Lesson RLC and Filters · RLC series and resonance
RLC series circuits
In an RLC series circuit one current flows through R, L, and C. Inductive and capacitive voltages are 180° apart, so XL and XC subtract. Impedance is Z = √[R² + (XL − XC)²]. Whichever reactance is larger sets whether current lags or leads the applied voltage.
Three voltages, one current
- ER in phase with IT
- EL leads IT by 90°
- EC lags IT by 90°
Because EL and EC are opposite, the net reactive voltage is |EL − EC|. The applied voltage ET completes the right triangle with ER and that net reactive voltage.
Impedance formula
Z = √[R² + (XL − XC)²]
If XL > XC, the circuit looks inductive overall (lagging PF). If XC > XL, the circuit looks capacitive overall (leading PF). If XL = XC, Z collapses toward R — resonance (next lesson).
Example pattern: R = 12 Ω, XL = 24 Ω, XC = 8 Ω:
Z = √[12² + (24−8)²] = √[144+256] = 20 Ω
For ET = 240 V: IT = 12 A. Component voltages can be large: EL = IT×XL = 288 V even though the line is 240 V — vector cancellation makes that possible.
Sign of (XL − XC)
If XC > XL, (XL − XC) is negative. Squaring removes the sign, so Z is still positive. You track lead vs lag separately by comparing XL and XC (or VARsL vs VARsC).
Powers
P = IT² × R VARsL = IT² × XL VARsC = IT² × XC
Net reactive = |VARsL − VARsC| VA = ET × IT
Why voltages can exceed the line
Ohm’s law on each reactance still holds: EL = I×XL. High current from partial cancellation of reactances drives high EL or EC. Insulation and meter ranges must respect those voltages — especially near resonance.
Field intuition
Motors (inductive) plus capacitors (capacitive) on the same series path — or intentional RLC filters — live here. Most power circuits are not pure series RLC, but the math trains your eye for net XL vs XC thinking that also appears in parallel form later.
Numbers you should be able to work cold
R=10 Ω, XL=50 Ω, XC=20 Ω, ET=100 V:
Z = √(10²+30²) = 31.6 Ω IT ≈ 3.16 A Net inductive → lagging PF EL = 158 V (can exceed ET) EC = 63 V
Always state which reactance won before stating lead/lag.
Power bookkeeping
P only in R. VARsL and VARsC oppose. Net VARs = |VARsL−VARsC|. VA = ET×IT. If someone reports huge VARsL and VARsC but modest VA, cancellation is doing real work in the math — and stressing the parts.
Field red flags in series-ish LC strings
Unexpectedly high voltage on a reactor or capacitor with modest supply voltage; fuse irregularities when frequency changes; PF flipping with small component swaps.
Field case
Situation. A series RLC test bench shows 400 V across an inductor on a 120 V supply. A helper assumes the meter is broken.
What happened. XL and XC partially cancel, current is high, EL = I×XL exceeds ET. The vector story is consistent.
Applied lesson. Before condemning meters, check whether net reactance is small. High component voltage with modest line voltage is a hallmark of series RLC.
### 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
RLC series circuits 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.
In the field
Symptom
Unexpected high EL or EC; hotter than expected inductor; odd PF
Where to look
Series choke + capacitor strings; intentional RLC branches
Likely causes
- Partial cancellation of XL and XC
- wrong component values
- near resonance
What to measure
- ET, IT, ER, EL, EC
- compute XL, XC
- verify Z
What not to do
- Add EL+EC+ER as scalars
- ignore overvoltage on L or C
Checklist
- I write Z = √[R²+(XL−XC)²]
- I compare XL vs XC for lag vs lead
- I accept that EL or EC can exceed ET
- I compute P and net VARs correctly
- I use one IT through all parts
- I preview resonance when XL≈XC