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| AC Circuit Containing Resistance, Inductance and Capacitance Circuit (LCR in series) |
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| When an AC source is connected in a circuit with a resistance and a reactance together, the current varies initially in a complex way. After sufficient time, a sinusoidally varying current persists in the circuit. This steady state current has a frequency equal to that of the source and may have a phase difference with the source voltage. |
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| As the three elements are in series, the current has the same amplitude and phase in all. Therefore voltage across R is in phase with the current. The voltage across 'L' leads the current by 90o and the voltage across 'C' lags the current by 90o. The phasor diagram is as follows |
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| If the LCR circuit is predominantly an inductive circuit [ i.e., EoL > EoC], then the effective value of E would be |
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| where, |
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| 'Z' represents the total effective opposition offered by LCR circuit to AC and is called impedance. |
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| q is the phase angle which indicates that effective EMF leads the current (provided EL > EC). |
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| If I = Iosinwt then the voltage in the LCR circuit would be |
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| E = Eo sin (wt+q) where |
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| Eo = IoZ |
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| q = tan-1[XL -XC]/R |
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| Note 1: |
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| When XL = XC , tan q = 0 |
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| Here voltage and current are in the same phase. The AC circuit is purely a resistive one. |
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| \I = Io sin wt |
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| E = Eo sin wt where Eo = IoR |
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| Note 2: |
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| When XL > XC, tanq>0. Here, the voltage leads the current and the AC circuit is the inductance-dominated circuit. |
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| \I = Io sin wt |
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| E = Eo sin (wt+q) where q = tan-1[XL -XC]/R |
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| Note 3: |
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| When XL < XC, tanq<0. Here, the voltage lags current and the AC circuit is capacitive dominated circuit. |
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| \I = Io sin wt |
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| E = Eo sin (wt - q) where = tan-1[XC -XL]/R |
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