Quality factor

The study material notes of Quality factor explain the important factor of resonance. The frequency response of a circuit or network operating at the resonant frequency is called the resonance. The voltage supply is constant for many applications. The supply voltage operates with a varying frequency in many communication systems. It is important to note the measurement of sharpness in the resonant circuits. For this measurement, we are calculating the factor called the Quality factor. In this Quality factor study material, we are going to discuss the formula and derivation of a Quality factor in the resonant circuit. 

Quality factor

The sharpness of tuning of a resonant circuit is called the Quality factor or Q-factor. In the series resonant circuit, the quality factor is the ratio of the voltage across the inductor or capacitor to the voltage.

Quality factor = voltage across inductor or capacitor/applied voltage   —(1)

We know that the voltage across VL = I0L   —(2)

In the above equation, 0 is the angular frequency of the ac at resonance. At the resonance, the applied voltage is the potential drop across the resistance. But, across inductance(L), the potential drop is the drop across capacitance. They are 1800 out of phase. Both the potential drop cancel out and the only potential drop across resistance will exist.

Applied voltage = IR   —(3)

We can substitute equations (2) and (3) in equation (1), we get

Q = I0L / IR

Q = 0L /R

Quality factor = (1/ RC ) L/R

The Quality factor is also defined as the ratio of the reactive power of the capacitor or inductor to the average power of the resistor at resonance.

Quality factor = reactive power of capacitor or inductor/average power of the resistor

For the inductive reactance XL at resonance,

Quality factor = I2XL / I2R 

= r L / R

For the capacitance reactance XC at resonance,

Quality factor = I2XC / I2R 

= 1 / r CR

Sharpness of resonance

In the series LCR circuit, the amplitude of the current is expressed as:

Im = vm / R2  + ( L – 1/C )2

When = 0 = 1 / LC

The maximum value will be i mmax  =  m / R

Other than 0 ,for values of , the maximum value is less than the amplitude of the current. 

Now, we are choosing the value of for which the current amplitude is 1/ 2   times the maximum value. 

At this value, the dissipated power value becomes half. 

There are two values of like 1  and 2 .

1  is greater than the 0  and 2 is smaller than 0. Both are symmetrical about 0.

1  = 0 + 

2  = 0 – 

The difference of  1  and 2= 2 is called the bandwidth of the circuit. The quality is regarded as a measure of the sharpness of resonance. The smaller the , the narrower or sharper the resonance. The current amplitude im is ( 12 ) i mmax  for 1  = 0 +  . 

Therefore, at , im = vm / R2 + ( 1L – 1/1C )2

= immax / 2 

= m /  R2

1L – 1 / 1C  = R

[(0 +  ) L] – [1 / (0 +  ) C ]= R

[0L ( 1 + / 0  )] – [1 / 0C ( 1 + / 0  )]  = R

Using  02= 1 / LC, we get

[0L ( 1 + / 0  )] – [0L ( 1 + / 0  )]  = R

We can approximate ( 1 + / 0 ) -1  as ( 1 + / 0 ) since  / 0 <<1

[0L ( 1 + / 0  )] – [0L ( 1 – / 0  )]  = R

 0L . 2 / 0   = R

= R / 2L

The sharpness of resonance is given by 

0 / 2 = 0L / R

The ratio 0L / R  is the quality factor or Q factor of the circuit.

Q =  0L / R

We see that, 2   = 0/ Q

When the Q value is large, the value of  2 is smaller and the resonance is sharper. 

Using 02= 1 / LC  is expressed as Q =  0L / R.

Frequency response curve

The quality factor is the unitless number having values from 10 to 100 for normal range frequencies. If the circuit has high Q values, then the frequency range will be very narrow. If the circuit has low Q values, then the frequency range will be very broad. The circuit with a low Q value has a flat resonance and the circuit with the high Q value is sharply tuned. 

If the value of resistance is large, the current frequency curve is very flat. And if the values of resistances decrease, then the curve becomes sharp. This curve is called the frequency response curve.

Quality factor and damping

The quality factor determines the behaviour of simple damped oscillators and affects the response within the filters. There are three conditions for the consideration of damping and quality factors.

• Underdamped condition

In this underdamped condition, the quality factor value would be greater than half. The step impulse is applied, the quality factor is only just over a half before the oscillations. 

Q > ½

• Over-damped condition

In this overdamped condition, the quality factor value would be less than half.

Q < ½

• Critical- damped condition

In the critically damped condition, the Q factor is equal to half or 0.5

Q = ½  or 0.5

Effects of Q factor

The Q factor affects the following parameters in the RF tuned circuits

• Wide bandwidth
• General spurious signals
• Bandwidth
• Oscillator phase noise
• Ringing

Conclusion

The quality factor study material gives a detailed summary of the Quality factor. Quality factor or Q factor is a performance measure of a capacitor or inductor or RF tuned circuit. It is a unitless number, which describes the circuit damping. We have also derived the sharpness response or analysis of resonance in resonant circuits and discussed the three damping conditions. We can find how the quality factor affects the RF tuned circuit in this study material notes on quality factor.