Derive an expression for the total impedance ($Z$) and phase angle ($\phi$) of a series LCR circuit connected to an alternating voltage source $V = V0 \sin(\omega t)$ using the phasor diagram method. Also, derive the condition for electrical resonance and write the expression for the resonant frequency.

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Series LCR Circuit Analysis Circuit Description and Phasor Diagram Method
Consider a resistor of resistance $R$, an inductor of inductance $L$, and a capacitor of capacitance $C$ connected in series across an alternating voltage source given by: $$V = V0 \sin(\omega t)$$
Let $I$ be the instantaneous current flowing through the circuit at any instant $t$. The potential difference across the resistor is $VR = I R$ (in phase with current $I$). The potential difference across the inductor is $VL = I XL$ (leading the current $I$ by a phase angle of $\pi/2$). The potential difference across the capacitor is $VC = I XC$ (lagging behind the current $I$ by a phase angle of $\pi/2$). 
Here, $XL = \omega L$ is the inductive reactance and $XC = \frac{1}{\omega C}$ is the capacitive reactance. --- Derivation of Expression for Impedance ($Z$)
Since $VL$ and $VC$ are in opposite phase ($180^\circ$ out of phase), their effective magnitude is $(VL - VC)$ assuming $VL > VC$.
Using the phasor diagram, the resultant supply voltage $V0$ is given by the Pythagorean theorem: $$V0^2 = VR^2 + (VL - VC)^2$$ 
Substituting the expressions for $VR$, $VL$, and $VC$: $$V0^2 = (I0 R)^2 + (I0 XL - I0 XC)^2$$ $$V0^2 = I0^2 \left R^2 + (XL - XC)^2 ight$$ $$\frac{V0}{I0} = \sqrt{R^2 + (XL - XC)^2}$$ 
The ratio $\frac{V0}{I0}$ is defined as the total effective opposition to the flow of AC, called the Impedance ($Z$) of the circuit: $$Z = \sqrt{R^2 + (XL - XC)^2}$$ $$Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}ight)^2}$$ --- Phase Angle ($\phi$)
From the phasor right-angled triangle, the phase angle $\phi$ between voltage and current is given by: $$	an\phi = \frac{VL - VC}{VR} = \frac{I0 XL - I0 XC}{I0 R}$$ $$	an\phi = \frac{XL - XC}{R} = \frac{\omega L - \frac{1}{\omega C}}{R}$$ $$\phi = 	an^{-1}\left( \frac{\omega L - \frac{1}{\omega C}}{R} ight)$$ --- Electrical Resonance and Resonant Frequency Definition: Electrical resonance occurs in a series LCR circuit when the circuit allows maximum current flow at a specific frequency of the applied AC voltage. Condition for Resonance: At resonance, the inductive reactance equals the capacitive reactance: $$XL = XC$$ $$\omegar L = \frac{1}{\omegar C}$$ $$\omegar^2 = \frac{1}{LC} \implies \omegar = \frac{1}{\sqrt{LC}}$$ 
Since angular frequency $\omegar = 2\pi fr$, the resonant frequency $fr$ is: $$2\pi fr = \frac{1}{\sqrt{LC}}$$ $$fr = \frac{1}{2\pi \sqrt{LC}}$$ 
At resonance, the impedance becomes minimum ($Z{min} = R$) and pure resistive, resulting in a maximum amplitude of current $I0 = \frac{V0}{R}$.

Series LCR Circuit Analysis

1. Circuit Description and Phasor Diagram Method\nConsider a resistor of resistance $R$, an inductor of inductance $L$, and a capacitor of capacitance $C$ connected in series across an alternating voltage source given by:

2. Derivation of Expression for Impedance ($Z$)\nSince $V_L$ and $V_C$ are in opposite phase ($180^\circ$ out of phase), their effective magnitude is $(V_L - V_C)$ assuming $V_L > V_C$.\nUsing the phasor diagram, the resultant supply voltage $V_0$ is given by the Pythagorean theorem:

3. Phase Angle ($\phi$)\nFrom the phasor right-angled triangle, the phase angle $\phi$ between voltage and current is given by:

4. Electrical Resonance and Resonant Frequency