Chapter I - Transient and Harmonic Analyses (of Linear Systems)Chapter I - Transient and Harmonic Analyses (of Linear Systems)

Chapter Outcomes

Joseph Fourier

Formulate time dependent differential equation descriptions of circuits and solve them using the Laplace Transform.
Formulate and solve circuit equations under harmonic excitation using the complex impedance/phasor representations.
Determine the appropriate method used to solve for each of the steady-state and transient solution for a given circuit.

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Transient analysis is the process of solving for the time domain evolution of signals (and system responses) as time functions.
Harmonic analysis is the process of solving for the harmonic (single frequency) steady state signals (and system responses).
The time domain is where we deal with all analysis procedures as functions of time. This type of analysis typically results in time dependent differential equations that require transformation to the Frequency domain (or the Laplace s‐domain).

Time Domain and Frequency Domain “Languages” of expression for the development of solutions “expressions” describing Transient and Steady “States”.
Typically, the Time Domain “language” is used for expressing transient state terms while the Frequency domain “language” is used for harmonic steady state expressions.
At times, or for special purposes, the opposite is applied.

Refer to Figure 1.7: The harmonic signal, $A \cos(\omega t + \phi)$ can be seen as the horizontal projection of a rotating line “phasor” at an angular frequency of $\omega$.
A complementary “quadrature” term, $A \sin(\omega t+\phi)$ coexists (does not contain additional information). A “pack” phasor expression $\bar{A} = A \cdot e^{j(\phi)}=A\cos(\phi)+j\cdot A\sin(\phi)$ is used for mathematical convenience without compromising the accuracy of the analysis and maintains the ability to recover the true harmonic expressions (see addendum) .

Start with a circuit model in the Frequency Domain.
Develop relevant equation in terms of voltage and current phasors, e.g. KVL, KCL, and device voltage‐current relations.
Solve the circuit “simultaneous” algebraic relations for the voltage/current phasors of interest.
Extract the time domain forms of relevant voltages and currents as the real parts of the product of the phasor expression and $exp(j\omega t)$, i.e.,
$f(t)=\text{Real Part of} \lbrace [F\cdot exp(j\phi)]\cdot [exp(j\omega t)] \rbrace$

The Laplace transform allows us to convert time domain expressions (signals as well as system responses) into the Laplace or s-Domain
$F(s)$=$\int_{0}^{\infty}f(t)e^{-st}dt; s=\sigma + j\omega$
Carrying the analysis in the s‐domain enables us to solve time domain transient and steady state problems through the conversion of complicated differential equations to much more manageable algebraic equations.
Letting $\sigma=0$, the $F(s)$ converges to $F(j\omega)$ which is the frequency domain phasor expression for $f(t)$

Phasors “Pack” both the harmonic form of a signal (or system response) $A \cos(\omega t)$ with its complementary quadrature term $A sin(\omega t)$ in a complex expression format: $A \cos(\omega t)+ jA \sin(\omega t)$
The use of the exponent form of the Euler’s formula $\bar{A}[=A\angle(\phi)]=A\cdot e^{j(\phi)}=A\cos(\phi)+j\cdot A\sin(\phi)$ offers the great convenience of dealing with exponential in contrast to potentially complicated trigonometric identities.
The imaginary coefficient $j$ {=sqrt(‐1)}, used for including the quadrature term, allows us to carry out relevant procedures while maintaining the physical harmonic terms required in the analysis. ”$j$” is an expression of a π/2 phase shift (90° CCW rotation.)