What is Q?

Problem:
What is Q?

Solution:
Q is defined as the Quality Factor of a bandpass or notch filter. It is defined as the center frequency of a filter divided by the bandwidth. The bandwidth is the frequency of the upper 3 dB roll-off point minus the frequnecy of the lower 3 dB roll-off point.

Three values of Q for a 10 kHz bandpass filter are shown in the diagram below:

• The -3 dB points are at about 1 kHz and 100 kHz for a Q of 0.1 and a center frequency of 10 kHz. The low and high frequency roll offs look exactly like what would be expected from a single pole high pass and single pole low pass. At one tenth the frequency of the lower -3 dB point and ten times the frequency of the upper 3 dB point, the response is down 20 dB from the center frequency. This means that the two pole filter bandpass filter is effectively putting a single pole on the low frequency end and a single pole on the high frequency end of the passband.
  
  
• The -3 dB points are at about 600 Hz and 1.6 kHz for a Q of 1 and a center frequency of 10 kHz. The -20 dB points, however, are now at about 1 kHz and 100 kHz, which are NOT at one tenth and 10 times the lower and upper -3 dB frequency, respectively. The shape of the curve is also different, looking like a rounded 90 degree angle more than a single pole characteristic. The single pole performance has been lost in the region between the -20 dB points, or within ten times the bandwidth. Outside of this region, however, the single pole response of the bandpass filter returns. Therefore, for Q values between 0.1 and 1, the response of a bandpass circuit will change to whatever is required to satisfy the requirements of the - 3 dB points, as determined by the Q, and an ultimate slope of - 20 dB per decade for the region between 10 and 100 times the bandwidth. This is a final value of slope, and will be maintained at higher multiples of the bandwidth.
  
  
• The response of the bandpass filter with a Q of ten dramatically illustrates this effect. Between the -20 dB points, the shape of the response is completely opposite what it was for a Q of 10. The initial -3 dB points are so close to the center frequency that they have not been highlighted, but the -20 dB points are the same as -3 dB points for a Q of 1. In the region between 10 and 100 times the bandwidth, the slope continues to change to its final value of -20 dB per decade at 100 times the bandwidth.
  
  

Three values of Q for a 10 kHz notch filter are shown in the diagram below:

• The -3 dB points are at about 1 kHz and 100 kHz for a Q of 0.1 and a center frequency of 10 kHz. At one tenth the bandwidth, approximately 600 Hz and 1.6 kHz, the amplitude is down 20 dB.
  
  
• The -3 dB points are at about 600 Hz and 1.6 kHz for a Q of 1 and a center frequency of 10 kHz. The -20 dB points, however, are not marked because they are so close to the center frequency.
  
  
• The response of the bandpass filter with a Q of ten is also shown. The -3 dB points, however, are not marked because they are so close to the center frequency - the same place as the -20 dB for a Q of 1.