Understanding of voltage vector and rotation of three-phase electric u, v, w

Function representation of three-phase electricity

According to the general formula y=A cos(wt+φ), it can be known that the peak value of the country's one-way electricity is $220\sqrt{3}$, the frequency f is 50Hz, the angular frequency w=2Π/T=2Π*f, where T is the period, and f is the frequency.
The initial phase difference of the three-phase electricity is 120°, which is 2Π/3 radian angles.

image representation

analyze

The time (angle) relationship of the magnitude of the three-phase voltage

It is also very difficult to analyze directly by looking at the above picture. But starting the analysis with a particular moment will be very easy to analyze . Here the analysis begins at time 0
, and the three corner points at time t=0 represent the value of each phase voltage at that time. But this seems to be different from the angle and amplitude analysis of general analysis .
The three angle values ​​at this time are 0, 2Π/3, -2Π/3 (or expressed as 4Π/3) . Since their frequencies are the same , the difference between these angles remains constant . That is, the angular difference of the trigonometric function of the three-phase voltage expressed at each moment is constant . Through this relationship, the magnitude of the two outer two-phase voltages can be obtained.

Through a phase voltage analysis

For example, using the simplest voltage analysis with an initial phase of 0 (as shown in the figure below), the phases at this time are a + $+220\sqrt{3}$and two $-220\sqrt{3}/2$ . The algebraic sum of the three values​​is exactly 0.For

the u-phase voltage, assuming that the phase angle at a certain point in time is φ, the voltage value at this timeis Acos φ, which is based on the geometric properties of the trigonometric cosine function. The value is taken as aline segment with length, then the modulus length of Acosφ isthe modulus length of the projection on the straight line with angle φ between this line segment, andthe positive or negative depends on the situation of this angle. This is considered in polar terms, considering the magnitude as constant A, and taking the angle φ as a variable. Then the direction at time 0 is taken as a positive value, and the angle φ is the angle of counterclockwise rotation around the pole. The other two phases aretwo phase angles of2Π/3 and 4Π/3 added to the φ phase angleThe correspondingreal-time voltagethe length projected in the direction of u voltage valueat time 0.
The above is still very long-winded. In fact, it isto define the amplitude modulus length and square phase of a voltage. Here, the defined amplitude is$220\sqrt{3}$, and the φ angle corresponds to the value of wt after time t . The trigonometric cosine function cos makes this Acos φ have a geometric meaning . Here, the voltage can be considered as a vector with direction, mainly the initial direction with reference .

In fact, this arrow is somewhat misleading, just to indicate that this voltage is positive or negative . The voltage value at a certain moment is a scalar, and the magnitude of this value is related to the amplitude A and the angle φ at the 0 moment . The projection of the amplitude on the straight line with the angle φ is the modulus length of the real-time voltage , and the positive and negative are related to the angular range of φ. The counterclockwise rotation around the origin of a modulus length in the coordinate system is just in line with the relationship between the modulus length and angle and the voltage scalar . If the voltage amplitude is R (for the convenience of understanding the calculated values ​​of the three vectors), then all three are R, then their sum is R*[cos(φ)+cos(φ+2Π/3)+cos( φ+4Π/3)] , according to periodicity, three sums R*[cos(φ)+cos(φ+2Π/3)+cos(φ-2Π/3)]=R*[cos( φ)+2cos(φ)cos(2Π/3)]=0 ,