Phasor Impedances And Complex Currents And Voltages-Applications of Power Electronics-Handout, Exercises of Power Electronics

Download Phasor Impedances And Complex Currents And Voltages-Applications of Power Electronics-Handout and more Exercises Power Electronics in PDF only on Docsity! Important Notes for Basic Electrical Engineering nZZZZZ 1 ........ 1111 321  21 21 ZZ ZZ Z   213132 321 ZZZZZZ ZZZ Z   321421431432 4321 ZZZZZZZZZZZZ ZZZZ Z   n n YYYY ZZZZZ  ........ 1 ........ 1111 321 321 where the real part of impedance is the resistance and the imaginary part is the reactance . Where it is required to add or subtract impedances the cartesian form is more convenient, but when quantities are multiplied or divided the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers. Ohm's law An AC supply applying a voltage , across a load , driving a current . The meaning of electrical impedance can be understood by substituting it into Ohm's law. V=IZ docsity.com Important Notes for Basic Electrical Engineering Complex voltage and current Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box. In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as and . )( VtjeVV    )( ItjeII    Impedance is defined as the ratio of these quantities. I V Z  )]()[( IV ttje I V Z    )( IVje I V Z    je I V Z  )( IV   Substituting these into Ohm's law we have Noting that this must hold for all t, we may equate the magnitudes and phases to obtain The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship. Phasors A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are docsity.com Important Notes for Basic Electrical Engineering ) 2 (     tj CC eII 22 ) 2 ( )( 1      jj C C tj C tj C C C C e C e I V eI eV I V Z    Cj e C Z j C   11 2   Inductor For the inductor, we have the relation: dt tdi Ltv LL )( )(  This time, considering the current signal to be )sin()( tIti PL  it follows that )cos( )( tI dt tdi P L  And thus )sin( ) 2 sin( )sin( )cos( )( )( t tL tI tLI ti tv P P L L        This tells us that the ratio of AC voltage amplitude LIP to AC current amplitude PI across an inductor is: L P P L L ZL I LI I V    Now in polar form: docsity.com Important Notes for Basic Electrical Engineering )( tj LL eVV  ) 2 (     tj LL eII 22 ) 2 ( )(      jj L L tj L tj L L L L Lee I V eI eV I V Z   LjLeZ j L    2 Generalized s-plane Impedance Impedance defined in terms of jω can strictly only be applied to circuits which are energised with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using complex frequency instead of jω. Complex frequency is given the symbol s and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the Laplace transform of the time domain expression of the signal. The impedance of the basic circuit elements in this more general notation is; Element Impedance expression Resistor R Inductor sL Capacitor sC 1 For a DC circuit this simplifies to s = 0. For a steady-state sinusoidal AC signal s = jω. Resistance vs Reactance Resistance and reactance together determine the magnitude and phase of the impedance through the following relations: docsity.com Important Notes for Basic Electrical Engineering 22* XRZZZ  )(tan 1 R X In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant. Resistance Resistance is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current. cosZR  Reactance Reactance is the imaginary part of the impedance; a component with a finite reactance induces a phase shift between the voltage across it and the current through it. sinZX  A purely reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance will not dissipate any power. Capacitive Reactance A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric. At low frequencies a capacitor is open circuit, as no charge flows in the dielectric. A DC voltage applied across a capacitor causes charge to accumulate on one side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero. Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current. CfC X C  2 11  Inductive Reactance Inductive reactance LX is proportional to the signal frequency f and the inductance L . LfLX L  2 An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf  (voltage opposing current) due to a rate-of-change of magnetic flux density B through a current loop. docsity.com