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RLC circuit analysis
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WHENEVER YOU SEE AC CIRCUITS WITH INDUCTANCE AND/OR CAPACITANCE AS well as resistance, you should switch your mind into 2D mode. You must be ready to deal with two-dimensional quantities. While you can sometimes talk and think about impedances as simple ohmic values, there are times you can t. If you re sure that there is no reactance in an ac circuit, then it s all right to say Z = 600 ohms, or This speaker is 8 ohms, or The input impedance to this amplifier is 1,000 ohms. As soon as you see coils and/or capacitors, you should envision the complex-number plane, either RX (resistance-reactance) or GB (conductance-admittance). The RX plane applies to series-circuit analysis. The GB plane applies to parallel-circuit analysis.
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Complex impedances in series
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When you see resistors, coils, and capacitors in series, you should envision the RX plane. Each component, whether it is a resistor, an inductor, or a capacitor, has an impedance that can be represented as a vector in the RX plane. The vectors for resistors are constant regardless of the frequency. But the vectors for coils and capacitors vary with frequency, as you have learned.
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Pure reactances
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Pure inductive reactances (XL) and capacitive reactances (XC) simply add together when coils and capacitors are in series. Thus, X XL XC. In the RX plane, their vectors add, but because these vectors point in exactly opposite directions inductive reactance upwards and capacitive reactance downwards the resultant sum vector will also inevitably point either straight up or down (Fig. 16-1).
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Complex impedanses in series 285
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16-1 Pure inductance and pure capacitance are represented by reactance vectors that point straight up and down.
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Problem 16-1
A coil and capacitor are connected in series, with jXL j200 and jXC j150. What is the net reactance vector jX Just add the values jX jXL jXC j200 ( j150) j(200 150) j50. This is an inductive reactance, because it is positive imaginary.
Problem 16-2
A coil and capacitor are connected in series, with jXL j30 and jXC jl10. What is the net reactance vector jX Again, add jX j30 ( jll0) j(30 110) j80. This is a capacitive reactance, because it is negative imaginary.
Problem 16-3
A coil of L 5.00 H and a capacitor of C 200 pF are in series. The frequency is f 4.00 MHz. What is the net reactance vector jX First calculate jXL Then calculate jXC Finally, add jX jXL jXC j126 ( jl99) j73 j(1/(6.28fC) j(l/(6.28 4.00 0.000200) j199 j6.28fL j(6.28 4.00 5.00) j126
286 RLC circuit analysis This is a net capacitive reactance. There is no resistance in this circuit, so the impedance vector is 0 j73.
Problem 16-4
What is the net reactance vector jX for the above combination at a frequency of f MHz First calculate jXL Then calculate jXC Finally, add jX jXL jXC j314 ( j79.6) j234 j(1/(6.28fC)) j(l/(6.28 10.0 0.000200) j79.6 j6.28fL j(6.28 10.0 5.00) j314 10.0
This is a net inductive reactance. Again, there is no resistance, and therefore the impedance vector is pure imaginary, 0 j234. Notice that the change in frequency, between Problems 16-3 and 16-4, caused the circuit to change over from a net capacitance to a net inductance. You might think that there must be some frequency, between 4.00 MHz and 10.0 MHz, at which jXL and jXC add up to j0 that is, at which they exactly cancel each other out, yielding 0 j0 as the complex impedance. Then the circuit, at that frequency, would appear as a short circuit. If, you suspect this, you re right. Any series combination of coil and capacitor offers theoretically zero opposition to ac at one special frequency. This is called series resonance, and is dealt with in the next chapter.
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