In Fig. 13.11 , R=2.2 kΩ and ν=V0cos(ω0t), where V0=25 V and f0=20 kHz equals the resonant frequency of the circuit. The reactive power dissipated in the inductor is QL=5 VAR. Find the inductance L and the capacitance C

Question

In Fig. 13.11 [latex], R=2.2 k \Omega[/latex] and [latex] u=V_0 \cos \left(\omega_0 t ight)[/latex], where [latex]V_0=25 V[/latex] and [latex]f_0=20 kHz[/latex] equals the resonant frequency of the circuit. The reactive power dissipated in the inductor is [latex]Q_L=5 VAR[/latex]. Find the inductance L and the capacitance C

Accepted Answer

The impedance of the circuit at resonance equals the resistance R. Thus the current through the series circuit is

[latex] ilde{I}=\frac{V_0}{R} \cong 11.36 mA \Rightarrow I_{r m s}=\frac{| ilde{I}|}{\sqrt{2}} \cong 8.04 mA [/latex]

The reactive powers dissipated in the inductor and in the capacitor are equal in magnitude and opposite in sign, and are given by

[latex] Q_L=I_{r m s}{}^2 \omega L, \quad Q_C=-\frac{I_{r m s}{}^2}{\omega C} [/latex]

Thus [latex]Q_C=-Q_L[/latex] and it follows that

[latex] \begin{aligned} L &=\frac{Q_L}{\omega_0 I_{r m s}{}^2} \cong \frac{5 VAR }{(2 \pi)(20 kHz )(8.04 mA )^2} \\ & \cong 616.2 mH , \\ C &=-\frac{I_{r m s}{}^2}{\omega_0 Q_C}=\frac{I_{r m s}{}^2}{\omega_0 Q_L} \\ & \cong \frac{(8.04 mA )^2}{(2 \pi)(20 kHz )(5 VAR )} \cong 102.8 pF , \\ ext { or } \\ C &=\frac{1}{\omega_0^2 L} \cong 102.8 pF .\end{aligned} [/latex]

Question 13.7: In Fig. 13.11 , R=2.2 kΩ and ν=V0cos(ω0t), where V0=25 V an...

Related Answered Questions

In the circuit shown in Fig. 13.37, the source output resistance is Ro=100Ω and the load resistance is RL=10Ω. The source voltage is sinusoidal, with frequency f=60 Hz . Find the values of the inductance L and capacitance C in the L-section matching circuit for which maximum power is ...

Verified Answer:

Refer to Fig. 13.3, where νS(t)=V0cos(ωt), V0=170 V , f=60 Hz R1=100 Ω, C=5 μF, R2=1 kΩ, and L=400 mH . Find the average power dissipated in the load. ...

Verified Answer:

In Fig. 13.8, the load impedance is Z=(15+j12)Ω at the frequency of the fixed amplitude source voltage Vrms=120 V∠0. The load requires real power P=500 W . What rms line current would be required if the load were Z=15 Ω ? What is the actual rms line current? ...

Verified Answer:

The (rms) voltmeter and ammeter readings in Fig. 13.9 are 13.67 V and 24.00 mA , respectively. Find the complex power, the apparent power, the real power, and the reactive power delivered by the independent source νS. ...

Verified Answer:

Show that complex power is conserved in the circuit of Example 13.2. ...

Verified Answer:

Refer to Fig. 13.6. Find the complex power dissipated in each element, including the source. ...

Verified Answer:

A voltage ν(t)=V0cos(ωt) with V0=25 V and f=1 kHz is applied to a load consisting of a resistor having resistance R=1.5 kΩ in parallel with an inductor having inductance L=50 mH . Find the average power dissipated in the load and the average power delivered by the source. ...

Verified Answer:

In Fig. 13.12, the rms amplitude of the sinusoidal current source is Irms=5 mA and the frequency of the source is 100 kHz, which is the resonant frequency of the circuit. The coil has quality factor Q=34 and effective resistance R=3.6 Ω at 100 kHz .Find the reactive power dissipated in the ...

Verified Answer:

A voltage ν(t)=V0cos(ωt) is applied to the series combination of a capacitor and a resistor. Obtain expressions for the power factor and the average power dissipated in the load. ...

Verified Answer:

Show that the real part of the impedance of a source-free R L C load is nonnegative. (We make use of this fact in the development following this example.) ...

Verified Answer: