Phase Angle (Phase angle) A positive phase angle shifts the cosine to the left. A negative phase angle shifts the cosine to the right.
Root Mean Square(RMS) Voltage (RMS value of a sine wave)
RMS Value of a Triangular Wave (Triangular wave)
Sinusoidal Response Trasient componentsteady-state component (Phasor analysis deals only with the steady-state component) Advantage: avoid solving diffEQ
Phasor Transform A phasor is a complex representation of a phse-shifted sine wave. If f(t) is equal to V m cos(ωt+Φ) then the phasor transform of f(t) is V m e jΦ Another way to write the phasor of f(t) is V m cos(Φ)+jV m sin(ωt+Φ) The phasor transform is useful in circuit analysis because it reduces the task of finding the maximum amplitude and phase angle of the steady state sinusoidal response to algebra of complex number.
Big Idea If a circuit is driven with a source f(t)=Acos(ωt+Φ), the frequency will be the same for all components in the circuit. Phasor transform allows us to focus on the phase shift and magnitude at a given ω
Properties f(t) ↔F df(t)/dt↔jωF (see notes) Integration of f(t) ↔F/(jω) (see notes)
Applications Voltage across an inductor – v=Ldi/dt – Phasor notation: V=L(jω)I – Impedance of an inductor Z=V/I=jωL Current of a Capacitor – i=Cdv/dt – Phasor notation: I=C(jω)V – Impedance of a capacitor Z=V/I=1/(jωC)
Application of Phasor Transform Assumption: in a linear circuit driven by sinusoidal sources, the steady-state response is also sinusoidal. The frequency of the sinusoidal response is the same as the frequency of the sinusoidal source.
Derivation of the Steady State Response (Assumption)