Lecture 26 Review Steady state sinusoidal response Phasor

Lecture 26 • Review • Steady state sinusoidal response • Phasor representation of sinusoids • Phasor diagrams • Phasor representation of circuit elements • Related educational materials: –Chapter 10. 3, 10. 4

Steady state sinusoidal response – overview • Sinusoidal input; we want the steady state response • Apply a conceptual input consisting of a complex exponential input with the same frequency, amplitude and phase • The actual input is the real part of the conceptual input • Determine the response to the conceptual input • The governing equations will become algebraic • The actual response is the real part of this response

Review lecture 25 example • Determine i(t), t , if Vs(t) = Vmcos(100 t). • Let Vs(t) be: • Phasor: • The phasor current is: • So that

Phasor Diagrams • Relationships between phasors are sometimes presented graphically • Called phasor diagrams • The phasors are represented by vectors in the complex plane • A “snapshot” of the relative phasor positions • For our example: • ,

Phasor Diagrams – notes • Phasor lengths on diagram generally not to scale • They may not even share the same units • Phasor lengths are generally labeled on the diagram • The phase difference between the phasors is labeled on the diagram

Phasors and time domain signals • The time-domain (sinusoidal) signals are completely described by the phasors • Our example from Lecture 25:

Example 1 – Circuit analysis using phasors • Use phasors to determine the steady state current i(t) in the circuit below if Vs(t) = 12 cos(120 t). Sketch a phasor diagram showing the source voltage and resulting current.

Example 1: governing equation

Example 1: Apply phasor signals to equation • Governing equation: • Input: • Output:

Example 1: Phasor diagram • Input voltage phasor: • Output current phasor:

Circuit element voltage-current relations • We have used phasor representations of signals in the circuit’s governing differential equation to obtain algebraic equations in the frequency domain • This process can be simplified: • Write phasor-domain voltage-current relations for circuit elements • Convert the overall circuit to the frequency domain • Write the governing algebraic equations directly in the frequency domain

Resistor i-v relations • Time domain: • Conversion to phasor: • Voltage-current relation:

Resistor phasor voltage-current relations • Phasor voltage-current relation for resistors: • Phasor diagram: • Note: voltage and current have same phase for resistor

Resistor voltage-current waveforms • Notes: Resistor current and voltage are in phase; lack of energy storage implies no phase shift

Inductor i-v relations • Time domain: • Conversion to phasor: • Voltage-current relation:

Inductor phasor voltage-current relations • Phasor voltage-current relation for inductors: • Phasor diagram: • Note: current lags voltage by 90 for inductors

Inductor voltage-current waveforms • Notes: Current and voltage are 90 out of phase; derivative associated with energy storage causes current to lag voltage

Capacitor i-v relations • Time domain: • Conversion to phasor: • Voltage-current relation:

Capacitor phasor voltage-current relations • Phasor voltage-current relation for capacitors: • Phasor diagram: • Note: voltage lags current by 90 for capacitors

Capacitor voltage-current waveforms • Notes: Current and voltage are 90 out of phase; derivative associated with energy storage causes voltage to lag current