Table of Contents
Introduction to Series AC Circuits
In AC circuits, components like resistors, inductors, and capacitors behave very differently compared to DC circuits. The voltage and current are no longer always in phase — they shift relative to each other depending on the reactive elements present. Understanding series AC circuits is fundamental to analysing power systems, filters, and resonant circuits used in real-world electrical engineering.
A series AC circuit is one where resistance (R), inductance (L), and capacitance (C) are connected end-to-end with a single AC source. The same current flows through every element, but the voltage across each component differs in magnitude and phase. This article covers R-L, R-C, and R-L-C series circuits along with impedance, phasor diagrams, resonance, power factor, and Q factor.
R-L Series Circuit
An R-L series circuit contains a resistor (R) and an inductor (L) connected in series with an AC voltage source. The inductor opposes changes in current, causing the current to lag behind the applied voltage.
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| R-L Series Circuit |
Key voltage relationships:
- Voltage across R: VR = I × R (in phase with current)
- Voltage across L: VL = I × XL (leads current by 90°)
- Inductive reactance: XL = 2πfL
Impedance of R-L Circuit
Since VR and VL are 90° apart, the applied voltage is their phasor sum:
Impedance, Z = √(R² + XL²) ohms
Phase angle, φ = tan⁻¹(XL / R)
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| Phasor Diagram — R-L Series Circuit |
The current lags the applied voltage by angle φ. The power factor is lagging:
R-C Series Circuit
An R-C series circuit contains a resistor and a capacitor in series with an AC source. The capacitor opposes changes in voltage, causing the current to lead the applied voltage.
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| R-C Series Circuit |
Key voltage relationships:
- Voltage across R: VR = I × R (in phase with current)
- Voltage across C: VC = I × XC (lags current by 90°)
- Capacitive reactance: XC = 1 / (2πfC)
Impedance of R-C Circuit
Impedance, Z = √(R² + XC²) ohms
Phase angle, φ = tan⁻¹(XC / R)
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| Phasor Diagram — R-C Series Circuit |
The current leads the applied voltage by angle φ. The power factor is leading:
This leading power factor property of capacitors is exactly why they are used for power factor improvement in industrial installations — they compensate for the lagging reactive power drawn by inductive loads.
R-L-C Series Circuit
The R-L-C series circuit is the most general case, containing all three passive elements. The behaviour depends on the relative magnitudes of inductive and capacitive reactance.
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| R-L-C Series Circuit |
Since VL and VC are 180° apart (directly opposing each other), the net reactive voltage is their difference:
Impedance, Z = √(R² + (XL − XC)²) ohms
Phase angle, φ = tan⁻¹((XL − XC) / R)
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| Phasor Diagram — R-L-C Series Circuit |
The circuit behaviour depends on which reactance dominates:
- XL > XC: Circuit is inductive — current lags voltage (lagging power factor)
- XC > XL: Circuit is capacitive — current leads voltage (leading power factor)
- XL = XC: Circuit is at resonance — current is in phase with voltage (unity power factor)
Series Resonance
Series resonance is a special condition in an R-L-C circuit where the inductive reactance equals the capacitive reactance (XL = XC). At this frequency, the reactive effects cancel each other completely, and the circuit behaves as a purely resistive circuit.
The resonant frequency is found by setting XL = XC:
f₀ = 1 / (2π√(LC)) Hz
At resonance:
- Impedance is minimum: Z = R (purely resistive)
- Current is maximum: I = V / R
- Power factor = 1 (unity)
- Voltage across L and C can be much greater than the supply voltage
- VL = VC (equal and opposite, cancelling each other)
This voltage magnification effect makes series resonant circuits useful in radio tuning, signal filtering, and frequency selection. However, in power systems, resonance can be dangerous — it can cause overvoltages that damage insulation and equipment.
Quality Factor (Q Factor)
The Quality Factor (Q) of a series resonant circuit measures the sharpness of the resonance peak and the voltage magnification at resonance. A higher Q means a narrower bandwidth and greater selectivity.
Q = VL / V = (I × XL) / (I × R) = XL / R
Q = (1/R) × √(L/C)
Q = ω₀L / R = 1 / (ω₀CR)
Practical significance of Q factor:
- High Q (> 10): Sharp resonance, narrow bandwidth — used in radio receivers and bandpass filters
- Low Q (< 1): Broad response, heavy damping — used in measurement circuits
- Bandwidth = f₀ / Q (the higher the Q, the narrower the bandwidth)
Comparison Table
Practical Applications
Series AC circuits are not just theoretical — they appear everywhere in electrical and electronics engineering:
- Radio Tuning: Series RLC circuits select a specific radio station frequency by resonating at that frequency
- Power Factor Correction: Series capacitors compensate inductive reactance in transmission lines
- Fluorescent Lamp Ballasts: R-L circuits limit current through the lamp
- Signal Filters: R-C and R-L circuits form the basis of low-pass and high-pass filters
- Voltage Multipliers: At resonance, voltage across L or C can be Q times the supply voltage
- Protective Relays: Impedance-based relays use series circuit principles for fault detection
Understanding impedance and phase relationships is essential for anyone working with AC power systems. The concepts covered here directly connect to basic laws of electrical engineering like Ohm's law and Kirchhoff's voltage law applied to AC circuits.
Frequently Asked Questions
What is impedance in a series AC circuit?
Impedance (Z) is the total opposition to current flow in an AC circuit. It combines resistance (R) and reactance (X) as a phasor sum: Z = √(R² + X²). Unlike pure resistance, impedance accounts for the phase difference between voltage and current. It is measured in ohms (Ω).
Why does current lag voltage in an R-L circuit?
An inductor opposes changes in current by generating a back-EMF. This means the current cannot change instantaneously — it takes time to build up. As a result, the current waveform is delayed (lags) relative to the voltage waveform. The lag angle depends on the ratio of inductive reactance to resistance.
What happens at resonance in a series RLC circuit?
At resonance (f₀ = 1/2π√LC), the inductive and capacitive reactances are equal and cancel each other. The impedance drops to its minimum value (Z = R), current reaches maximum, power factor becomes unity, and the voltage across L or C can be many times greater than the supply voltage.
How is power factor related to series AC circuits?
Power factor (cos φ) equals R/Z in any series AC circuit. It indicates what fraction of the apparent power is actually doing useful work. In R-L circuits it is lagging, in R-C circuits it is leading, and at resonance it is unity. Low power factor means higher current for the same real power, increasing losses.
What is the difference between reactance and impedance?
Reactance (X) is the opposition offered by inductors or capacitors alone — it causes a 90° phase shift. Impedance (Z) is the combined effect of resistance and reactance. Reactance is purely imaginary in phasor notation, while impedance has both real (R) and imaginary (X) components.