Lesson 10 of 10 · 15 min
Chapter review
Watch a class
The whole chapter on YouTube
Loading the full lesson
Must-know facts
18 facts
- 01v = vm sin ωt; in a resistor i = (vm/R) sin ωt, in phase with v.
- 02The average of an ac current over a cycle is zero, but the average of i² is not.
- 03rms values: I = im/√2 = 0.707 im and V = vm/√2 = 0.707 vm.
- 04Average power in a resistor: P = I²R = V²/R = IV.
- 05Mains 220 V is rms; its peak is 311 V.
- 06A phasor rotates anticlockwise at ω; its length is the peak value and its vertical projection is the instantaneous value.
- 07Inductor: XL = ωL; current lags voltage by π/2.
- 08Capacitor: XC = 1/ωC; current leads voltage by π/2.
- 09Average 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
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.