Q-factor, L.C.R. Circuit

Q-factor or Quality factor or Figure of Merit

The inductive and capacitive reactance are the enrgy storing device. The Q-factor of a circuit containing such type of energy storing device is defined as\begin{align*}Q=2\pi .\frac{energy\space stored}{energy\space loss\space per\space cycle} \end{align*}

Q-factor of L-R circuit

Consider an inductor of inductance L is connected to a source of emf \( E=E_0 sin\omega t\) as shown in figure. If \( I=I_0 sin(\omega t+\phi)\) be the current stored in inductor is \(\frac12 LI_0^2 \) and energy loss per cycle through resistor R is \({I_{rms}^2}RT=\frac{I_0^2}{2}RT \)

where T is the time period of Oscillation of AC source.

\begin{align*}Q-factor=2\pi .\frac{energy\space stored}{energy\space loss\space per\space cycle} \end{align*}\begin{align*}=2\pi .\frac{\frac12 LI_0^2}{\frac{I_0^2}{2}RT} \end{align*}\begin{align*}=\frac{2\pi }{T}.\frac{L}{R} \end{align*}\begin{align*}Q=\frac{L\omega }{R} \end{align*} This is the value of Q-factor for L-R circuit.

Q-factor for R-C circuit

Consider a capacitor of capacitance 'C' is connected to a source of emf \(E=E_0 sin\omega t\) as shown in figure.
The energy stored in capacitor \(=\frac12 CE_0^2\)
The energy losss per cycle due to resistor R \(=\frac{E_{rms}^{2}}{R}\times T \)
where 'T' is the time period of the ac source.

Q-factor of R-C circuit,\(Q=2\pi .\frac{\frac12 CE_0^2}{\frac{E_{rms}^{2}}{R}\times T}\)
\(Q=\omega CR\)
This is the Q-factor for capacitive circuit.

L.C.R. circuit(Resonance circuit)

Consider an emf source of emf \(E=E_0 sin\omega t\) is connected to the inductor and capacitor in series to a resistor as shown in figure.

Total impedance of the circuit\(=R+j(L\omega -\frac{1}{\omega C})\rightarrow 1\)
When L and C are inductance and capacitance of the inductor and capacitor respactively. The phenomenon of cancellation of inductive and capacitive reactance when they are in series is called series resonance and corresponding frequency is called series resonance frequency \(\omega_{r}\).
At the resonance frequency \(\omega=\omega_{r}\)
\begin{align*}L\omega -\frac{1}{\omega C}=L\omega_{r} -\frac{1}{\omega C_{r}}=0 \end{align*}\begin{align*}or,\omega_{r}^2=\frac{1}{LC} \end{align*}
\begin{align*}\omega_{r}=\frac{1}{\sqrt{LC}} \end{align*}\begin{align*}f_{r}=\frac{1}{2\pi \sqrt{LC}} \end{align*}
This is the value of series resonance frequency. AT resonance frequency, the curren in the circuit has maximum value and is given by
\begin{align*}I_0=\frac{E_0}{R} \end{align*}
The variation of current with frequency in series resonance is shown in figure.

The lower \(\omega_1\), higher \(\omega_2\) frequencies in which the current equals to \(\frac{I_0}{\sqrt{2}}\) are called lower cut off frequency and upper cut off frequency respectively.
The difference between these two frequencies is called bandwidth of the resonance circuit.
At the resonance frequency, Voltage drop across inductor L\begin{align*}V_{L}=I_0 j \omega_{r}L=jI_0 L \omega_{r} \rightarrow 1 \end{align*}
Voltage drop across C
\begin{align*}V_{C}=I_0 (\frac{-j}{c\omega_{r}})=-jI_0 L \omega_{r}\rightarrow 2 \end{align*}
From equations 1 and 2, it is seen that at the resonance frequency, the voltage drop across the inductor and capacitor is equal in magnitude and opposite in sign so they cancel each other.

Sharpness in Resonance Circuit

Consider a frequency \(\omega \) closed to the resonance frequency \(\omega_{r}\) such that
\begin{align*}\omega =(\omega_{r}+\Delta \omega ) \end{align*}
Now consider the term
\begin{align*}L\omega -\frac{1}{\omega C}=L(\omega_{r}+\Delta \omega ) -\frac{1}{(\omega_{r}+\Delta \omega ) C} \end{align*}
\begin{align*}=L\omega_{r}(1+\frac{\Delta \omega }{\omega_{r}})-\frac{1}{C\omega_{r}}(1+\frac{\Delta \omega }{\omega_{r}})^{-1}\rightarrow 3 \end{align*}
Using Binomial expansion and neglecting the higher order terms in \(\Delta \omega \),
\begin{align*}(1+\frac{\Delta \omega }{\omega_{r}})^{-1}=(1-\frac{\Delta \omega }{\omega_{r}})\rightarrow 4 \end{align*}
From equation 3 and 4,
\begin{align*}L\omega -\frac{1}{\omega C}=L\omega_{r}(1+\frac{\Delta \omega }{\omega_{r}})-\frac{1}{C\omega_{r}}(1-\frac{\Delta \omega }{\omega_{r}}) \end{align*}
\begin{align*}= L\Delta \omega +\frac{\Delta \omega}{C}\omega_{r}^2=L\omega_{r}\frac{\Delta \omega }{\omega_{r}}+\frac{1}{C\omega_{r}}\frac{\Delta \omega }{\omega_{r}} \end{align*}
\begin{align*}=(L\omega_{r}+\frac{1}{\omega_{r}}).\frac{\Delta \omega }{\omega_{r}} \end{align*}\begin{align*}=L\omega_{r}\frac{2\Delta \omega }{\omega_{r}} \end{align*}
If \(\omega \) represents cut off frequency then
\begin{align*}L\omega -\frac{1}{\omega C}=R \end{align*}\begin{align*}2\Delta \omega=Band\space width \end{align*}
\begin{align*}\frac{2\Delta \omega }{\omega_{r}}=\frac{R}{L\omega_{r}} \end{align*}
The ratio of Band width to the resonance frequency is called sharpness of resonance and given by\begin{align*}S=\frac{2\Delta \omega }{\omega_{r}}=\frac{1}{\frac{\omega_{r}}{2\Delta \omega }}=\frac{1}{Q} \end{align*}
Where Q is the quality factor or Q-factor of resonance circuit. From this relation, it is seen that the sharpness of resonance is inversely proportional to Q-factor of circuit.

Parallel Resonance Circuit[Anti-resonance circuit]

Consider a source of emf \(E=E_0 sin\omega t\) which is applied to a circuit consists of an inductor and a capacitor in parallel as shown in figure. The reactance or impedance of inductive branch
\begin{align*}Z_{L}=R+j\omega L \end{align*}
The impedance of reactance of capacitive branch\begin{align*}X_{C}=\frac{-j}{\omega C} \end{align*}
The reciprocal of impedance or reactance is called admittance. So, admittance of inductive branch,
\begin{align*}Y_{L}=\frac{1}{Z_{L}}=\frac{1}{R+j\omega L}=\frac{R-j\omega L}{R^2+\omega^2 L^2}\rightarrow 1 \end{align*}
Admittance of the capacitive branch\begin{align*}Y_{C}=\frac{1}{X_{C}}=\frac{1}{\frac{-j}{\omega C}}=j\omega C \rightarrow 2 \end{align*}
The phenomenon of cancellation of inductive and capacitive admittance when inductor and capacitor are in parallel is known as anti-resonance and corresponding frequency is called anti-resonance frequency \(\omega_{r}\).
\begin{align*}Y=Y_{L}+Y_{C}=0 \end{align*}\begin{align*}\frac{R-j\omega_{r} L}{R^2+\omega_{r}^2 L^2}+j\omega_{r} C=0 \end{align*}
\begin{align*}or,\space \frac{R}{R^2+\omega_{r}^2 L^2}+j( \frac{-\omega_{r} L}{R^2+\omega_{r}^2 L^2}+\omega_{r} C)=0 \end{align*}\begin{align*}or,\space ( \frac{-\omega_{r} L}{R^2+\omega_{r}^2 L^2}+\omega_{r} C)=0 \end{align*}
\begin{align*}or,\space C=\frac{R-j\omega L}{R^2+\omega_{r}^2 L^2} \end{align*}\begin{align*}or,\space R^2+\omega_{r}^2 L^2=\frac{L}{C} \end{align*}
\begin{align*}or,\space \omega_{r}^{2}=\frac{1}{LC}-\frac{R^2}{L^2} \end{align*}\begin{align*}\omega_{r}=\sqrt{\frac{1}{LC}-\frac{R^2}{L^2}} \end{align*}
For the frequency to be real,\(\frac{R}{L}<1\). So neglecting higher order term in \(\frac{R}{L}\) we get
\begin{align*}\omega_{r}=\sqrt{\frac{1}{LC}}\space or,\space f_{r}=\frac{1}{2\pi }=\sqrt{\frac{1}{LC}} \end{align*}
This is the value of anti-resonance frequency. The variation of current with frequency in anti-resonance circuit is as shown in figure.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Griffiths, D.J., Introduction to Electrodynamics, PHI Learning Private Limited, 2013