Derive an expression for the impedance ($Z$) and phase angle ($ \phi$) of a series LCR alternating current circuit using the phasor diagram method. Also, define electrical resonance, derive the expression for the resonant frequency, and explain the Quality Factor ($Q$-factor) of the circuit.

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Series LCR Circuit Analysis 
When a resistor ($R$), an inductor ($L$), and a capacitor ($C$) are connected in series to an alternating voltage source $V = V0 \sin(\omega t)$: Current $I$ is same in all three components. Voltage across resistor: $VR = I R$ (in phase with current $I$). Voltage across inductor: $VL = I XL$ (leads current $I$ by $\pi/2$), where $XL = \omega L$. Voltage across capacitor: $VC = I XC$ (lags current $I$ by $\pi/2$), where $XC = \frac{1}{\omega C}$. --- Expression for Impedance ($Z$)
Since $VL$ and $VC$ are in opposite phase ($180^\circ$ apart), their resultant is $(VL - VC)$ assuming $VL > VC$. 
Using the phasor diagram, total peak voltage $V0$ is given by the Pythagorean theorem: $$V0^2 = VR^2 + (VL - VC)^2$$ 
Substituting $VR = I0 R$, $VL = I0 XL$, and $VC = I0 XC$: $$V0^2 = (I0 R)^2 + (I0 XL - I0 XC)^2$$ $$V0^2 = I0^2 \left R^2 + (XL - XC)^2 ight$$ $$V0 = I0 \sqrt{R^2 + (XL - XC)^2}$$ 
The total effective resistance of the circuit is called Impedance ($Z$): $$Z = \frac{V0}{I0} = \sqrt{R^2 + (XL - XC)^2} = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}ight)^2}$$ --- Phase Angle ($\phi$)
From the phasor right-angled triangle, the phase difference $\phi$ between voltage and current is: $$	an \phi = \frac{VL - VC}{VR} = \frac{I0 XL - I0 XC}{I0 R} = \frac{XL - XC}{R}$$ $$\phi = 	an^{-1}\left( \frac{\omega L - \frac{1}{\omega C}}{R} ight)$$ --- Electrical Resonance & Resonant Frequency
Electrical resonance occurs in a series LCR circuit when the inductive reactance equals the capacitive reactance ($XL = XC$), causing the impedance to become minimum ($Z = R$) and the current amplitude to reach its maximum. 
At resonance: $$XL = XC$$ $$\omegar L = \frac{1}{\omegar C}$$ $$\omegar^2 = \frac{1}{LC} \implies \omegar = \frac{1}{\sqrt{LC}}$$ 
Since $\omegar = 2\pi fr$, the resonant frequency $fr$ is: $$fr = \frac{1}{2\pi \sqrt{LC}}$$ --- Quality Factor ($Q$-factor)
The Quality Factor ($Q$-factor) measures the sharpness of resonance in an LCR circuit. It is defined as the ratio of the voltage drop across the inductor or capacitor at resonance to the applied voltage (voltage across $R$): $$Q = \frac{VL}{VR} = \frac{I \omegar L}{I R} = \frac{\omegar L}{R}$$ 
Substituting $\omegar = \frac{1}{\sqrt{LC}}$: $$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$$ A higher $Q$-factor indicates a sharper resonance peak and better selectivity of the circuit.

Series LCR Circuit Analysis

1. Expression for Impedance ($Z$)\nSince $V_L$ and $V_C$ are in opposite phase ($180^\circ$ apart), their resultant is $(V_L - V_C)$ assuming $V_L > V_C$.

2. Phase Angle ($\phi$)\nFrom the phasor right-angled triangle, the phase difference $\phi$ between voltage and current is:

3. Electrical Resonance & Resonant Frequency\nElectrical resonance occurs in a series LCR circuit when the inductive reactance equals the capacitive reactance ($X_L = X_C$), causing the impedance to become minimum ($Z = R$) and the current amplitude to reach its maximum.

4. Quality Factor ($Q$-factor)\nThe Quality Factor ($Q$-factor) measures the sharpness of resonance in an LCR circuit. It is defined as the ratio of the voltage drop across the inductor or capacitor at resonance to the applied voltage (voltage across $R$):