Logs & Exponents: Bode Plots

The Bode plot is an example of logarithmic graphing used by
electrical engineers.

The voltage gain of a passive or active filter, as a function of
frequency, is non-linear. For the **high pass** R-C
filter, the magnitude of the voltage gain, as a function of
frequency, is expressed by the formula:

where "fc" is called the **cutoff
frequency** and "f" is the frequency in Hertz.
"Av " is the voltage gain, (in this case, a loss) for
the passive filter circuit.

Likewise, the graph of the voltage gain for a typical
**low-pass** R-C filter is shown in the figure (1).
A logarithmic scale is used to plot the frequency along the
horizontal axis.

For low frequencies, the gain of the circuit is equal to 1.
This is called the **pass-band region**. The upper
limit for the pass-band region is located at the cutoff
frequency, "fc". The Bode db gain at the cutoff
frequency is:

Given a high pass filter and R= 1.0 kΩ and C = 0.1 μF:

- Calculate the cutoff frequency.
- Calculate the Bode gain at the cutoff frequency.
- Plot the Bode gain verses frequency, in Hertz, where the Bode gain is given by the formula:
- Sketch the straight-line approximation or
**"idealized Bode plot"**.

- The cutoff frequency for a high-pass R-C filter is:
- The Bode gain at the cutoff frequency is:
- The Bode plot versus frequency is shown below. Excel is used to generate the data.
- The idealized Bode plot is obtained by considering the Bode
gain:
The following approximations can be made:

**Case (i):**The Bode gain in the

**pass-band**region is zero.**Case (ii):**For the

**stop-band**region, the Bode gain is a straight line with slope equal to 20 db per decade. The Bode gain is negative, intersecting with the**pass-band**approximation at the cutoff frequency.

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Written by Ross Bradbeer, September 24, 1997