by Struggling with the series RLC circuit derivation? You’re not alone. But what if you could derive impedance, current, and phase angle in under 3 minutes—and understand why each step matters?
In this complete, exam-optimized guide, we walk through the full derivation of current and impedance in a series RLC circuit driven by AC voltage—using phasor diagrams, Ohm’s law for AC, and trigonometric reasoning that actually sticks.
Perfect for NCERT Class 12 Physics (Chapter 7), JEE Main/Advanced, and NEET prep!
⚡ The Series RLC Circuit: What’s Connected?
A series RLC circuit contains three components connected end-to-end with an AC source:
- Resistor (R) – dissipates energy
- Inductor (L) – opposes change in current
- Capacitor (C) – stores electric energy
All share the same current 
, but voltage drops differ in phase.
The applied AC voltage is:
![\[ v(t) = V_0 \sin(\omega t) \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-926ffe61b3271d30d91bf2d9dfb5d1e9_l3.png?resize=126%2C19&ssl=1 "Rendered by QuickLaTeX.com")
🔍 Step 1: Voltage Across Each Component
Since current is common, we express each voltage in terms of 
, where 
is the phase difference between source voltage and current.
- Resistor:
→ in phase with current - Inductor: → leads current by 90°
- Capacitor: → lags current by 90°
→ in phase with current
→ leads current by 90°
→ lags current by 90°Where:
(inductive reactance)
(capacitive reactance)
📐 Step 2: Phasor Diagram – The Visual Key
Because voltages are out of phase, we cannot add them algebraically. Instead, we use phasors (rotating vectors):
- Draw along the x-axis (reference, in phase with I)
- Draw upward (90° ahead of I)
- Draw downward (90° behind I)
upward (90° ahead of I)
downward (90° behind I)Net reactive voltage = 
(vertical component)
Resistive voltage = 
(horizontal component)
The source voltage phasor 
is the vector sum:
![\[ \vec{V} = \vec{V}_R + (\vec{V}_L - \vec{V}_C) \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-d1f5767af1ec934acb1d6dd539a172f8_l3.png?resize=158%2C23&ssl=1 "Rendered by QuickLaTeX.com")
✅ Exam Tip: In phasor diagrams, current is the reference for series circuits (since it’s common).
🧮 Step 3: Deriving Impedance (Z)
From the phasor diagram, the magnitude of total voltage is:
![\[ V_0 = \sqrt{V_R^2 + (V_L - V_C)^2} \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-bc7af3a6176fee13d80e3e8cbf06af5c_l3.png?resize=189%2C32&ssl=1 "Rendered by QuickLaTeX.com")
Substitute 
, 
, 
:
![\[ V_0 = I_0 \sqrt{R^2 + (X_L - X_C)^2} \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-dd58565d3d43a20553ef9082ea6956b8_l3.png?resize=213%2C22&ssl=1 "Rendered by QuickLaTeX.com")
This resembles Ohm’s law! So we define impedance 
as:
![\[ \boxed{Z = \sqrt{R^2 + (X_L - X_C)^2}} \quad (\Omega) \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-85283afa88f9128f45727ffab9b75aee_l3.png?resize=247%2C33&ssl=1 "Rendered by QuickLaTeX.com")
Thus:
![\[ V_0 = I_0 Z \quad \text{or} \quad V_{\text{rms}} = I_{\text{rms}} Z \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-9c1e7679d35d65976dac2190a242d430_l3.png?resize=223%2C14&ssl=1 "Rendered by QuickLaTeX.com")
🧭 Step 4: Phase Angle (φ) Derivation
The phase difference between source voltage and current is given by the angle of the impedance triangle:
![\[ \tan \phi = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{V_L - V_C}{V_R} = \frac{X_L - X_C}{R} \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-e36293deb7da7e2bbe29e28c78c9fa06_l3.png?resize=326%2C41&ssl=1 "Rendered by QuickLaTeX.com")
So:
![\[ \boxed{\phi = \tan^{-1}\left( \frac{X_L - X_C}{R} \right)} \]](https://i0.wp.com/physicsqanda.com/wp-content/ql-cache/quicklatex.com-81d3c14a660e6a7785cdf2a2e70af841_l3.png?resize=196%2C54&ssl=1 "Rendered by QuickLaTeX.com")
Interpretation:
- If → → voltage leads current (inductive circuit)
- If → → current leads voltage (capacitive circuit)
- If → → resistive behavior (resonance!)
→ 
→ voltage leads current (inductive circuit)
→ 
→ current leads voltage (capacitive circuit)
→ 
→ resistive behavior (resonance!)📊 Impedance Triangle – Visual Summary
Imagine a right triangle:
- Base =
- Height =
- Hypotenuse =
- Angle at base =
This triangle encodes everything: magnitude of opposition (Z) and timing (φ).
🎯 Why This Derivation is Crucial for Exams
- NCERT explicitly derives Z and φ in Example 7.5 and Exercise 7.11
- JEE Advanced often asks for instantaneous current or phasor-based proofs
- NEET tests conceptual questions like: “What happens to φ if C is increased?”
💡 Pro Tip: Always draw the phasor diagram first in exams—it earns partial marks even if math goes wrong!
🔗 Go Deeper on PhysicsQanda.com
- AC Voltage Applied to R, L & C: Visual Comparison
- RMS Value Explained: Why It’s More Important Than Average Value in AC Circuits
- Resonance in Series LCR Circuits: Full Explanation
- Power in AC Circuits: Real, Apparent & Power Factor
✨ Final Thought
The series RLC circuit isn’t just a formula—it’s a symphony of phase, opposition, and balance. Master this derivation, and you’ve unlocked the heart of AC circuit analysis.
Save this guide. Practice the phasor sketch. Own your next physics exam.
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