Series resonance

From Z = √[R² + (XL−XC)²], when XL = XC:

Z = R

IT = ET / R (maximum)

EL and EC can each be many times ET because I is large and each reactance is nonzero — they cancel in the sum but still exist on the components.

XL = 2πfL XC = 1/(2πfC)

Set equal and solve:

fr = 1 / (2π√(LC))

Only one frequency (for fixed L and C) satisfies resonance. Change L or C and fr moves.

A small R means a sharp current peak (high Q). A larger R broadens and lowers the peak. In radio tuning, sharpness selects stations. In power systems, a sharp accidental resonance can amplify harmonic currents.

• Harmonic frequencies can hit LC combinations in PF banks and cable capacitance
• Filters intentionally use resonance to pass or reject bands
• Series resonant paths can blow fuses even when “60 Hz kVAR” looked fine

• Current much higher than R alone would suggest at that voltage
• Capacitor and inductor voltages both very high
• PF near unity while VARs on L and C are individually huge
• Small frequency shift causes large current change

L = 0.1 H, C = 40 µF:

fr = 1/(2π√(0.1×40e-6)) ≈ 79.6 Hz

At 60 Hz this pair is not resonant; XC and XL differ. Sweep frequency (or change C) until XL=XC to see the current peak in a lab.

At resonance with R=5 Ω and ET=50 V: IT=10 A. If XL=XC=100 Ω, EL=EC=1000 V — lethal teaching example of why resonance demands respect.

Smaller R → taller, narrower current peak. Larger R → flatter peak. Harmonic problems love high-Q accidental resonances.

“Series resonance: easiest path for current at one frequency. Great for tuning. Bad when the power system finds it by accident.”

Situation. A plant installs capacitors for PF. At certain VFD speeds, capacitor-fuse events spike. Analyzers show a harmonic near the LC resonant frequency of the bank plus transformer inductance.

What happened. Series/parallel resonant conditions with harmonics created high current at a frequency that was not 60 Hz.

Applied lesson. Resonance is a frequency story. Measure spectrum, not only RMS at line frequency, when caps and inductances share a bus with nonlinear loads.

### Teaching pause — say this out loud

Before you leave this lesson, explain the main idea to an imaginary first-month helper in under one minute. If you need the book open to do it, reread How it works once more. Field diagnosis only helps after the concept is yours.

Also sketch the key diagram from memory (triangle, wye/delta, filter shape, or charge curve — whichever this lesson used). Labels beat artistic skill.

### Why this lesson matters on Monday morning

Series resonance is not trivia. You will meet it when a meter reading looks “impossible,” when a replacement part is almost right, or when a helper asks why the book uses √3 or lead/lag. Master the model here so the next call is pattern recognition, not panic.

Common Monday uses: verify a nameplate against clamps, explain a PF or capacitor change to a customer, or catch a miswired series/parallel or wye/delta assumption before energizing.