CIRCUITS and SYSTEMS – part I Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 3 Power in electric circuits

Instantaneous power p(t) At sinusoidal signals: u(t) = U m sin(ωt), i(t) =I m sin(ωt-φ) Definition

For sinusidal signals - power coefficient Unit of active power: wat [W] Active (average) power P

Active power (cont.) Resistor (φ=0) Inductor and capacitor Active power on reactive elements (L & C) is always zero.

Reactive power Q Definition Unit of reactive power: war [var] Resistor Inductor Capacitor Reactive power dissipated in resistor is always zero

Apparent complex power S Definition Unit of apparent power: [VA] Phasor diagram for circuit of the a) induktive charakter, b) capacitive charakter a) b)

Apparent power (cont.) Module = apparent power Power coefficient

Principle of conservation of power Power flow from sources (S s ) in a network equals the power flow (S l) into the other elements of the network S s =S l In this principle the directions of currents and voltages are the same for sources and opposite for passive RLC elements. If we assume the unified directions of currents and voltages (currents and voltages directed opposite irrespective of the type of element) the principle of conservation of power may be written in the form: S s +S l =0

Determine the currents and powers of elements in the circuit at sinusoidal excitation e(t)=141 sin(ωt+45 o ), at ω=1. Assume: R=Ω, C=0.5F, L=1H. Example

Solution Complex description of circuit elements: Equivalent impedances:

Solution (cont.) The succeding steps of solution

Phasor diagram circuit

Powers of elements Apparent power of the source Powers of passive elements Balance of powers : –Receivers –Source

Energy stored in capacitor General expression Energy of capacitor At zero initial condition of capacitor

Energy stored in inductor General expression Energy of inductor At zero initial condition of inductor