Chapter 5 Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances. 3. Compute power

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Chapter 5 Steady-State Sinusoidal Analysis1.
Identify the frequency, angular frequency, peak
value, rms value, and phase of a
sinusoidal signal.2. Solve steady-state ac
circuits using phasors and complex
impedances.3. Compute power for steady-state ac
4. Find Thévenin and Norton equivalent
circuits. 5. Determine load impedances for
maximum power transfer.
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5. Steady-State Sinusoidal Analysis

• In most circuits, the transient response (i.e.,
  the complimentary solution) decays rapidly to
  zero, the steady-state response (i.e., the forced
  response or the particular solution) persists.
• In this chapter, we learn efficient methods for
  finding the steady-state responses for sinusoidal
  sources.

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5. Steady-State Sinusoidal Analysis

• 5.1 Sinusoidal Currents and Voltages
• 5.1.1 Phase and Phase Angle
• Consider the sinusoidal voltage as shown,

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5. Steady-State Sinusoidal Analysis 5.1
Sinusoidal Current and Voltage

• 5.1.1 Phase and Phase Angle
• We usually use cosine function to model a
  sinusoidal signal.
• In case there is a sine function, we can use
  the following conversion

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5. Steady-State Sinusoidal Analysis 5.1
Sinusoidal Current and Voltage

• 5.1.2 Root-Mean-Square (RMS) Values (or Effective
  Values)

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5. Steady-State Sinusoidal Analysis 5.1
Sinusoidal Current and Voltage

• 5.1.3 RMS Value of a Sinusoid

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5. Steady-State Sinusoidal Analysis 5.1
Sinusoidal Current and Voltage

• Example 5.1 Power delivered to a resistance by
  a sinusoidal source

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5. Steady-State Sinusoidal Analysis

• 5.2 Phasors
• 5.2.1 Definition
• A phasor is a vector in complex number plane
  that represents the magnitude and phase of a
  sinusoid.
• In ac circuit analysis, voltages and currents
  are usually represented as phasors.

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.1 Definition
• Eulers Identity
• In phasor application

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.4 Phasor vs. Sinusoids
• The phasor is simply a snapshot of a
  rotating vector at t0.

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.2 Phasor Summation

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.2 Phasor Summation
• Now we use Phasor notation to simplify our
  calculation
• Note In using phasors to add sinusoids, all of
  the terms must have the same frequency.

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• Example 5.2 Using phasors to Add Sinusoids

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.3 Fundamental Phasor Operations

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5. Steady-State Sinusoidal Analysis 5.2 Phasors

• 5.2.5 Phase Relationships

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• 5.3.1 Impedance
• Impedance means complex resistance.
• The impedance concept is equivalent to stating
  that capacitors and inductors act as
  frequency-dependent resistors.
• By using impedances, we can solve sinusoidal
  steady-state circuit with relatively ease
  compared to the methods of Chapter 4.
• Except for the fact that we use complex
  arithmetic, sinusoidal steady-state analysis is
  the same as the analysis of resistive circuits.

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• 5.3.2 Inductance

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• 5.3.3 Capacitance

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• ELI the ICE man.
• Impedance that are pure imaginary are called
  reactance.
• 5.3.4 Resistance

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances
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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• Quiz - Exercises 5.6, 5.7, 5.8

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• Quiz - Exercises 5.6, 5.7, 5.8

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• Additional Example represent the circuit shown
  in the frequency domain using impedances and
  phasors.

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5. Steady-State Sinusoidal Analysis 5.3 Complex
Impedances

• Additional Example represent the circuit shown
  in the frequency domain using impedances and
  phasors.

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• 5.4 Circuit Analysis
• Impedance circuit analysis is the same as
  resistive circuit analysis, we can directly apply
  KCL, KVL, nodal analysis, mesh analysis,
• The above phasor approach can only apply for
  steady state with sinusoids of the same
  frequency.

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Example 5.3 Steady-State AC Analysis of a
  Series Circuit
• Find the steady-state current, the phasor
  voltage across each element, and construct a
  phasor diagram.

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• Example 5.4 Series/Parallel Combination of
  Complex Impedances
• Find the voltage across the capacitor, the
  phasor current through each element, and
  construct a phasor diagram

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Example 5.5 Steady-State AC Node-Voltage
  Analysis
• Find the voltage at node 1 using nodal analysis

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Exercise 5.11 Steady-State AC Mesh-Current
  Analysis
• Solve for the mesh currents

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example Commercial Airliner Door
  Bridge Circuit

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.4 Circuit
Analysis

• Quiz Exercise 5.10
• Find the phasor voltage and phasor current at
  each element

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5 Power in AC Circuit
• 5.5.1 Voltage, Current and Impedance

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.2 Voltage, Current and Power for a Resistive
  Load
• (1) Current is in phase with voltage.
• (2) Energy flows continuously from
• source to load.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.3 Voltage, Current and Power for an Inductive
  Load
• (1) Current lags the voltage by 90 degree (ELI)
• (2) Half of the power is positive, energy is
  delivered to the inductance and stored in the
  magnetic field the other half of the power is
  negative, the inductance returns energy to the
  source.
• (3) The average power is zero, and we say
  reactive power flows back and forth in-between
  the source and the load.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.4 Voltage, Current and Power for a Capacitive
  Load
• (1) Current leads the voltage by 90 degree (ICE)
• (2) The average power is zero reactive power
  flows back and forth in-between the source and
  the load.
• (3) Reactive power is negative (positive) for a
  capacitance (inductance).
• (4) Reactive power in inductance and in
  capacitance cancel each other.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.5 Power Calculation for a General (RLC) Load
• Active (Real) load due to R
• Reactive load due to L, C

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.5 Power Calculation for a General (RLC) Load

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.5 Power Calculation for a General (RLC) Load
• Average Power
• Power Factor
• Power factor is often stated as percentage, e.g.,
• 90 lagging (i.e., current lags voltage,
  inductive load)
• 60 leading (i.e., current leads voltage,
  capacitive load)
• Reactive Power
• The last term in power formula is the power
  flowing back and forth between the source and the
  energy-storage elements. Reactive power is its
  peak power.
• Apparent Power
• Note 5kW load is different from 5kVA load.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.5 Power Calculation for a General (RLC) Load

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.6 Impedance triangle and Power Triangle
• The impedance triangle
• The Power triangle
• Apparent power, average (real) power, and
  reactive power form a triangle.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.6 Impedance triangle and Power Triangle

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.6 AC Power Calculation

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.6 AC Power Calculation

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.7 Using Power Triangles
• Find power, reactive power, and power factor for
  the source and the phasor currents as shown.
• We first find the power and reactive power for
  each load, then sum over to obtain the power and
  reactive power for the source.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.7 Using Power Triangles

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.7 Using Power Triangles

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• 5.5.7 Power-Factor Correction

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Example 5.8 Power-Factor Correction
• A 50kW load operates from a 60-Hz 10kV-rms line
  with a power factor of 60 lagging. Compute the
  capacitance that must be placed in parallel with
  the load to achieve a 90 lagging power factor.

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5. Steady-State Sinusoidal Analysis 5.5 Power
in AC Circuit

• Quiz - Exercise 5.12 Power in AC Circuits
• .

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• 5.6 Thevenin and Norton Equivalent Circuits
• A two terminal circuit composed of sinusoidal
  sources (of the same frequency), resistances,
  capacitances, and inductances can be simplified
  to Thevenin or Norton equivalent circuit.
• 5.6.1 Thevenin Equivalent Circuits
• The Thevenin impedance can also be
• obtained by zeroing sources.
• 5.6.2 Norton Equivalent Circuits

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Example 5.9 Thevenin and Norton Equivalents

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Additional Example

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton
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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• 5.6.1 Maximum Power Transfer

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Example 5.10 Maximum Power Transfer

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Quiz Exercise 5.14 and Exercise 5.15

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Quiz Exercise 5.14

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Quiz Exercise 5.15

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5. Steady-State Sinusoidal Analysis 5.6
Thevenin and Norton

• Quiz Exercise 5.15

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5. Steady-State Sinusoidal Analysis SUMMARY
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5. Steady-State Sinusoidal Analysis SUMMARY
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5. Steady-State Sinusoidal Analysis SUMMARY
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Chapter 6Frequency Response, Bode Plots,and
Resonance1. State the fundamental concepts of
Fourier analysis.2. Determine the output of a
filter for a given input consisting of
sinusoidal components using the filters
transfer function.3. Use circuit analysis to
determine the transfer functions of simple
circuits.4. Draw first-order lowpass or highpass
filter circuits and sketch their transfer
functions.5. Understand decibels, logarithmic
frequency scales, and Bode plots.6.
Calculate parameters for series and parallel
resonant circuits.
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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• 6.1 Fourier Analysis, Filters, and Transfer
  Functions
• 6.1.1 Fourier Analysis
• Most real-world information-bearing electrical
  signals are not sinusoidal.
• Fourier theorem tells that a non-sinusoidal
  signal can be expressed by the summation of
  sinusoidal functions in the form.

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• 6.1.1 Fourier Analysis
• All real-world signals are sums of sinusoidal
  components having various frequencies,
  amplitudes, and phases.
• The square wave is a special example
• Most of the real-world signals are
• confined to finite range of frequency.
• It is important to learn how the circuits
• respond to components having
• different frequencies.

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• 6.1.2 Filters
• Filters process the sinusoidal components of an
  input signal differently depending of the
  frequency of each component.
• Often, the goal of the filter is to retain
  the components in certain frequency ranges and
  reject components in other frequency ranges.

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• 6.1.3 Filters and Transfer Functions
• Since the impedances of inductances and
  capacitances change with frequency, RLC circuits
  provide one way to realize electrical filters.
• The transfer function of a two-port filter is
  defined as

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• Example 6.1 Using Transfer Function to Find
  Output
• For the transfer functions shown, find the output
  signal,
• given the input

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• Example 6.2 Multi-input components,
  Superposition Principle
• The input involves two components

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6. Frequency Response 6.1 Fourier Analysis,
Filters, Transfer Functions

• Example 6.2 Multi-input components,
  Superposition Principle

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• Ideal Filters

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• 6.2 First Order Low-Pass Filters
• A low-pass filter is designed to pass
  low-frequency components and reject
  high-frequency components. In other words, for
  low frequencies, the output magnitude is nearly
  the same as the input while for high
  frequencies, the output magnitude is much less
  than the input.
• 6.2.1 Transfer Function

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• 6.2.2 Magnitude and Phase Plots of the Transfer
  Function

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• Example 6.3 Calculation of RC Low-pass Output

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• Example 6.3 Calculation of RC Low-pass Output

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6. Frequency Response 6.2 First-Order Low-Pass
Filters

• Quiz Exercise 6.4 Another First-Order
  Low-Pass Filter
• This is also a low-pass filter

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6. Frequency Response 6.3 Decibels and the
Cascade Connection

• 6.2 Decibels and the Cascade Connections
• 6.3.1 Decibels
• We usually express the ratio of voltage (or
  power) amplitude in decibels.

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6. Frequency Response 6.3 Decibels and the
Cascade Connection

• 6.3.2 Cascade two-Port Networks

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6. Frequency Response 6.4 Bode Plots

• 6.4 Bode Plots

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6. Frequency Response 6.4 Bode Plots

• 6.4 Bode Plots

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6. Frequency Response 6.5 First-Order High-Pass
Filters

• 6.5 First-Order High-Pass Filters
• 6.5.1 Transfer Function

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6. Frequency Response 6.5 First-Order High-Pass
Filters

• 6.5.2 Bode Plots

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6. Frequency Response 6.5 First-Order High-Pass
Filters

• Exercise 6.13 Another First-Order High-Pass
  Filter

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6. Frequency Response 6.5 First-Order Filters

• First-Order Low-Pass Filters
• First-Order High-Pass Filters

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6. Frequency Response 6.6 Series Resonances

• 6.6.1 Resonant Circuits (Second Order)
• The resonance circuits forms the basis of
  second-order filters that have better performance
  than the first-order filters.
• When a sinusoidal source of the proper
  frequency is applied to a resonant circuit,
  voltage much larger then the source can appear.

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6. Frequency Response 6.6 Series Resonances

• 6.6.1 Resonant Circuits (Second Order)

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6. Frequency Response 6.6 Series Resonances

• 6.6.1 Resonant Circuits (Second Order)

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6. Frequency Response 6.6 Series Resonances

• 6.6.2 Series Resonant Circuits as Band-Pass Filter

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6. Frequency Response 6.6 Series Resonances

• 6.6.2 Series Resonant Circuits as Band-Pass Filter

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6. Frequency Response 6.6 Series Resonances

• Example 6.5 Series Resonant Circuit

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6. Frequency Response 6.6 Series Resonances

• Example 6.5 Series Resonant Circuit

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6. Frequency Response 6.7 Parallel Resonances

• 6.7 Parallel Resonance

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8 Ideal and Second-Order Filters
• 6.8.1 Ideal Filters

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8.1 Ideal Filters

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8.2 Second-Order Low-Pass Filter

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8.2 Second-Order High-Pass Filter

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8.2 Second-Order Band-Pass Filter

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• 6.8.2 Second-Order Band-Reject (Notch) Filter

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• Example 6.7 Filter Design
• Design a second-order filter with L50mH that
  passes components higher in frequency than 1kHz,
  rejects components lower than 1kHz.
• We need a high-pass filter.
• To obtain a approximately
• constant transfer function
• In the pass-band, we choose

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• The Popular Sallen-Key Filters

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• Higher-order Filters using Cascade of 2nd-order
  Filters

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6. Frequency Response 6.8 Ideal and
Second-Order Filters

• Higher-order Filters using Cascade of 2nd-order
  Filters