Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Low Pass Filter

Figure 7.49 shows a circuit of a simple RC filter. Determine:

(a) The phasor form of the frequency response, *H(jω)* in terms of ω, R and C.

(b) The cutoff frequency of the RC filter.

**The string**:

S_{7}P_{5}A_{51} (Physical change).
**The math**:

Pj Problem of interest is of type *change*. Frequency problems are *change problems*. They are similar to velocity, acceleration and duration problems which are also *change problems*.

In general, *frequency response* is a measure of the variation in a load-related parameter as a function of the frequency of the excitation element. In electric circuits, the *load-related* parameter is usually the voltage across a load or the current through it and the excitation element is usually a sinusoidal signal. Consequently, any of the following is an acceptable definition of the frequency response of a circuit:

H_{V}(jω) = V_{L}(jω)/V_{s}(jω)

Where H_{V}(jω) is frequency response of load; V_{L}(jω) is voltage across load; V_{s}(jω) is frequency dependent voltage source.

H_{I}(jω) = I_{L}(jω)/I_{s}(jω)

Where H_{I}(jω) is frequency response of load; I_{L}(jω) is current through load; I_{s}(jω) is frequency dependent current source.

(a) H(jω) = V_{o}(jω)/V_{i}(jω)

By the voltage divider rule,:

V_{o}(jω) = V_{i}(jω)[(1/jωC)/(R + 1/jωC)] = V_{i}(jω)[1/(1 + jωRC)].

So, H(jω) = [1/(1 + jωRC)]----(1)

In phasor form:

1 = 1e^{j0}

And

1 + jωRC = [1 + (jωRC)^{2})]^{1/2}(e^{jarctan(ωCR})

So, in Phasor form:

H(jω) = [1/(1 + (jωRC)^{2})]^{1/2}(e^{-jarctan(ωCR})----(2)

Where |H(jω)| = [1/(1 + (jωRC)^{2})]^{1/2}

And the phase angle is:

<H(jω) = -arctan(ωCR).

Equations (1) and (2) reveals that at ω = 0,

V_{o}(jω = 0) = V_{i}(jω = 0).

That is, no filtering at ω = 0 (DC signal).

As the signal frequency increases, the magnitude of the frequency response decreases.

(b) *Cutoff frequency, ω _{0}* = 1/RC.

At low frequencies (When ω << 1/RC), filter is called a

The value of |H(jω)| at the cufoff frequency, is 1/√2 = .707.

The cufoff frequency depends only on the values of R and C. As a result filtering characteristics can be selected for various values of R and C.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings