Expressions Of Pj Problems

Pj Problems - Overview

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COHN - Natures Engineering Of The Human Body

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Differential Calculus

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High Pass Filter

Figure 7.50 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)Frequency response, *H(jω)* = V_{o}/V_{i}(jω)

By the voltage divider rule:

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

So, H(jω) = V_{o}/V_{i}(jω) = [jωRC/(1 + jωRC)]----(1)

Phasor form:

jωRC = (ωRC)e^{jπ/2}

Phasor form:

1 + jωRC = (1 + (jωRC)^{2})^{1/2}(e^{jarctanωCR}

So, phasor form:

H(jω) = [ωRC/(1 + (jωRC)^{2}))^{1/2}](e^{j[π/2-arctan(ωCR)]}----(2)

where |H(jω)| = ωRC/(1 + (jωRC)^{2})^{1/2}

And
the phase angle:

<H(jω) = π/2-arctan(ωCR)

Equations (1) and (2) reveal 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 approaches infinity, the magnitude of the frequency response asymptotically approach 1.

(b) Cutoff frequency, ω_{0} = 1/RC.

When ω >> 1/RC, filter is called a *high pass filter*. A *high-pass filter* passes signals at high frequencies and filters out signals at low frequencies (ω << 1/RC).

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

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

In essence the *filter effect* results in the *scaling* and *phase angle shifting* of the input signal. In mathematical terms, for the following input signal in phasor form:

V_{i} = |V_{i}|e^{jφi}

The scaling of the output signal becomes:

V_{o} = |H|(V_{i}).

And the phase angle of the output signal becomes:

φ_{o} = <H + φ_{i}.

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

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