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AC circuits: inductive reactance, capacitive reactance, impedance and admittance

CHM6158C - Lecture 18

Notes from Gibilisco; Teach Yourself Electricity and Electronics, McGraw-Hill, Chapters 13-15

• voltage-current profiles of L, C, RL, RC and circuits• impedance of RL, RC and RLC circuits

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Inductive reactance (XL)

inductance = property of circuit where change in electric current through the circuit induces an electromotive force that opposes the change in current= …..

Increase AC frequency –what will be the effect?

fLX L 28.6=Inductive reactance:

Inductance (Henrys = Ωs)

Frequency (Hz)

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Examples

• inductance = 0.50 H, frequency = 60 Hz, what is XL?

• for the same circuit as above, we use a battery with 12 VDC, what is XL?

What are the units of XL?

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Voltage-current profile

Current lags driving (source) voltage

For pure inductance circuit For pure resistance circuit

Current in phase with driving (source) voltage

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Inductive reactance and resistance

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Combined inductance and resistance

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Phase angle

Apply Pythagoras to obtain hypotenuse = impedance

sinφ =

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Capacitive reactance (XC)

Similarly to inductance, capacitance has an effect on impedance

By convention, capacitive reactance is expressed as a negative ohmic value

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AC and capacitance

What is the voltage-current profile going to look like?

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Current leads voltage

)28.6/(1 fCXC −=Capacitive reactance:

FHz

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Inverse relationship

Intuitive understanding:Larger C ⇒ more movement of charge ⇒ smaller resistanceHigher f ⇒ more movement of charge ⇒ approach short-circuit

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Examples

• capacitor = 0.001 μF, frequency = 1 MHz, what is XC?

• for the same circuit as above, we use a battery with 12 VDC, what is XC?

For the second question, what this mean for transmission on DC across a capacitor?

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Combined capacitive reactance and resistance

Work out phase angle

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Impedance

Impedance (Z) = resistance to AC current, composed of resistance, inductive reactance and capacitive reactance

Inductive reactance

Capacitive reactance

Resistance

Absolute value of impedance

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RLC in-series circuit

Examples:

• R = 50 Ω, XL = 22 Ω, XC = -33 Ω

• R = 50 Ω, XL = 444 Ω, XC = -444 Ω

Z = R + j(XL + XC)

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Inverse definitions

DC circuit

Resistance (R) ≠ conductance (G)

AC circuit

Impedance (Z) ≠ admittance (Y)

Reactance (X) ≠ susceptance (B)

BUT, note sign inversion for imaginary numbers:BC = 1/XC, but due to imaginary numbers:

jBC = -j/XC

Example:

XC = 10 Ω, what is jBC?

jBC = 1/(j10) = (1/j)(1/10) = -j 0.1

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RLC in-parallel

GB planeY = G + j(BL + BC)

Example:• G = 0.1 Siemens, BL = -j0.010, BC = j0.020