AC Circuit Analysis

Exponential Function Properties of exponential function: Essentially all waveforms encountered in practice can be expressed as a sum of exponential functions (2) The response of a “Linear Time-Invariant” (LTI) system to an exponential function e st is also an exponential function, H(s)e st So we only need to know H(s) – the transfer function.

The Complex Plane  j RHP LHP - What about negative frequencies? -j exponentially decreasing signals increasing - What about negative frequencies? -j

Imaginary axis projection with phase t unit circle A phasor can represent either voltage or current.

Rotating Phasor at radian frequency  Im axis Re axis Projection onto real axis imaginary axis Cosine Function Sine Function Phasor

? Voltage and current are in phase (0 ) Current leads voltage by 90 (= /2) Voltage leads current by 90 ? Resistor Capacitor Inductor Voltage in blue Current in red Phasors shown projected onto real axis.

A phasor is a complex number whose magnitude is the magnitude of a corresponding sinusoid, and whose phase is the phase of that corresponding sinusoid. A phasor is complex, and does not exist. Voltages and currents are real, and do exist. A voltage is not equal to its phasor. A current is not equal to its phasor. A phasor is a function of frequency, w. A sinusoidal voltage or current is a function of time, t. The variable t does not appear in the phasor domain. The square root of –1, or j, does not appear in the time domain. Phasor variables are often given as upper-case boldface variables, with lowercase subscripts. For hand-drawn letters, a bar are typically placed over the variable to indicate that it is a phasor.

Example: R-L Circuit i(t) v(t) R L

General Form of H(s)

A representative pole-zero diagram

Plot of H(s) 3 poles 2 zeros jw  http://lpsa.swarthmore.edu/Representations/SysRepZPK.html