Lesson 10 of 10 · 15 min
Chapter review
Watch a class
The whole chapter on YouTube
Phasors and LCR circuits, full chapter
Rakshak Sir Science · Hinglish · Whole chapter · Open on YouTube
NCERT revision of the whole chapter
NCERT Wallah · Hinglish · Whole chapter · Open on YouTube
Must-know facts
18 facts
- 1v = vm sin ωt; in a resistor i = (vm/R) sin ωt, in phase with v.
- 2The average of an ac current over a cycle is zero, but the average of i² is not.
- 3rms values: I = im/√2 = 0.707 im and V = vm/√2 = 0.707 vm.
- 4Average power in a resistor: P = I²R = V²/R = IV.
- 5Mains 220 V is rms; its peak is 311 V.
- 6A phasor rotates anticlockwise at ω; its length is the peak value and its vertical projection is the instantaneous value.
- 7Inductor: XL = ωL; current lags voltage by π/2.
- 8Capacitor: XC = 1/ωC; current leads voltage by π/2.
- 9Average power into a pure L or pure C is zero.
- 10Series LCR: Z = √(R² + (XC − XL)²), im = vm/Z, tan φ = (XC − XL)/R.
- 11Voltages across R, L and C must be added as phasors, not as plain numbers.
- 12Resonance: XL = XC at ω₀ = 1/√(LC); Z = R and current is greatest.
- 13Resonance needs both L and C.
- 14Average power P = VI cos φ; cos φ is the power factor.
- 15Wattless current: current with cos φ = 0, which uses no power.
- 16Ideal transformer: Vs/Vp = Ns/Np = Ip/Is.
- 17Step-up raises voltage and lowers current; step-down does the reverse.
- 18Transformer losses: flux leakage, winding resistance, eddy currents, hysteresis.
Common traps
Where marks are lost
Adding VR, VL and VC as plain numbers to get the source voltage.
Using the peak value in P = I²R.
Thinking a capacitor's reactance rises with frequency.
Saying the current in an inductor leads the voltage.
Assuming no current flows when no power is used.
Expecting an RC or RL circuit to resonate.
Thinking a step-up transformer gives more power.
Treating 220 V mains as the peak value.
Formulas
10 to know
AC voltage and current in R
v = vm sin ωt, i = (vm/R) sin ωt
In phase.
RMS values
I = im/√2, V = vm/√2
0.707 times the peak.
Average power in R
P = I²R = V²/R = IV
rms values.
Inductive reactance
XL = ωL = 2πνL
Current lags by π/2.
Capacitive reactance
XC = 1/ωC = 1/2πνC
Current leads by π/2.
Impedance of series LCR
Z = √(R² + (XC − XL)²)
im = vm/Z.
Phase angle
tan φ = (XC − XL)/R
φ > 0: current leads; φ < 0: current lags.
Resonant frequency
ω₀ = 1/√(LC), ν₀ = 1/2π√(LC)
Z = R at resonance.
Average power
P = VI cos φ
cos φ is the power factor.
Ideal transformer
Vs/Vp = Ns/Np = Ip/Is
Input power = output power.
Key terms
16 terms
- Alternating voltage
- A voltage that varies sinusoidally with time, v = vm sin ωt.
- Amplitude (peak value)
- The largest value vm or im reached in a cycle.
- RMS value
- Peak value divided by √2; the steady value giving the same average heating.
- Phasor
- A rotating arrow whose length is a peak value and whose vertical projection is the instantaneous value.
- Phase angle (φ)
- The angle by which the current leads (φ > 0) or lags (φ < 0) the source voltage.
- Inductive reactance (XL)
- ωL, the opposition an inductor offers to ac; in ohm.
- Capacitive reactance (XC)
- 1/ωC, the opposition a capacitor offers to ac; in ohm.
- Impedance (Z)
- √(R² + (XC − XL)²), the total opposition of a series LCR circuit; in ohm.
- Impedance triangle
- Right triangle with sides R and XC − XL and hypotenuse Z.
- Resonance
- The condition XL = XC in a series LCR circuit, where Z = R and the current is largest.
- Resonant frequency
- ω₀ = 1/√(LC).
- Power factor
- cos φ, the fraction of VI that is actually used as power.
- Wattless current
- Current 90° out of phase with the voltage, which on average uses no power.
- Transformer
- Two coils on an iron core that change an ac voltage by mutual induction.
- Step-up / step-down
- A transformer with more / fewer secondary turns than primary turns.
- Laminated core
- A core made of thin insulated sheets, which cuts eddy-current loss.