# barcode in vb.net 2005 Cosine of phase angle in Software Printer Denso QR Bar Code in Software Cosine of phase angle

Cosine of phase angle
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Recall that in a circuit having reactance and resistance, the current and the voltage are not in phase. The extent to which they differ in phase is the phase angle. If there is no reactance, then the phase angle is 0 degrees. If there is a pure reactance, then the phase angle is either 90 degrees or 90 degrees. The power factor is equal to the cosine of the phase angle. You can use a calculator to find this easily.
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Problem 17-1
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A circuit contains no reactance, but a pure resistance of 600 . What is the power factor Without doing any calculations, it is evident that PF 1, because PVA PT in a pure resistance. So PT/PVA 1. But you can also look at this by noting that the phase
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PF PF% PT/PVA 100PT/PVA
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Calculation of power factor 311 angle is 0 degrees, because the current is in phase with the voltage. Using your calculator, find cos 0 1. Therefore, PF 1 100 percent. The vector for this case is shown in Fig. 17-5.
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17-5 Phase angle for a pure resistance 600 j0.
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Problem 17-2
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A circuit contains a pure capacitive reactance of 40 , but no resistance. What is the power factor Here, the phase angle is 90 degrees (Fig. 17-6). A calculator will tell you that cos 90 0. Therefore, PF 0. This means that PT/PVA 0 0 percent. That is, none of the power is true power, and all of it is imaginary or reactive power.
17-6 Phase angle for a pure capacitive impedance 0 j40.
312 Power and resonance in ac circuits
Problem 17-3
A circuit contains a resistance of 50 and an inductive reactance of 50 in series. What is the power factor The phase angle in this case is 45 degrees (Fig. 17-7).The resistance and reactance components are equal, and form two sides of a right triangle, with the complex impedance vector forming the hypotenuse. Find cos 45 0.707. This means that PT/PVA 0.707 70.7 percent.
17-7 Phase angle for impedance 50 j50.
Ratio R/Z
Another way to calculate the power factor is to find the ratio of the resistance R to the absolute-value impedance Z. In Fig. 17-7, this is visually apparent. A right triangle is formed by the resistance vector R (the base), the reactance vector jX (the height) and the absolute-value impedance Z (the hypotenuse). The cosine of the phase angle is equal to the ratio of the base length to the hypotenuse length; this represents R/Z.
Problem 17-4
A circuit has an absolute-value impedance Z of 100 , with a resistance R 80 . What is the power factor Simply find the ratio: PF R/Z 80/100 0.8 80 percent. Note that it doesn t matter whether the reactance in this circuit is capacitive or inductive. In fact, you don t even have to worry about the value of the reactance here.
Problem 17-5
A circuit has an absolute-value impedance of 50 , purely resistive. What is the power factor Here, R Z 50. Therefore, PF R/Z 50/50 1 100 percent.
How much of the power is true 313
Resistance and reactance
Sometimes you ll get data that tells you the resistance and reactance components in a circuit. To calculate the power factor from this, you can either find the phase angle and take its cosine, or find the absolute-value impedance and take the ratio R/Z.
Problem 17-6
A circuit has a resistance of 50 and a capacitive reactance of 30 . What is the power factor Use the cosine method. The tangent of the phase angle is equal to X/R. Therefore, the phase angle is arctan (X/R) arctan ( 30/50) arctan ( 0.60) 31 degrees. The power factor is the cosine of this angle; PF cos ( 31) 0.86 86 percent.
Problem 17-7
A circuit has a resistance of 30 and an inductive reactance of 40 . What is the power factor Use the R/Z method. 1600 Find the absolute-value impedance: Z2 R2 X2 302 402 900 2500; therefore Z 50. The power factor is therefore PF R/Z 30/50 0.60 60 percent. This problem is represented very nicely by a 3:4:5 right triangle (Fig. 17-8).
17-8 Illustration for Problem 17-7.