Complex Impedances: Simplifying Circuit Analysis, Study notes of Electrical and Electronics Engineering

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Complex Impedances

ECE 3710, Fall 2011

Text Key Sec<ons 5.2 and 5.3 in the book

Impedance of an Inductor i L ( t ) =^ I m sin(! t + !) Suppose we have the current through an inductor given by: Voltage across the inductor can be expressed as: v L ( t ) =^ L^ di L ( t ) dt

=! L I

m cos(! t + !) Now we can express the phasors for the current and voltage: I L

= I

m

V

L

=! L I

m ! " =(! L! 90 °) " I m

Impedance of an Inductor V L

=! L I

m ! " = (! L! 90 °) " I m

V L =! L! 90 ° " I L The phasor for the current V L = j! L! I L V L = Z L ! I L " Z L = j! L Ohm’s Law in phasor form! The impedance of an inductor

Impedance of a Capacitor Solving the phasor voltage and current for a capacitor, we can similarly find an expression for the impedance of a capacitor V C = Z C I C Z C = 1 ! C ! " 90 ° = 1 j! C

What do Phases in the Impedance Mean? Z L =! L! 90 ° 2π Remember that one period of sinusoid is 2π (Think of the rota<ng vector!) v L ( t ) i L ( t ) 90 ° For an inductor, the current lags the voltage by 90°

Impedance of a Resistor Does a resistor have a complex impedance? 2π v L ( t ) i L ( t ) V R = RI R No , the voltage and current are in-­‐phase.

Phasor Diagrams V C = V M !! I C = I M !! + 90 ° The current is leading the voltage by 90° (pure capacitance)

Inductor Example (A) V L

= 100! 0 ° I

L

V

L I L 100 2