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The transistor generator and active resistance

Lessons 13–14 · 2 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 11, 1st edition. Niso Poligraf Publishing House, Tashkent, 2018
14

Active resistance in an alternating-current circuit

Textbook: pp. 51–54
GoalUse i = Iₘcosωt in an AC circuit with active resistance, the fact that current and voltage are in phase, the effective values Iₑf = Iₘ/√2, Uₑf = Uₘ/√2 and the mean power P = Iₑf·Uₑf.
New words
active resistance · aktiv qarshilikeffective (rms) value · effektiv qiymatmean power · o‘rtacha quvvatangular frequency · siklik chastota
Explanation

Active resistance is a resistance in which electrical energy is completely converted into heat, light or other energy (a resistor, an incandescent lamp, an iron). If a voltage u = Uₘcosωt is applied (ω = 2πν is the angular frequency), Ohm's law gives i = u/R = Iₘcosωt, with Iₘ = Uₘ/R: current and voltage oscillate in phase, the phase shift is zero. The instantaneous power p = i·u = Pₘcos²ωt is never negative and its mean value is Pₘ/2 = Iₘ·Uₘ/2. The effective value is the direct current that releases the same heat in an active resistance in the same time as the alternating current: Iₑf = Iₘ/√2 ≈ 0.707·Iₘ, Uₑf = Uₘ/√2. Hence P = Iₑf·Uₑf = Iₑf²·R. The 220 V and 50 Hz of the mains are the effective voltage and frequency; the amplitude is ≈ 311 V. Voltmeters and ammeters normally show effective values. Mains voltage is life-threatening: never touch it yourself.

Worked examples
u = 100cos(100πt) V, R = 20 Ω. ν = ω/(2π) = 50 Hz, Iₘ = 100/20 = 5 A, mean power P = Iₘ·Uₘ/2 = 5·100/2 = 250 W. Check: Iₑf = 5/√2 ≈ 3.54 A, P = Iₑf²R = 12.5·20 = 250 W.
An electric heater is connected to the 220 V (effective) mains, R = 44 Ω. Iₑf = 220/44 = 5 A, P = 220·5 = 1100 W; the voltage amplitude is Uₘ = 220·√2 ≈ 311 V and the current amplitude Iₘ ≈ 7.1 A.
Class activity

In your notebook sketch u(t), i(t) and p(t) over one period (T = 20 ms) for the 220 V mains with R = 44 Ω. Show why p(t) is never negative and why its mean is half its maximum. Do not touch mains appliances.

Practice
1
What is the phase shift between current and voltage in an active resistance?
2
Active resistance R = 25 Ω and Iₑf = 4 A. What is the mean power in W?
3
In an active resistance with Iₘ = 6 A and Uₘ = 50 V what is the mean power in W?
4
Why do we not notice the flicker even though a lamp goes dim and bright 100 times a second on the mains?