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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT Inductive reactance (XL ) causes the total current to lag the applied voltage. Capacitive reactance (XC ) causes the total current to lead the applied voltage. Therefore, XL and XC tend to offset each other.

The magnitude of the total reactance in the series circuit:

The impedance :

Using Ohm s Law:

Voltage drop across the capacitor:

Voltage drop across the inductor:

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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT Voltage drop across the resistor:

Total Voltage :

a) When XC > XL (the circuit is capacitive) Impedance phasor diagram:

From the diagram, the value of can be calculated:

where : X = |XL - XC|

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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT a) When XC > XL (the circuit is capacitive) Expressing the diagram in term of voltages:

From the diagram, the value of also can be calculated by using:

where VCL = VC - VL b) When XL > XC (the circuit is inductive)Impedance phasor diagram:

Expressing the diagram in term of voltages:

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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT c) When XC = XL (the circuit is at resonance)

When XC = XL : the total reactance is zero.

Therefore:

At the resonant frequencyVC and VL > VSVC and VL cancel leaving 0V because their voltages are equal in magnitude but opposite in phaseZT = R, at the minimum value because XT = 0Thus I at the maximum valueVR = VS

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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT d) ANALYSIS OF SERIES RLC CIRCUITS

At low frequencyXC XL Circuit predominantly capacitiveAs f XC and XL until XL = XC

At high frequencyXL XC Circuit predominantly inductive

At resonance frequencyXL = XCthe two reactance cancelCircuit purely resistiveseries resonance XL is a straight line since,XL = 2fLGeneral straight line formula, y = mx + CThus y = XL; slope, m = 2L and C = 0

XC curve is called hyperbolaGeneral hyperbola formula; xy = kWhere x = f; y = XC and k = 1/2C

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Chap[17] RLC Circuit. 17.1 SERIES RLC CIRCUIT

Determine the total impedance and the phase angle for RLC circuitFind the voltage across R, L and C and draw the voltage phasor diagram.For the series RLC circuit, determine Xc , XT and Z at resonance.

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Chap[17] RLC Circuit. 17.2 PARALLEL RLC CIRCUIT

The parallel RLC circuit as show. The total impedance can be calculated using the sum of reciprocals method:

(BCL = |BC BL |) Conductance:

Capacitive susceptance:

Inductive susceptance:

Admittance:

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Chap[17] RLC Circuit. 17.2 PARALLEL RLC CIRCUIT CURRENT RELATIONSHIPSIn parallel RLC circuit, the current in the capacitive branch and the current in the inductive branch are always 180 degree out of phase with each other. For this reason, IC and IL subtract from each other. The total current can be expressed as:

where ICL = IC - IL Following the Ohm s Law:

Therefore, the total current IT:

Current at the inductor branch, IL :

Current at the capacitor branch, IC :

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17.1 PARALLEL RLC CIRCUIT

Chap[17] RLC Circuit.a) When IC > IL (the circuit is capacitive)

where ICL = IC - ILb) When I C XLXL > XCcapacitveinductiveXC = XLfrZZ = RZffrXL < XCXC < XL0