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ECE 210 Review Session10/15/16
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OverviewPart 1
● Steady/Transient State, Zero-State/Zero-Input
● Phasor Circuit Analysis
● Available power
● Frequency Response, Resonance
● Periodic Signals
● Fourier Series
Part 2
● Past Exam Problems
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Steady State & Transient ResponsesSystem response - sum of the steady state response and the transient response: y(t) = yss(t) + ytr(t)
Steady State Response - yss(t)
● The component of the system response that is time invariant
● Constants: e.g. 5, j
● Sinusoids: e.g. cos(ωt), sin(ωt)
● yss(t) = lim (t→inf) y(t)
Transient Response - ytr(t)
● The component of the system that decays to 0 as time proceeds
● Exponentials: e.g. ce-at
● ytr(t) = y(t) - yss(t)
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Zero-State & Zero-Input ResponsesSystem response - sum of the zero-state response and the zero-input response: y(t) = yzs(t) + yzi(t)
Given the ODE describing the system:
Zero-State Response - yzs(t)
● Solution to the ODE given the condition: f(t) = f(t), y(t0) = 0
Zero-Input Response - yzi(t)
● Solution to the ODE given the condition: f(t) = 0, y(t0) = k1
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Phasor Circuit AnalysisPhasor - Vector spinning at an arbitrary angular velocity (w). Usually written in polar form: |F|ej∠F
Conversion of a sinusoid to phasor form requires finding its magnitude and phase, ex: f(t) = 2cos(2t + pi/3); magnitude = 2, phase = pi/3 rad -> F = 2ej∠pi/3
Impedance (Z) - Ratio of phasor Voltage to phasor Current passing through a circuit component. Z = V/I (extension of Ohm’s Law to complex domain)
Impedance of electrical components
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From Chapter 2:
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Resonant Frequency● Possible existence of steady-state co-sinusoidal oscillations in a source-free
circuit● In series resonance, equivalent impedance of L and C is effective short circuit● In parallel resonance, equivalent impedance of L and C is effective open
circuit
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Periodic Signals
● A signal is said to be periodic if there exists some delay t0 such that:
f(t-t0) = f(t)
● The period of the signal (T): the smallest non-zero value of t0.
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Harmonically related frequencies
● All frequencies are positive integer multiples of the frequency of the original wave, known as the fundamental frequency (also called 1st harmonic).
● Note: Fourier series frequency components are harmonically related!
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Fourier Series
A way to represent a periodic function as the sum of a set of simple sinusoid or complex exponential waves.
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Fourier Series (Cont’d)
● Absolutely Integrable: ∫T |f(t)|dt < ∞
if the above inequality is satisfied, then the Fourier
Coefficients Fn must be bounded.
● Orthogonality
∫ (e^jnwt)(e^jmwt)*dt = 0 (m ≠ n)
This condition is crucial to Fourier Series representation!
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(Spring 2016 Exam 2 Problem 5)
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Questions?
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Acknowledgement● A large portion of this slide is taken from HKN ECE210 Review Session slide
from Spring 2016.● Special thanks to Kaidong Peng, Julian Michaels, Seungjun Cho