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Zorluk: ZorAlternating Current (AC) Circuits

A 150 V150\text{ V} (RMS) AC generator is connected across a series combination of a 12 Ω12\ \Omega resistor, an inductor with an inductive reactance of 20 Ω20\ \Omega, and a capacitor with a capacitive reactance of 11 Ω11\ \Omega. What is the average electrical power consumed by this circuit?

  1. A
    900 W900\text{ W}
  2. 1200 W1200\text{ W}Cevap
  3. C
    1500 W1500\text{ W}
  4. D
    1875 W1875\text{ W}

Cevap

The average electrical power consumed by the circuit is 1200 W1200\text{ W}.
To find the average real power dissipated in an AC series circuit, first determine the net reactance X=XL−XC=20 Ω−11 Ω=9 ΩX = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega. Next, calculate total impedance using Z=R2+X2=122+92=15 ΩZ = \sqrt{R^2 + X^2} = \sqrt{12^2 + 9^2} = 15\ \Omega. The RMS current in the circuit is Irms=VrmsZ=150 V15 Ω=10 AI_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}. Since energy is dissipated only by resistance, the average real power is P=Irms2R=(10)2×12=1200 WP = I_{\text{rms}}^2 R = (10)^2 \times 12 = 1200\text{ W}.

Adım Adım Çözüm

1
Calculate the net reactance of the circuit.
X=XL−XC=20 Ω−11 Ω=9 ΩX = X_L - X_C = 20\ \Omega - 11\ \Omega = 9\ \Omega
Inductive and capacitive reactances oppose each other in phase by 180∘180^\circ.
2
Calculate the total impedance of the series RLC circuit.
Z=R2+(XL−XC)2=122+92=144+81=225=15 ΩZ = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\ \Omega
Resistance and net reactance add vectorially at a right angle.
3
Determine the RMS current flowing through the circuit.
Irms=VrmsZ=150 V15 Ω=10 AI_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{150\text{ V}}{15\ \Omega} = 10\text{ A}
Ohm's law for AC circuits states that current is the supply RMS voltage divided by total impedance.
4
Calculate the average real power dissipated by the circuit.
Pavg=Irms2R=(10 A)2×12 Ω=1200 WP_{\text{avg}} = I_{\text{rms}}^2 R = (10\text{ A})^2 \times 12\ \Omega = 1200\text{ W}
In an AC circuit, average power is only dissipated by resistive components, as ideal inductors and capacitors consume zero net real power over a complete cycle.

Anahtar Kavram

Power Dissipation in AC Series Circuits
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Alternating Current (AC) Circuits

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