AC Power

1.Explain the concepts of effective values of alternating current and voltage, and the power of alternating current.

Solution:

Measuring instruments do not measure instantaneous or maximum values of current and voltage, but the "effective values" of current and voltage.
fyzika-vykon-striedaveho-prudu-1.gif


Power of AC current in a circuit with a resistor:

P=RIm2sin⁡2(ωt),[P]=WP = R I_m^2 \sin^2 (\omega t), \qquad [P] = \text{W}

Power of AC current in a circuit with impedance:

a) Active (real) power of AC current

P=IUcos⁡φ[P]=1 WP = I U \cos \varphi \qquad [P] = 1\,\text{W}

b) Reactive power of AC current

PZ=IU[PZ]=1 V⋅AP_Z = I U \qquad [P_Z] = 1\,\text{V⋅A}

c) Apparent power of AC current

PQ=IUsin⁡φ[PQ]=1 V⋅ArP_Q = I U \sin \varphi \qquad [P_Q] = 1\,\text{V⋅Ar}

I,U are effective (RMS) values.

Relationship between the individual AC power components:

PZ2=P2+PQ2P_Z^2 = P^2 + P_Q^2

Power factor:

cos⁡φ=PIU,0<φ<π2\cos \varphi = \frac{P}{I U}, \qquad 0 < \varphi < \frac{\pi}{2}

Efficiency of an electric motor:

η=P2P1⋅100%\eta = \frac{P_2}{P_1} \cdot 100\% = \frac{U I \cos \varphi}{P_{Z2}} \cdot 100\

P1=active (real) power of the electric current
P2=power of the electric motor

2. A resistor with resistance 20Ώ is connected to an AC voltage source Uef = 24V and f = 50Hz. Write the equation for the instantaneous value of the current in the circuit. Determine the effective value of the current in the circuit.

Solution:

Analysis:

R = 20Ώ,  Uef = 24V  f = 50s-1,  Ief = ?

fyzika-vykon-striedaveho-prudu-2n.gif
 

  • The equation for the instantaneous value of current in the circuit is i = 1.7.sin(100π.t) [A]
  • The effective value of the current in the circuit is Ief = 1.2A.

3.Determine the time moments during one period of AC voltage at which the instantaneous voltage equals the effective voltage.

Solution:

fyzika-vykon-striedaveho-prudu-3.gif 

The instantaneous voltage equals the effective voltage at t1 =T/8 and t2 = 3T/8.

4.The AC voltage has an effective value Uef = 156V. Determine the amplitude of the voltage. After what time from the initial moment will the instantaneous value of the AC voltage reach the effective value, if f = 50Hz.

Please log in to view the solution.

5.For instantaneous values of current and voltage in an AC circuit, the following equations hold: i = 1.sin(100π.t – π/4) [A] and u = 150.sin(100π.t) [V]. Determine the active, apparent, and reactive power of this current. Show that it holds:

fyzika-vykon-striedaveho-prudu-5za.gif
Please log in to view the solution.

6.Calculate the active power of alternating current connected to an AC source (U = 220V, f = 50Hz, cos φ = 0.8), if the circuit contains a load with the following properties:

  • a.) coil with inductance L = 0.1H
  • b.) capacitor with capacitance C = 10-3F
Determine how the active power of alternating current changes with frequency.
Please log in to view the solution.

7. Calculate the current flowing through a single-phase AC motor if it has a power of 5kW and is connected to a 220V power supply. The motor power factor is 0.85 and its efficiency is 80%.

Please log in to view the solution.

8. At a voltage of 48V, the circuit current is 6A. The active power in the circuit is 200W. Calculate the power factor. Assess how the active power of the device would change if the power factor were improved to 0.98.

Please log in to view the solution.

9. An AC voltage of 3000V with f = 50Hz is connected in series with a resistor R = 40Ώ and a coil with inductance L = 95·10-3H. Determine the current in the circuit, the phase shift between current and voltage, and the active power of the current.

Please log in to view the solution.

10. A single-phase motor consumed current 10A at voltage 220V for 5 minutes. The electricity meter measured 0.125 kWh during that time. Determine the motor power factor.

Please log in to view the solution.

11.The voltage and current in a coil change according to the equations u = 160.sin(100πt) (V) and i = 3.sin(100π – π/3) (A). Determine the effective values of voltage and current in the circuit, the power factor, and the active power.

Please log in to view the solution.

12.A resistor with resistance 20Ω is connected to an AC voltage source Uef = 24V and frequency f = 50Hz. Write the equation for the instantaneous value of the current in the circuit. Calculate the instantaneous current at time 0.001s.

Please log in to view the solution.

13.An AC voltage has an effective value Uef = 156V. Determine the amplitude of the voltage. After what time from the initial moment will the instantaneous value of the AC voltage reach the effective value, if f = 50Hz? At the initial moment, the value of the AC voltage is zero.

Please log in to view the solution.

14.Let P be the active power of alternating current, PZ the apparent power, and PQ the reactive power. Show that the following holds:

vykon-striedaveho-prudu-14z.gif
Please log in to view the solution.

15.The label of an electric motor states U = 230V, I = 20A, cosφ = 0.6. Calculate the active, apparent and reactive power and perform a check.

Please log in to view the solution.

16.For the instantaneous values of voltage and current in an AC circuit, the following holds:

vykon-striedaveho-prudu-16z

  • a.) Calculate the frequency of the alternating current
  • b.) Calculate the active power of the alternating current
  • c.) Determine whether the circuit has inductive or capacitive characteristics.
Please log in to view the solution.

17.How will the active power of alternating current in a device change when the frequency increases, if the device has inductive characteristics?

Please log in to view the solution.

18.How will the active power of alternating current in a device change when the frequency increases, if the device has capacitive characteristics?

Please log in to view the solution.

19.The active power of a power plant at voltage 6kV, cosφ = 1, is P = 3 MW. Calculate how the current changes if the power factor changes to cosφ = 0.8

Please log in to view the solution.

20.What capacitance must a capacitor connected in series with a light bulb with rated voltage U/ = 12V and power input 60W have, so that we can connect the bulb to a voltage of U = 240V and frequency 50Hz, and the bulb shines at full brightness?

vykon-striedaveho-prudu-20z
Please log in to view the solution.