Complex Impedances Sinusoidal Steady State Analysis

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Complex ImpedancesSinusoidal Steady State
Analysis

• ELEC 308
• Elements of Electrical Engineering
• Dr. Ron Hayne
• Images Courtesy of Allan Hambley and Prentice-Hall

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Complex Impedances

• Inductance and Capacitance represented as Complex
  Numbers

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Inductance
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Inductance
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Impedance

• Ohms Law in phasor form
• Phasor voltage equals impedances times the phasor
  current
• Impedance is COMPLEX, in general
• Can be strictly REAL
• Impedance Resistance
• Can be strictly IMAGINARY
• Impedance Reactance
• Both inductances and capacitances

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Capacitance
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Capacitance
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Resistance

• The phasors are related by
• VR RIR

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Exercise 5.7
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Exercise 5.8
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Steady-State Circuit Analysis

• Circuit Analysis Using Phasors and Impedances
• Replace the time descriptions of voltage and
  current sources with corresponding phasors.
• All of the sources must have the SAME frequency!
• Replace inductances, capacitances, and
  resistances with their corresponding impedances.
• Analyze the circuit using any of the techniques
  from Chapters 1 and 2 by performing the
  calculations with complex arithmetic.

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Example 5.4

• Find the steady-state current for the circuit
  shown below. Also, find the phasor voltage
  across each element and construct a phasor
  diagram.

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Phasor Diagram
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Example 5.5

• Series and Parallel Combinations of Complex
  Impedances
• Find the voltage vc(t) in steady state. Find the
  phasor current through each element, and
  construct a phasor diagram showing the currents
  and source voltage.

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Phasor Diagram
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Exercise 5.9

• Find i(t) in the circuit below. What is the
  phase relationship between vs(t) and i(t)?

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Exercise 5.9

• Find i(t) in the circuit below. What is the
  phase relationship between vs(t) and i(t)?

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Exercise 5.10

• Find the phasor voltage and current for each
  circuit element.

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Summary

• Complex Impedances
• Inductance
• Capacitance
• Resistance
• Sinusoidal Steady State Analysis
• Ohms Law
• KVL (Mesh-Current Analysis)
• KCL (Node-Voltage Analysis)