Chapter 14 – Basic Elements and Phasors Lecture 15 by Moeen Ghiyas 09/08/2015 1

Chapter 14 – Basic Elements and Phasors

Introduction (Basic Elements & Phasors) – Ch 14 The Derivative Response of Basic R, L, And C Elements to a Sinusoidal Voltage or Current

In this chapter, we will study How frequency will affect the “opposing” characteristic of each element. Phasor notation will then be introduced to establish a method of analysis of ac circuits that permits a direct correspondence with a number of the methods and concepts introduced in the dc chapters. 09/08/2015 4

We know that the derivative dx/dt is defined as the rate of change of x with respect to time If x fails to change at a particular instant, dx = 0, and the derivative is zero The derivative dx/dt is actually the slope of the graph at any instant of time

At 0 and 2π, x increases at its greatest rate, and the derivative is given a positive sign since x increases with time At π, dx/dt decreases at the same rate as it increases at 0 and 2π, but the derivative is given a negative

A plot of the derivative shows that the derivative of a sine wave is a cosine wave

The derivative of a sine wave has the same period T and frequency f as the original sinusoidal waveform but peak value of derivative (resulting cosine wave) is directly proportional to frequency (of original sine wave)

For the sinusoidal voltage e(t) = E m sin(ωt ± θ), the derivative is Note that the peak value of the derivative, 2πfEm, is a function of the frequency of e(t), and the derivative of a sine wave is a cosine wave

Response of Resistor For power-line frequencies and frequencies up to a few hundred kilohertz, resistance is, for all practical purposes, unaffected by the frequency of the applied sinusoidal voltage or current. For this frequency region, the resistor R can be treated as a constant, and Ohm’s law can be applied.where

for a given i, Where A plot of v and i reveals that for a purely resistive element, the voltage across and the current through the element are in phase, with their peak values related by Ohm’s law

Response of Inductor – v L = V m sin(ωt ± θ) we are interested in determining the opposition of the inductor as related to the resistance of a resistor The inductive voltage, is directly related to the frequency (thus ω of sinusoidal ac current through the coil) and the inductance L of the coil. For increasing values of f and L the (opposition and thus) magnitude of v L will increase.

Above equation reveals that the larger the inductance of a coil (with N fixed), the larger will be the instantaneous change in flux Φ linking the coil due to an instantaneous change in current I through the coil. We can write Faraday’s Law as Using both equations we have Therefore, greater the inductance L or the rate of change of current through coil, the greater will be the induced voltage. 09/08/

Since polarity of voltage across an inductor will always oppose the source that produced it, the notation v L is used: Thus, instead of If the current through the coil fails to change at a particular instant, the induced voltage across coil will be zero. For dc applications, after the transient effect has passed, di/dt = 0, and the induced voltage is 09/08/

For the inductor, now we know Applying differentiation on rate of change of current Therefore orwhere Thus for an inductor, v L leads i L by 90°, or i L lags v L by 90° and that peak value of v L is directly related to ω ( =2πf ) and L 09/08/

For an inductor, v L leads i L by 90°, or i L lags v L by 90°. If a phase angle is included in the sinusoidal expression for i L, such as i L = I m sin(ωt ± θ) then v L = V m sin(ωt ± θ + 90°) =

For resistor, we know thatV m = I m R Drawing analogy for inductorV m = I m X L Where X L is called Inductive reactance, opposition to the flow of current, which results in the continual interchange of energy between the source and the magnetic field of the inductor, unlike resistance (which dissipates energy in the form of heat)

Remember cause and effect relationship Substituting values The quantity ωL, called the reactance (from the word reaction) of an inductor, is symbolically represented by X L and is measured in ohms; that is,

Our investigation so far Resistor opposes flow of current Inductor opposes the instantaneous change in current through the coil. Now for response of capacitor, the voltage across the capacitor is limited by the rate at which charge can be deposited on, or released by, the plates of the capacitor during the charging and discharging phases, respectively. In other words, an instantaneous change in voltage across a capacitor is opposed by the fact that there is an element of time required to deposit charge on (or release charge from) the plates of a capacitor, and V = Q/C and I = Q/t or I = Cdv/dt. 09/08/

We already know Since the current i = Q / t, Thus current i C is associated with the capacitance and derivative of voltage across the capacitor wrt time i.e. where dv C /dt is a measure of the change in v C in a vanishingly small period of time. If the voltage fails to change at a particular instant, then and in turn 09/08/

Response of Capacitor – we are interested in determining the opposition of the capacitor as related to the resistance of a resistor and ωL for the inductor. Since an increase in current corresponds to a decrease in opposition, and i C is proportional to ω and C, the opposition of a capacitor is inversely related to ω ( =2πf ) and C.

For the capacitor, now we know Applying differentiation on rate of change of voltage Therefore orwhere Thus for a capacitor, i L leads v L by 90°, or v L lags i L by 90° and that peak value of i C is directly related to ω ( =2πf ) and C 09/08/

For a capacitor, i C leads v C by 90°, or v C lags i C by 90°. If a phase angle is included in the sinusoidal expression for v C, such as then i C = I m sin(ωt ± θ + 90°) =

For resistor, we know thatV m = I m R Drawing analogy for capacitorV m = I m X C Where X C is the capacitive reactance, opposition to the flow of charge, which results in the continual interchange of energy between the source and the electric field of the capacitor. Like the inductor, the capacitor does not dissipate energy in any form (ignoring the effects of the leakage) resistance).

Remember cause and effect relationship Substituting values The quantity 1/qC, called the reactance of a capacitor, is symbolically represented by X C and is measured in ohms; that is,

In the inductive circuit, or mathematically In the capacitive circuit, or mathematically 09/08/

Since we now have an equation for the reactance of an inductor or capacitor, we do not need to use derivatives or integration generally. Simply applying Ohm’s law, I m = E m /X L (or X C ), and keeping in mind the phase relationship between the voltage and current for each element, will be sufficient to analyse general problems. 09/08/

Further, it is possible to determine whether a network with one or more elements is predominantly capacitive or inductive by noting the phase relationship between the input voltage and current. If the source current leads the applied voltage, the network is predominantly capacitive, and if the applied voltage leads the source current, it is predominantly inductive. 09/08/

Introduction (Basic Elements & Phasors) – Ch 14 The Derivative Response of Basic R, L, And C Elements to a Sinusoidal Voltage or Current

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