2. Single Pha Power
Consider a resistor and inductor connected in ries
A single frequency ac voltage v is applied across the two elements and a current i flows through
them. The impedance equivalent circuit reprentation is单词搜索
Phasor values are ud to reprent the quantities of this circuit. They implicitly take into
account the ac waveform. In general complex terms
v = V ∠φ = e + jf
camouflagewhere V = (e 2 + f 2)½ is the magnitude of the applied voltage
and φ = tan -1⎟⎠
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⎞⎜⎝⎛e f is the voltage angle with respect to some reference.
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The voltage itlf is often chon as the reference vector
v = V + j0 = V ∠0o
The impedance is
Z = |Z |∠θ = R + jX L
where |Z | = (R 2 + X L 2)½
and θ = tan -1⎟⎠
⎞⎜⎝⎛R X L is the angle of the impedance
Thus, the current through the impedance is
i = Z v = ⎟⎟⎠⎞⎜⎜⎝⎛+L jX R V 1 = ()L L
jX R X R V −+22
= a – jb
In phasor notation
i = θ
∠Z V = I ∠-θ
The current is said to lag the voltage by an angle θ. The phasor reprentation of the voltage and current is
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In this reprentation, the voltage phasor is vertical. This is the direction of the voltage drawn across the two terminals of the impedance Z . It conveys that the voltage increas from the lower terminal (often earthed or grounded) to the higher terminal, akin to the depiction of a voltage of a transmission line conductor above the ground.
The current phasor is parated from the voltage phasor by an angle θ. This depiction shows the current flowing in the general direction to that depicted in the equivalent circuit, into a resistance/inductance impedance.
In phasor notation
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i = θ∠∠Z V 0 = θ−∠Z
V
= θcos Z V - j θsin Z
圣诞歌英文版V
= a – jb
The complex power absorbed by the impedance is the product of the voltage and the complex conjugate of the current
s = vi *
= V (a + jb) = VI ∠θ
= ()L L
jX R X R V ++222
= P + jQ
A phasor reprentation of P and Q often ud by electric power engineers is
P is the real power and Q is the reactive power of the circuit, measured in Watts and VArs
respectively. P is measure of the power associated with the resistor and Q is a measure of the transfer of energy stored in the magnetic field of the inductor (inductive reactance). An inductor is said to absorb reactive power.
By convention, the complex conjugate of the current is ud in determining P and Q , to define the terms as being positive or flowing into a resistor/inductor circuit.
Also s = S ∠θ
where S = VI is the apparent power
hence s = VI cos θ + jVI sin θ
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Thus P = VI cos θ
Alternatively
cos θ = VI
P
cos θ is the ratio of the real power to the apparent power, and is called the power factor. It is also given the symbol pf .
pf = VI
P
For the ca where the current lags the voltage (as for a resistor/inductor impedance), the power factor is lagging. In a power system, a load reprented by a resistor and an inductor has a lagging power factor, pf .
The angle θ is the power factor angle.
θ = tan -1⎟⎠
⎞⎜⎝⎛P Q
suThus the power factor angle is the impedance angle, and the phasor relationship of the power quantities is
This has been drawn in the same orientation as the voltage and current phasors reprenting the
impedance.
In power systems, the main goal is to deliver real power to customers so that uful work can be done. However, becau of the nature of induction motors in particular, but also the transformers and transmission lines of the system, there is always some inductive reactance associated with the resistance that the real power is associated with.
Thus for any given pf , to get a required amount of real power, reactive power must also be supplied. Therefore, equipment must be rated to take the phasor sum of P and Q . This is S , the apparent power. It is important to have high pf for high real power transfer utilization of equipment.
2.1 CAPACITOR
C onsider a voltage applied to a capacitor
The current i C leads the voltage v by 90°.
The impedance of the capacitor is
lookintoZ C = C
j ω1 = -jX C = X C ∠-90o
Hence i C = C Z v = C
X v ∠90o
The power associated with the capacitor is
s = vi C * = C
X v 2∠-90o = -jQ C
By convention, when the current leads the voltage, the reactive power is negative. Esntially a capacitor generates reactive power.
