, recovers to −33° at 30 rad/s as the zero lifts it, and heads for −90° after the last pole. Four lines and three corrections, and you can see the system.

💡Resonance and Q

A resonant system — a tuned circuit, a guitar string, an aircraft wing in flutter — has a sharp peak at

\omega_0

. Its quality factor

Q

is the ratio of the peak frequency to the width of the peak between its −3 dB points:

Q = \frac{\omega_0}{\Delta\omega_{3\,\mathrm{dB}}} \approx \frac{1}{2\zeta}.

High

Q

means selectivity (a radio tuner picks one station) and also long ringing (the same circuit, kicked, rings for about

Q

cycles). A series RLC circuit has

\omega_0 = 1/\sqrt{LC}

and

Q = \frac{1}{R}\sqrt{L/C}

Magnitude decides phase — sometimes. For a stable system with no zeros in the right half of the complex plane — a minimum-phase system — the phase is fixed by the magnitude curve: a slope of

-20n

dB/decade held over a wide band comes with about

-90n

degrees of phase. Bode proved this gain–phase relation, and it is why a single magnitude plot often tells a control engineer whether a loop will be stable. Systems with right-half-plane zeros or pure delays have extra phase lag the magnitude does not show.

when you can measure what you are speaking about, and express it in numbers, you know something about it

– William Thomson, Lord Kelvin, lecture on "Electrical Units of Measurement" (3 May 1883), Popular Lectures Vol. I, quoted on Wikiquote

🕰Straight lines at Bell Labs

Bode's asymptotes came out of telephone engineering: designing feedback amplifiers for long-distance lines, where many stages had to be stabilised together. His gain–phase relations, worked out alongside Harry Nyquist's stability criterion, became the foundation of classical control (Lesson 7-4) (Hendrik Wade Bode).

⚠Power decibels and amplitude decibels

Decibels measure power ratios:

10\log_{10}(P_2/P_1)

. Amplitudes square to give power, so for voltages, currents and transfer functions it is

20\log_{10}|H|

. A gain of 2 in amplitude is +6 dB; a gain of 2 in power is +3 dB. Mixing the two conventions is the classic slip.

💡Where you will use this

Lesson 4-4 designs filters on these axes. Lesson 7-3 connects poles and zeros to the shape of the plot, and Lesson 7-4 reads stability margins straight off it. In discrete time the same reading applies to

H(e^{j\Omega})

(Lesson 8-3). Datasheets for amplifiers, sensors and loudspeakers are mostly Bode plots.

📚Further reading

Oppenheim and Willsky, Signals and Systems, sections 6.2–6.5 — log-magnitude plots of first- and second-order systems.
Hendrik W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, 1945) — the source.
Bode plot on Wikipedia — asymptotes and corrections in detail.

↓But there is a problem

We can read any frequency response and sketch one from its factors. Now turn the problem round: start from what you want — pass everything below 1 kHz, reject everything above 2 kHz — and find a response that does it. The perfect answer, a brick wall, turns out to be impossible to build. In 1930 a British engineer found the best compromise in one particular sense: as flat as possible where it matters.