Lecture 26 Review Steady state sinusoidal response Phasor representation of sinusoids Phasor diagrams Phasor representation of circuit elements Related educational modules: –Section 2.7.2, 2.7.3

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 V s (t) = V m cos(100t). Let V s (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 V s (t) = 12cos(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: Voltage-current relation: 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: Voltage-current relation: 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: Voltage-current relation: 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