barcode printing using vb.net Vectors in the RL Plane in Software

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Vectors in the RL Plane
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Engineers sometimes like to represent points in the RL plane (and in other types of coordinate planes, too) as vectors. This gives each point a definite magnitude and a precise direction.
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236 Inductive reactance
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13-6 Four points in the RL impedance plane. See text for discussion.
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In Fig. 13-6, there are four different points shown. Each one is represented by a certain distance to the right of the origin (0,0), and a specific distance upwards from the origin. The first of these is the resistance, R, and the second is the inductive reactance, XL. Thus, the RL combination is a two-dimensional quantity. There is no way to uniquely define RL combinations as single numbers, or scalars, because there are two different quantities that can vary independently. Another way to think of these points is to draw lines from the origin out to them. Then you can think of the points as rays, each having a certain length, or magnitude, and a definite direction, or angle counterclockwise from the resistance axis. These rays, going out to the points, are vectors (Fig. 13-7). You ve already been introduced to these things. Vectors seem to engender apprehension in some people, as if they were invented by scientists for the perverse pleasure of befuddling ordinary folks. What are you taking this semester asks Jane. Vector analysis! Joe shudders (if he s one of the timid types), or beams (if he wants to impress Jane). This attitude is completely groundless. Just think of vectors as arrows that have a certain length, and that point in some direction. In Fig. 13-7, the points of Fig. 13-6 are shown as vectors. The only difference is that there is some more ink on the paper.
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Current lags voltage 237
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13-7 Four vectors in the RL impedance plane.
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Current lags voltage
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Inductance, as you recall, stores electrical energy as a magnetic field. When a voltage is placed across a coil, it takes awhile for the current to build up to full value. When ac is placed across a coil, the current lags the voltage in phase.
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Pure inductance
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Suppose that you place an ac voltage across a low-loss coil, with a frequency high enough so that the inductive reactance, XL, is much larger than the resistance, R. In this situation, the current is one-quarter of a cycle behind the voltage. That is, the current lags the voltage by 90 degrees (Fig. 13-8). At very low frequencies, large inductances are normally needed in order for this current lag to be a full 1 4 cycle. This is because any coil has some resistance; no wire is a perfect conductor. If some wire were found that had a mathematically zero resistance, and if a coil of any size were wound from this wire, then the current would lag the voltage by 90 degrees in this inductor, no matter what the ac frequency. When the value of XL is very large compared with the value of R in a circuit that is, when there is an essentially pure inductance the vector in the RL plane points straight up along the XL axis. Its angle is 90 degrees from the R axis, which is considered the zero line in the RL plane.
238 Inductive reactance
13-8 In a pure inductance, the current lags the voltage by 90 degrees.
Inductance and resistance
When the resistance in a resistance-inductance circuit is significant compared with the inductive reactance, the current lags the voltage by something less than 90 degrees (Fig. 13-9). If R is small compared with XL, the current lag is almost 90 degrees; as R gets larger, the lag decreases. A circuit with resistance and inductance is called an RL circuit. The value of R in an RL circuit might increase relative to XL because resistance is deliberately placed in series with the inductance. Or, it might happen because the ac frequency gets so low that XL decreases until it is in the same ball park with the loss resistance R in the coil winding. In either case, the situation can be schematically represented by a coil in series with a resistor (Fig. 13-10). If you know the values of XL and R, you can find the angle of lag, also called the RL phase angle, by plotting the point R jXL on the RL plane, drawing the vector from the origin 0 j0 out to that point, and then measuring the angle of the vector, counterclockwise from the resistance axis. You can use a protractor to measure this angle, or you can compute its value using trigonometry. In fact, you don t need to know the actual values of XL and R in order to find the angle of lag. All you need to know is their ratio. For example, if L 5 and R 3 , you will get the same angle as you would get if XL 50 and R 30 , or if XL 20 and R 12 . The angle of lag will be the same for any values of XL and R in the ratio of 5:3. It s easy to find the angle of lag whenever you know the ratio of R to XL. You ll see some examples shortly.
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