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2.6 The full Bode plot: gain and phase

In Section 2.5, you heard how a full Bode plot would show not only how the gain changes with frequency, but also how the phase difference between output and input changes with frequency. Conventionally the phase plot is put under the gain plot, with their respective frequency axes aligned, as in Figure 10. This example is for a first-order low-pass filter.

Described image
Figure 10 Normalised first-order low-pass frequency response showing gain and phase
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This figure consists of two graphs, one above the other. They have the same horizontal axis, showing normalised frequency on a logarithmic scale from 0.01 to 100.

The upper graph is gain G against normalised frequency, where the vertical axis for gain is on a logarithmic scale from 0.01 to 1. The graph line is the same shape as the one shown in Figure 9. It starts at a gain of 1 for a normalised frequency of 0.01 and remains at this value up to a normalised frequency of just over 0.1. The graph line then begins to curve gently downwards, until it becomes a diagonally descending straight line with a gain of 0.2 at a normalised frequency of 5 and a gain of 0.02 at a normalised frequency of 50.

The lower graph is phase difference against normalised frequency, where the vertical axis for phase difference is on a linear scale from minus 90 degrees to zero. The graph line starts at a phase difference of zero for a normalised frequency of 0.01, then curves down at first gently and then more steeply until it becomes a diagonally descending straight line that reaches a phase difference of minus 45 degrees at a normalised frequency of 1. After this, the graph line continues to curve down at first steeply and then more gently, until it flattens out again at a phase difference of minus 90 degrees for a normalised frequency of 100. The overall shape of the graph line is a flattened, backwards S with rotational symmetry around the central point (the steepest part of the curve) at normalised frequency 1, phase difference minus 45 degrees.

Figure 10 Normalised first-order low-pass frequency response showing gain and phase

Figure 10 shows that in the passband, the output is virtually in phase with the input. As frequency increases towards the cut-off frequency (1 on the normalised frequency axis), a phase difference opens between the output and the input, with the output lagging the input (the negative values of phase angle indicate lagging phase). At the cut-off frequency, the phase difference is −45°. This means that for a sinusoidal input, the output lags the input by 45°. By the time you are well into the stop band, the phase difference levels off at −90°.

That concludes the discussion of first-order filters. To end Section 2 you will consider some commonly found higher-order filters and have a look at their characteristics.