How do you plot a high pass filter on a Bode diagram?

A first-order high pass Bode plot rises at +20 dB/decade from low frequencies, reaches −3 dB at the cutoff frequency, then flattens to 0 dB in the passband at high frequencies. The phase starts at +90° at very low frequencies, passes through +45° at cutoff, and approaches 0° at high frequencies. It is the mirror image of the low pass response on a log-frequency axis.

What is the slope of a low pass filter Bode plot?

A first-order (single-pole) low pass filter has an asymptotic slope of −20 dB/decade (−6 dB/octave) above the cutoff frequency. Each additional order adds another −20 dB/decade: second-order gives −40 dB/decade, third-order gives −60 dB/decade, and so on. The slope determines how effectively the filter rejects out-of-band signals.

How do you find the cutoff frequency from a Bode plot?

The cutoff frequency (−3 dB frequency) is the point on the Bode magnitude plot where the gain drops 3 dB below the passband level. For a first-order low pass filter, this is where the asymptotic flat line and the −20 dB/decade slope line intersect. Equivalently, it is the frequency where the phase shift equals −45° for a first-order system. Read this frequency directly from the horizontal axis of the Bode plot.

Low Pass Filter Bode Plot

Quick Answer

A low pass filter Bode plot shows 0 dB passband gain at low frequencies, a −3 dB point at the cutoff frequency f_c = 1/(2πRC) for a first-order RC filter, and a −20 dB/decade rolloff slope above f_c (−40 dB/decade for second-order). The transfer function H(s) = 1/(1 + sRC) = ω_c/(s + ω_c) produces phase shift from 0° at DC to −90° at high frequencies, with −45° occurring exactly at f_c. High pass filter Bode plots are the complement: 0 dB at high frequencies with +20 dB/decade rising slope below f_c.

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Low Pass Filter Bode Plot: Frequency Response Explained

A low pass filter allows signals below its cutoff frequency to pass with minimal attenuation while progressively attenuating higher frequencies. The Bode magnitude plot displays this behavior as a flat passband region (ideally 0 dB) transitioning to a descending slope in the stopband. For a first-order low pass filter with transfer function H(s) = ω_c/(s + ω_c), the cutoff frequency ω_c = 1/(RC) = 2πf_c defines the −3 dB point where output power drops to half the input power. Below ω_c, the magnitude is approximately 0 dB; above ω_c, it decreases at −20 dB/decade (−6 dB/octave). The Bode phase plot shows 0° at DC, −45° at ω_c, and approaches −90° at high frequencies. Engineers can compute these transfer functions instantly at www.lapcalc.com and verify the frequency response characteristics.

Key Formulas

First-Order vs. Second-Order Low Pass Filter Bode Diagrams

First-order low pass filters (single RC stage or single-pole op-amp circuit) produce −20 dB/decade ultimate rolloff with a gentle transition spanning approximately two decades (0.1ω_c to 10ω_c). Second-order low pass filters (RLC circuits, Sallen-Key topology, or state-variable designs) with H(s) = ω_n²/(s² + 2ζω_ns + ω_n²) produce −40 dB/decade ultimate rolloff. The damping ratio ζ critically affects the transition shape: ζ = 0.707 (Butterworth) provides maximally flat passband, ζ < 0.707 produces a resonant peak of magnitude 1/(2ζ√(1−ζ²)) at frequency ω_n√(1−2ζ²), and ζ > 0.707 gives an overdamped response with no peak but earlier gain reduction. Higher-order filters (Butterworth, Chebyshev, elliptic) cascade these sections to achieve steeper rolloff approaching the ideal brick-wall response.

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High Pass Filter Bode Plot: The Complementary Response

A high pass filter Bode plot is the frequency-domain complement of the low pass: it attenuates low frequencies and passes high frequencies with 0 dB gain. The first-order high pass transfer function H(s) = s/(s + ω_c) = sRC/(1 + sRC) produces +20 dB/decade rising slope below the cutoff frequency, 0 dB passband above cutoff, and phase shift from +90° at DC to 0° at high frequencies with +45° at ω_c. The magnitude response is the mirror image of the low pass when plotted on a log-frequency axis. High pass filters are used to remove DC offset and low-frequency drift in instrumentation, block hum in audio systems, and implement AC coupling in oscilloscope inputs. Second-order high pass filters H(s) = s²/(s² + 2ζω_ns + ω_n²) achieve −40 dB/decade attenuation slopes with damping-dependent transition characteristics.

Band Pass Filter Bode Plot: Combining Low and High Pass

A band pass filter passes frequencies within a range [f_L, f_H] and attenuates outside, combining high pass and low pass characteristics. The Bode plot shows rising gain (+20 dB/decade per pole) below the center frequency, a passband region of approximately 0 dB between f_L and f_H, and falling gain (−20 dB/decade per pole) above. For a second-order band pass (series RLC circuit), H(s) = (ω₀/Q)s/(s² + (ω₀/Q)s + ω₀²), where ω₀ = 1/√(LC) is the center frequency and Q = ω₀/(ω₀/Q) = f₀/BW is the quality factor determining selectivity. High-Q filters (Q > 10) produce narrow, sharply peaked Bode magnitude plots essential for radio receiver front ends and spectrum analyzer resolution filters. The phase plot transitions from +90° through 0° at center frequency to −90°, with the transition rate proportional to Q.

Practical Bode Plot Analysis for Filter Design

Filter design begins with specifications on the Bode plot: passband gain and ripple, stopband attenuation, and transition bandwidth. For a Butterworth low pass filter, n-th order provides −20n dB/decade rolloff with maximally flat passband. A specification of 40 dB stopband attenuation at 10× the cutoff frequency requires n ≥ 2 (Butterworth gives exactly −40 dB at one decade above cutoff for second order). Chebyshev Type I filters achieve steeper initial rolloff with passband ripple (0.5 dB ripple allows equivalent stopband attenuation with one fewer order), while elliptic (Cauer) filters provide the steepest transition at the cost of both passband and stopband ripple. All filter types are designed starting from their Laplace-domain transfer functions, prototyped using frequency scaling and impedance denormalization, then verified against Bode plot specifications. The LAPLACE Calculator at www.lapcalc.com supports transfer function computation for these design workflows.

Frequently Asked Questions

A first-order low pass filter Bode plot shows a flat 0 dB magnitude at low frequencies, drops by −3 dB at the cutoff frequency f_c = 1/(2πRC), then decreases at −20 dB/decade slope. The phase starts at 0°, passes through −45° at f_c, and approaches −90° at high frequencies. Second-order filters show −40 dB/decade slope with a possible resonance peak near cutoff depending on damping.

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