Understanding Filter Order and Gain Roll-off

The order of an electronic filter is a crucial parameter that describes the steepness of its frequency response, particularly how rapidly its gain changes in the transition band or stopband. This rate of change is typically measured in decibels per decade (dB/decade) or decibels per octave (dB/octave).

Filter Gain Decrease Rate Explained

For passive or active filters, there's a direct relationship between the filter's order and its asymptotic gain roll-off rate in the stopband. Each order of a filter contributes an additional 20 dB/decade (or 6 dB/octave) to the attenuation rate.

• A first-order filter (e.g., a simple RC or RL circuit) has a roll-off rate of 20 dB/decade.
• A second-order filter (e.g., an RLC circuit or cascaded first-order filters) has a roll-off rate of 40 dB/decade.
• A third-order filter will exhibit a gain roll-off of 60 dB/decade.
• In general, an \(\text{N}^{\text{th}}\) order filter will have a gain roll-off rate of \(\text{N} \times 20\) dB/decade.

Calculating Filter Order

The question states that the gain decreases at a rate of 60 dB/decade. To find the order of the filter, we can use the following formula:

\(\text{Filter Order (N)} = \frac{\text{Given Gain Decrease Rate}}{\text{Rate per Order}}\)

\(\text{N} = \frac{60 \text{ dB/decade}}{20 \text{ dB/decade per order}}\)

\(\text{N} = 3\)

Therefore, the filter is a third-order filter.

Identifying Filter Type: Low Pass Filter Characteristics

The phrase "gain decreases" specifically refers to the attenuation of the filter. For a low-pass filter, the gain remains relatively constant in the passband (low frequencies) and then starts to decrease sharply as the frequency increases beyond the cutoff frequency. This decrease in gain as frequency increases is the characteristic behavior of a low-pass filter's stopband.

Given the calculated third order and the description of gain decreasing, this aligns perfectly with the characteristics of a third-order low-pass filter. Such filters are designed to pass low frequencies while significantly attenuating higher frequencies, with the attenuation rate determined by their order.