This post provides definitions for important terms used in the analysis and design of first- and second-order filters. The terms are organized into three groups: Frequency Response, System behaviour, and Circuit Elements and Properties. This collection of terms forms the foundational vocabulary for understanding and designing first- and second-order filters.

Please note that many of the terms in this list are not used in the course and book, however I decided to include them for the sake of making the list as complete as possible.

Frequency Response Terms

dB (decibel)

A logarithmic unit that expresses the ratio between two values of a quantity, often power or voltage. For voltages:

[math]\text{Gain (dB)} = 20 \log_{10}\left(\frac{V_{out}}{V_{in}}\right)[/math]

Decade

A tenfold increase or decrease in frequency. For example, from 100 Hz to 1 kHz is one decade.

Octave

A twofold increase or decrease in frequency. For example, from 1 kHz to 2 kHz is one octave.

dB/decade

The rate at which a filter's gain changes with frequency, measured in decibels per decade. For instance, a first-order low-pass filter attenuates at -20 dB/decade beyond its cutoff frequency.

dB/octave

Similar to dB/decade but based on octaves. A slope of -6 dB/octave is equivalent to -20 dB/decade.

Cutoff Frequency ([math]f_c[/math])

The frequency at which the output power drops to half of the input power, corresponding to a -3 dB point in the magnitude response:

[math]|H(f_c)| = \frac{1}{\sqrt{2}} \approx 0.707[/math]

Bandwidth

The range of frequencies over which the filter passes signals with acceptable attenuation. For a band-pass filter, it is:

[math]\text{Bandwidth} = f_2 - f_1[/math]

where [math]f_1[/math] and [math]f_2[/math] are the lower and upper -3 dB points.

Center Frequency ([math]f_0[/math])

The frequency at the midpoint of a band-pass or band-stop filter, usually given by the geometric mean:

[math]f_0 = \sqrt{f_1 f_2}[/math]

Gain

The ratio of output to input, expressed either as a simple ratio or in decibels.

Attenuation

The reduction in amplitude of a signal as it passes through a filter, typically measured in dB.

Slope

The rate of increase or decrease of the filter's magnitude response after the cutoff frequency. A first-order filter has a slope of ±20 dB/decade.

Frequency Response

The behaviour of a circuit as a function of frequency, typically represented by a plot of gain and phase versus frequency.

Magnitude Response

The plot or expression showing how the amplitude of the output varies with frequency.

Phase Response

The plot or expression showing how the phase of the output signal shifts with frequency.

System behaviour Terms

Transfer Function ([math]H(s)[/math])

The ratio of output to input in the Laplace domain:

[math]H(s) = \frac{V_{out}(s)}{V_{in}(s)}[/math]

where [math]s = j\omega[/math].

Poles

Values of [math]s[/math] that cause the denominator of the transfer function to become zero. Poles largely determine the filter's stability and frequency response.

Zeros

Values of [math]s[/math] that cause the numerator of the transfer function to become zero.

Order

The highest power of [math]s[/math] in the denominator of the transfer function. It determines the filter's roll-off rate.

Natural Frequency ([math]\omega_n[/math])

The frequency at which a second-order system oscillates when not damped:

[math]\omega_n = \sqrt{\frac{k}{m}}[/math]

Damping Ratio ([math]\zeta[/math])

A dimensionless measure of how oscillations in a system decay after a disturbance:

• [math]\zeta < 1[/math]: Underdamped
• [math]\zeta = 1[/math]: Critically damped
• [math]\zeta > 1[/math]: Overdamped

Quality Factor (Q)

A dimensionless parameter that describes the sharpness of the resonance peak:

[math]Q = \frac{f_0}{\text{Bandwidth}}[/math]

Higher [math]Q[/math] indicates a narrower bandwidth and sharper resonance.

Resonance

The condition where a system oscillates with maximum amplitude at the natural frequency, especially when [math]Q[/math] is high.

Critical Damping

The exact amount of damping that prevents oscillations and allows the system to return to equilibrium as quickly as possible without overshooting.

Underdamping

A condition where the damping is insufficient to prevent oscillations.

Overdamping

A condition where the system returns to equilibrium without oscillating but more slowly than in critical damping.

Circuit Elements and Properties

Reactance ([math]X[/math])

The opposition to AC current flow by capacitors or inductors:

• Capacitive reactance: [math]X_C = \frac{1}{\omega C}[/math]
• Inductive reactance: [math]X_L = \omega L[/math]
• Impedance ([math]Z[/math])*

The total opposition to AC, combining resistance and reactance:

[math]Z = R + jX[/math]

where [math]j[/math] is the imaginary unit.

Bode Plot

A graphical representation of a system's frequency response, with separate plots for magnitude (in dB) and phase (in degrees) versus logarithmic frequency.

Asymptotic Approximation

A technique used in Bode plots where straight lines approximate the curve behaviour at low and high frequencies.

Phase Margin

The additional phase shift required to bring a system to the verge of instability. Relevant for active filters and feedback systems.

Gain Margin

The amount by which the gain can be increased before a system becomes unstable.

Summary Table: Key Terms in Filter Analysis

Formula map

Here is a formula map that shows the relationships between [math]RC[/math], [math]\omega_c[/math], [math]\omega[/math], and how substitution works in transfer functions:

1. Cutoff angular frequency ([math]\omega_c[/math]) definition: [math]\omega_c = \frac{1}{RC}[/math]
2. Rearranging for [math]RC[/math]: [math]RC = \frac{1}{\omega_c}[/math]
3. Transfer function before substitution (example: low-pass filter): [math]H(j\omega) = \frac{1}{1 + j\omega RC}[/math]
4. Substituting [math]RC[/math] in terms of [math]\omega_c[/math]: [math]H(j\omega) = \frac{1}{1 + j\left( \frac{\omega}{\omega_c} \right)}[/math]
5. Relationship between [math]\omega[/math] and [math]f[/math]: [math]\omega = 2\pi f[/math] [math]\omega_c = 2\pi f_c[/math]

Thus: [math]\frac{\omega}{\omega_c} = \frac{f}{f_c}[/math]

Here is the final Simplified Transfer Functions (in terms of frequency ratio):

• Low-pass filter: [math]H(f) = \frac{1}{1 + j\left( \frac{f}{f_c} \right)}[/math]
• High-pass filter: [math]H(f) = \frac{j\left( \frac{f}{f_c} \right)}{1 + j\left( \frac{f}{f_c} \right)}[/math]

Try to remember these points:

• Substituting [math]RC[/math] with [math]1/\omega_c[/math] simplifies expressions.
• The ratio [math]f/f_c[/math] or [math]\omega/\omega_c[/math] always appears in transfer functions.
• This ratio controls the behaviour of the filter across different frequencies.

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