Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

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Energy

Light

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Sound

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Language

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Nuclear Reactions

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Energy Conversion

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Human-cells

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Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

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Photosynthesis

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Differential Calculus

Antiderivative

Integral Calculus

Economies

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Money Supply

Painting

Closed Loop Gain Of An Inverting Operational Amplifier

**(a)** What is an *ideal amplifier* and what is an *op-amp*?
**(b)** What is an *open loop op-amp*?
**(c)** What is a *closed loop op-amp*?
**(d)** What is an *ideal op-amp*?
**(e)** Show that the inverting amplifier closed-looped gain equals -R_{F}/R_{s}. Where R_{F} is the feedback impendance and R_{s} is the impedance associated with the voltage source.
**(f)** An inverting amplifier uses two 10% tolerance resistors: R_{F} = 33kΩ and R_{s} = 1.2 kΩ.

(i) What is the nominal gain of the amplifier?

(ii) What is the maximum value of |A_{v}|?

(iii) What is the minimum value of |A_{v}|?

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

Pj Problem of Interest is of type *change* (change - physical).

**(a)** An *amplifier* (figure 8.1.3) is an electronic device use to amplify low level electric signals (e.g. voltage and currents). For example, the sound produced by a loud speaker is the result of amplication. Basically an ideal amplifier recieves a signal v_{s}(t) and outputs a signal:

v_{o}(t)= Av_{s}(t) (also called the load voltage)-------(1)

Where A is called the *gain* of the amplifier.

An *operational amplifier* (op-amp) is an *integrated circuit* (an aggregation of many electronic and electric circuits on a wafer of a semi-conductor such as silicon). The op-amp amplifies the difference between two input signals. In figure 8.1, v_{d} is the difference signal. The relationship between the difference voltage that is input and the output voltage is as follows:

v_{o} = A_{OL}v_{d}-------(2)

Where A_{OL} is called the *open-loop gain*.
**(b)** The op-amp of figure 8.2.1 is open looped. That is, the output signal is not fed back as input.
**(c)** The inverting op-amp (output signal is inverted) of figure 8.2 is closed looped. That is, output signal is fedbacked as input. This is why i_{F} is called the *feed back current*.
**(d)** An op-amp is ideal if it satifies the following characteristics:

- The open-loop voltage gain is negatively infinite.

- The input current i_{in}m = 0 which implies input impedance R_{in} is infinitely large.

- The output impedance R_{0} = 0, which implies the output voltage is independent of the load.

**(e)** Consider figure 8.2. By KCL, i_{1} + i_{F} = i_{in}-------(3)

So. (v_{s} - v_{d})/R_{1} + (v_{o} - v_{d})/R_{F} = v_{d}/R_{d}------(4)

By eq (2), v_{d} = v_{o}/A_{OL}

So, substituting v_{o}/A_{OL} in eq(4) and allowing A_{OL} tend to - infinity, we have:

v_{s}/R_{1} + V_{o}/R_{F} = 0

So, inverting amplifier closed loop gain, A_{v} = v_{o}/v_{s} = - R_{F}/R_{1}.

This derivation uses the first characteristics of the ideal op-amp: that open loop gain A_{OL} tends to - infinity.

If we use the characteristic that i_{in} = 0, we gat the same result thus:

i_{in} = 0, implies i_{1} = i_{F}

So, v_{s}/R_{1} + v_{o}/R_{F} = 0

So, A_{v} = v_{o}/v_{s} = -R_{F}/R_{1}.

**(fi)** Nominal gain = A_{v} = -R_{F}/R_{1} = - -(33/1.2) = -27.5
**(fii)** Max value of |A_{v}| = (33 + 10% of 33)/(1.2 - 10% of 1.2) = 36.3
**(fiii)** Min value of |A_{v}| = (33 - 10% of 33)/(1.2 + 10% of 1.2) = 22.5

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.

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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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