LectR2EEE 2021 Exam #2 Review Dr. Holbert March 26, 2008

LectR2EEE 2022 Don’t Forget the Essentials Verify voltage polarity and current direction Obey the passive sign convention The Fundamentals: Ohm’s Law; KCL; KVL Series/Parallel Impedance combinations

LectR2EEE 2023 Circuit Analysis Techniques All these circuit analysis techniques have wide applicability: DC, AC, and Transient Voltage and Current Division Nodal and Loop/Mesh Analyses Source Transformation Superposition Thevenin’s and Norton’s Theorems

LectR2EEE 2024 Transient Circuit Analysis First and second order circuit responses Differential equation approach Laplace transform approach Inspection (step-by-step) method Bottom line: Using appropriate techniques can you find v(t) and/or i(t) in transient RLC circuits?

LectR2EEE 2025 RLC Characteristics ElementV/I RelationDC Steady-State ResistorV = I R CapacitorI = 0; open circuit InductorV = 0; short circuit ELI and the ICE man

LectR2EEE 2026 Circuit ODE Solutions Determine the circuit differential equation(s) Find the forced (particular) and natural (complementary) solutions First-order vs. second-order circuits –First-order: find time constant (  =RC;  =L/R) –Second-order: Compute the natural frequency,  0, and the damping ratio,  (or compute the roots, s 1,2, of the characteristic equation) Transient and steady-state waveforms

LectR2EEE 2027 Damping Summary Damping Ratio Poles (s 1, s 2 )Damping ζ > 1Real and unequalOverdamped ζ = 1Real and equalCritically damped 0 < ζ < 1Complex conjugate pair set Underdamped ζ = 0Purely imaginary pairUndamped

LectR2EEE 2028 Laplacian Domain Determining the Laplace transform from –The defining integral –Transform pairs in conjunction with properties u(t) ↔ 1/se -at ↔ 1/(s+a) –Circuit element representations in s domain Finding the transfer function Performing the inverse Laplace transform to find the time-domain response –Three possible cases based on poles

LectR2EEE 2029 Laplacian of Circuit Elements Using Ohm’s Law, impedance (Z) can be defined via: V = I Z Circuit Element Impedance (ohms) ResistorR Capacitor1 / (sC) InductorsL 1/sC + – VC(s)VC(s) +–+– v(0) s IC(s)IC(s) i(0) s + – VL(s)VL(s) sLsL IL(s)IL(s)

LectR2EEE Transfer Function The transfer function, H(s), is the ratio of some output variable (y) to some input variable (x) The transfer function is shown in block diagram form as (where h(t) is the impulse response) H(s) ↔ h(t) X(s) ↔ x(t)Y(s) ↔ y(t) InputOutput System

LectR2EEE Some Terminology & Quantities Our vocabulary has expanded with several new terms, including: Phasor & impedance Impulse (delta) and step functions Transfer function Impulse response Poles and zeros Initial and final value theorems Linearity and time invariance Convolution integral Period, frequency, and amplitude Characteristic equation Over/under damped