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Chapter 32: Q. 58 (page 926)

Show that the power factor of a series RLC circuit iscosϕ=RZ

Short Answer

Expert verified

We have prove that,

Power factor, cosϕ=RZ

Required proof is given below.

Step by step solution

01

Given information

We need to find out that,

Power factor is, cosϕ=RZ

02

Simplify

We know that,

Power dissipated by LCR circuit is, P=I2Zcosϕ

Here, Iis current in LCR circuit

And Zis impedance

And cosϕis power factor

Also, we know that power, P=VI

Where Vis voltage across resistor as in series voltage remain same across resistor, inductor and capacitor

Therefore, V=IR

Where Ris resistance

Therefore,

P=I2Zcosϕ=VII2Zcosϕ=IRII2Zcosϕ=I2Rcosϕ=RZ

Hence we get the require expression i.e.cosϕ=RZ

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Most popular questions from this chapter

A series RLC circuit consists of a 50Ω resistor, a localid="1650177582139" 3.3mH inductor, and a localid="1650177586647" 480nF capacitor. It is connected to an oscillator with a peak voltage of localid="1650177591401" 5.0V. Determine the impedance, the peak current, and the phase angle at frequencies :

(a) localid="1650177630395" 3000Hz(b) localid="1650177635339" 4000Hzand (c) localid="1650177638990" 5000Hz.

In FIGURE, what is the current supplied by the emf when

(a) the frequency is very small and

(b) the frequency is very large

You’re the operator of a 15000Vrms, 60HZ electrical substation. When you get to work one day, you see that the station is delivering 6MWof power with a power factor of0.90.

a. What is the rms current leaving the station?

b. How much series capacitance should you add to bring the power factor up to 1.0?

c. How much power will the station then be delivering?

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs E1=E0cosvt,E2=E0cos1vt+120°,andE3=E0cos1vt-120°over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of 450MWof electric power, a realistic value.

a. What would be the current in each wire if the transmission voltage were E0=120Vrms?

b. In fact, transformers are used to step the transmission-line voltage up to 500kVrms. What is the current in each wire?

c. Big transformers are expensive. Why does the electric company use step-up transformers?

A high-pass filter consists of a 1.59mFcapacitor in series with a 100Ωresistor. The circuit is driven by an ACsource with a peak voltage of 5.00V.

a. What is the crossover frequency fc?

b. What is VRwhen f=12fc,fc, and 2fc?
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