enee204Lectures_07_08_Gomez

enee204Lectures_07_08_Gomez - Welcome to ENEE 204 Basic...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
1 9/28/2005 1 Welcome to ENEE 204 Basic Circuit Theory Lecture 7 Mid 1: Oct. 6, 2005 (Thursday) Chapter 3: Solving steady state AC circuits using phasors 9/28/2005 2 Recap: ‘Phasors’ are complex numbers used to represent Harmonic AC signals v(t) = V m cos ( ω t + φ v ) Typical AC signal v(t) = Re {V m e j ( t + v ) } } e { j Re t ω ˆ V = v(t) = Re {V m e j v e j t } Phasor is the complex signal at t=0 : Phasor ˆ V V m e j v The ‘phasor’ is a complex the contains the value of the amplitude and the phase .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 9/28/2005 3 Example phasor real signal V = 2 + 2 j v(t) = 2.83 cos ( ω t + π /4) I = 5 e -j π /6 i (t) = 5cos ( ω t - π /6) 83 . 2 8 2 2 2 2 = = + (convert first to polar) 9/28/2005 4 Converting from a phasor to a real signal B j A V ˆ + = ) t cos( v(t)= V m φ ω+ e V ˆ j m V V m 2 = A 2 +B 2 B/A tan = SKILL : Converting from one form to the next should be natural.
Background image of page 2
3 9/28/2005 5 Z i (t) + _ v(t) constant v ) ( ) ( t t i and m m I V However, : Impedance = complex constant (in general) Impedance for resistors, capacitors and inductors in phasor form constant + = I V φ I V j j e e Z m m I V I ˆ V ˆ = generalized ‘resistance’ for AC 9/28/2005 6 Impedance of a ideal resistor is real [ ] [] } Re{ ) ( ) ( : Relation Terminal } Re{ ) ( t j i m t j i m e R e I R t i t v e e I t i I I ω = = = R I V Z R = = ˆ ˆ i (t) + _ v(t) R Z e I R e I I V R i m i m I I = = = that, So ˆ ˆ Z Now, R
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4 9/28/2005 7 Impedance of a capacitor is negative imaginary C j C j I V Z C ω = = 1 ˆ ˆ [ ] [] } Re{ } Re{ ) ( ) ( : Relation Terminal } Re{ ) ( t j i m t j i m t j i m e e CV j dt e e V d C dt t dv C t i e e V t v V V V φ = = = = i(t) + _ v(t) C j C j Z j e CV e V I V C i m i m C V V = = = = 1 that, So } { ˆ ˆ Z Now, 9/28/2005 8 Impedance of an inductor is positive imaginary L j I V Z L = ˆ ˆ [ ] } Re{ } Re{ ) ( ) ( : Relation Terminal } Re{ ) ( t j i m t j i m t j i m e e LI j dt e e I d L dt t di L t v e e I t i I I I = = = = i(t) + _ v(t) L j Z e I j e LI I V L i m i m L I I = = = that, So ˆ ˆ Z Now,
Background image of page 4
5 9/28/2005 9 Steady-state AC circuits can be solved by using Kirchhoff’s laws and impedance in Phasor Form i 1 (t) i 2 (t) i 3 (t) i 4 (t) i 5 (t) = k k 0 ˆ I = k k 0 ˆ V I v s (t) v 1 (t) v 2 (t) v 6 (t) z 1 z 2 z 3 z 5 I 1 ^I 2 ^ I 3 ^ I 5 ^ I 4 ^ ac V 1 ^ I + Z 1 Z 2 Z 6 V s ^ V 2 ^ V 6 ^ L j Z C j 1 Z R Z ω = = = L C R 9/28/2005 10 Impedance contains Re and Im components R R G 1 2 2 X R G + = Z = R + j X Impedance Resistance Reactance Admittance Conductance Susceptance jB G Z Y + = = 1 Admittance is reciprocal of impedance Note with care: 2 2 1 X R X B X B + =
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6 9/28/2005 11 Numerical example SAY: v s (t) = 170 cos (2 π 10 4 t) v s (t) R = 10 C = 1 µ F Find: 1. Z 2. Resistance 3. Reactance 4. Z in polar form 5. Phasor V 6. Phasor I 7. i(t) 9/28/2005 12 1. Calculating the impedance C j R Z ω 1 + = R = 10 C = 1 µ F Z Since: v s (t) = 170 cos (2 10 4 t) Then: ω = 2 10 4 = 6.28 x 10 4 () ( ) + = 6 4 10 10 28 . 6 1 10 j Z + = + = j j j j Z 92 . 15 10 92 . 15 10 = 92 . 15 10 j Z Resistance Reactance o j e 2 / C 92 . 15 Z capacitor of Impedance =
Background image of page 6
7 9/28/2005 13 2. Expressing Z in polar form.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 39

enee204Lectures_07_08_Gomez - Welcome to ENEE 204 Basic...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online