WHAT IS A BODE PLOT AND ITS SIGNIFICANCE?

It is essential to understand the behavior of filter circuits over a wide range of frequencies to make judicious use of filters for various applications. Magnitude of output (gain) and change of phase at output stage in relation to input, with a change in frequency, are very important considerations for study. All filters are studied for sine wave response, since sine waves are most common, and most irregular waveforms can be split up into a combination of sine waves. Frequency response of filters is analyzed over a wide range of frequencies, from very low to extremely high frequencies.

Frequency Response and Bode Plot

It is convenient to represent frequency response of filters as two plots representing amplitude (gain) and phase change between input and output over the frequency range. An American engineer Hendrick Bode from Bell Labs invented a way to represent this plot, which is named after him. Bode plots are a very useful to represent gain and phase of a system as function of frequency (frequency domain behavior of a system). 

Transfer Function of a Filter

Relation of output of a linear time variant system to input is denoted by a ratio (output/input). It is obtained by applying Laplace Transforms to differential equations of output and input. In the absence of data, it can also be obtained by actual measurements of input and output. They allow usage of simple algebraic equations for analyzing and designing of systems.

Transfer function of Low pass Filter

In a simple RC Low Pass circuit above, transfer function is

As an example, we may take C=2μF and R = 1MΩ. The symbol s may be replaced by jω. The above equation the becomes

Now that the transfer function is known, V_out can be calculated if V_in is known.

Transfer function of High Pass RC Filter is

Output is out of phase with respect to input, the phase shift is given by Φ= tan^-1ωRC

If ωt << 1, |g| ≈ ωt and Φ ≈ π/2, i.e., phase shift is 90°, and output amplitude is smaller than input. If ωt >> 1, gain approaches 1 and phase shift is near zero. If ωt = 1, gain is 1/√2 = 0.707 and phase shift is π/4 (45°), In high-pass filters, frequencies below ω_c get attenuated, and those above ω_c are passed with minimal phase shift or change. Phase shift in high-pass filters is positive, while it is negative in low-pass filters.

Bode Plot of Filter Circuits

In Bode plot, frequencies are on horizontal axis, and amplitudes on vertical axis, both in log scale (log-log scale). In phase-angle plot, phase angle is in linear scale on vertical axis (lin-log scale). Active part of gain begins at corner (or cut-off) frequency f_c, where gain falls by -3 dB. Exactly halfway between possible extremes of 0 and 90°, i.e.  phase lag of 45° represents corner frequency, being phase-lag of low-pass filter output, or phase-lead of high-pass filter output. Frequencies below this corner point are not affected by filter, while they get increasingly attenuated at constant rate of 6 dB per octave, i.e., signal output voltage drops by half (-6 dB) for every doubling (an octave) of input frequency.

Gain fall off (reduction in gain) line, starting from corner frequency (cut-off frequency), is a slope 20 dB/ decade, since a tenfold increase in frequency sees a tenfold decrease in gain. For example, gain may fall from 1 dB to 0.1 dB when frequency changes from 1 kHz to 10 kHz.  The slope is similar for both low-pass and high-pass filters- only their direction of fall (increasing or decreasing) changes. The straight line approximations shown are very useful for calculation and analysis.

For an RC filter, the cutoff frequency is given by

For a RL filter, the cutoff frequency is

Basic frequency response for low-pass RC filter rolls off by 6 dB per octave, while it increases by 6 dB per octave for high-pass filter. Band pass and band-notch filters have Bode plot a combination of the low- and high-pass filters. Frequency response of stop filter and notch filter gives the following approximations of Bode plots.

Phase angle plot is not shown, but may be drawn on same principles. Similar treatment can be given for all types of filters, including second and higher orders of filters.  The sharper the slopes, the nearer the curve is to an ideal desired response. The shape and complexity can differ with type of circuit and order of filter, and whether it is a passive or active filter.

Designers try to strike a balance between the ideal and the economically and technically practical solutions. This is done by judicious design of passive / active circuit (or a combination) to get the best results.