Fundamentals of Electrical Engineering 2 Grundlagen der Elektrotechnik 2

Fundamentals of Electrical Engineering 2 Grundlagen der Elektrotechnik 2 Chapter: Sinusoidal Steady State Analysis / Netzwerkanalyse bei harmonischer Erregung Michael E. Auer Source of figures: Alexander/Sadiku: Fundamentals of Electric Circuits, McGraw-Hill

Course Content Sinusoids and Phasors / Harmonische Funktionen und Zeiger Sinusoidal Steady State Analysis / Netzwerkanalyse bei harmonischer Erregung Frequency Response Analysis / Frequenzgang Analyse AC Power / Leistung in Wechselstromkreisen Magnetically Coupled Circuits / Magnetisch verkoppelte Kreise

Chapter Content Introduction / Einführung Thevenin and Norton Equivalent Circuits / Spannungs- und Stromquellenersatzschaltung Source Transformation / Quellentransformation Nodal and Mesh Analysis / Knotenspannungs- und Maschenstromanalyse Superposition Theorm / Überlagerungssatz OpAmp AC Circuits / OV Schaltungen bei Wechselstrom Summary / Zusammenfassung

Basic Approach 1. Transform the circuit to the phasor or frequency domain. 2. Solve the problem using circuit techniques (nodal analysis, mesh analysis, superposition, etc.). 3. Transform the resulting phasor to the time domain. Time to Freq Solve equations in Freq domain Freq to Time

Basic Example (1) Time domain 1 jωc Frequency domain V V s 0

Basic Example (2) Time domain Frequency domain jωl 1 jωc Vs V0

Thevenin and Norton Equivalent Circuits (1) Thevenin transform Norton transform

Thevenin and Norton Equivalent Circuits (2) Example R =1Ω L =1H v ( t) = 5cost V C =1F R jωl V 1 jωc

Thevenin and Norton Equivalent Circuits (3) Find the Thevenin equivalent at a - b: 30 < 20 V Solution: Z Th = 12.4 j3.2 ; V Th = 18.97 < -51.57 V

Source Transformation (1)

Source Transformation (2) Use source transformation to obtain V x : 5,52 < -28 V

Mesh Analysis

Nodal Analysis

Superposition (1) for linear circuits only applies to ac circuits the same way it applies to dc circuits circuits with more than one source sources operating at different frequencies possible

Superposition (2) Example dc voltage source ac voltage source ac current source

OpAmp AC Circuits (1) Assumptions: The OpAmp is operating in the linear region. No current enters either of its input terminals. The voltage across its input terminals is zero

OpAmp AC Circuits (2) Time domain circuit Frequency domain 1 jωc 1 1 jωc 2 V s V 0

Summary We apply nodal and mesh analysis to ac circuits by applying KCL and KVL to the phasor form of the circuits. In solving for steady-state response of a circuit that has independent sources with different frequencies, each independent source must be considered separately. The most natural approach to analyzing such circuits is to apply the superposition theorem. A separate phasor circuit for each frequency must be solved independently, and corresponding response should be obtained in the time domain. The overall response is the sum of the time domain responses of all the individual phasor circuits. The Thevenin equivalent of an ac circuit consists of a voltage source V Th in series with the impedance Z Th. The Norton equivalent of an ac circuit consists of a current source I N in parallel with the impedance Z N = Z Th. The concept of source transformation is also applicable in the frequency domain.