Lecture 03: AC RESPONSE ( REACTANCE N IMPEDANCE )

OBJECTIVES  Explain the relationship between AC voltage and AC current in a resistor, capacitor and inductor.  Explain why a capacitor causes a phase shift between current and voltage (ICE).  Define capacitive reactance. Explain the relationship between capacitive reactance and frequency.  Explain why an inductor causes a phase shift between the voltage and current (ELI).  Define inductive reactance. Explain the relationship between inductive reactance and frequency.  Explain the effects of extremely high and low frequencies on capacitors and inductors.

AC RESISTOR

AC V AND I IN A RESISTOR  Ohm’s Law still applies even though the voltage source is AC.  The current is equal to the AC voltage across the resistor divided by the resistor value.  Note: There is no phase shift between V and I in a resistor.

v R (t) AC V AND I IN A RESISTOR PHASE ANGLE FOR R,  =0 

AC CAPACITOR

CURRENT THROUGH A CAPACITOR  The faster the voltage changes, the larger the current.

 The phase relationship between “V” and “I” is established by looking at the flow of current through the capacitor vs. the voltage across the capacitor. PHASE RELATIONSHIP

Graph v C (t) and i C (t) 90° v c (t) i c (t) Note: Phase relationship of I and V in a capacitor

 In the Capacitor (C), Voltage LAGS charging current by 90 o or Charging Current (I) LEADS Voltage (E) by 90 o  I. C. E. PHASE RELATIONSHIP

CAPACITIVE REACTANCE  In resistor, the Ohm’s Law is V=IR, where R is the opposition to current.  We will define Capacitive Reactance, X C, as the opposition to current in a capacitor.

CAPACITIVE REACTANCE  X C will have units of Ohms.  Note inverse proportionality to f and C. Magnitude of X C

Ex. Ex: f = 500 Hz, C = 50 µF, X C = ?

 Capacitive reactance also has a phase angle associated with it.  Phasors and ICE are used to find the angle PHASE ANGLE FOR X C

 If is our reference wave:  If V is our reference wave: I.C.E

AC INDUCTOR

 The phase angle for Capacitive Reactance (X C ) will always = -90°  X C may be expressed in POLAR or RECTANGULAR form.  ALWAYS take into account the phase angle between current and voltage when calculating X C or

VOLTAGE ACROSS AN INDUCTOR  Current must be changing in order to create the magnetic field and induce a changing voltage.  The Phase relationship between V L and I L (thus the reactance) is established by looking at the current through vs the voltage across the inductor.

Graph v L (t) and i L (t) Note the phase relationship v L (t ) i L (t) 90°

 In the Inductor (L), Induced Voltage LEADS current by 90 o or Current (I) LAGS Induced Voltage (E) by 90 o.  E. L. I. V C I C 90

INDUCTIVE REACTANCE  We will define Inductive Reactance, X L, as the opposition to current in an inductor.

INDUCTIVE REACTANCE  X L will have units of Ohms (  ).  Note direct proportionality to f and L. Magnitude of X L

Ex1. f = 500 Hz, L = 500 mH, X L = ?

PHASE ANGLE FOR X L  If is our reference wave:  If V is our reference wave: E.L.I

 The phase angle for Inductive Reactance (X L ) will always = +90°  X L may be expressed in POLAR or RECTANGULAR form.  ALWAYS take into account the phase angle between current and voltage when calculating X L or

COMPARISON OF X L & X C  X L is directly proportional to frequency and inductance.  X C is inversely proportional to frequency and capacitance.

SUMMARY OF V-I RELATIONSHIPS ELEMENTTIME DOMAINFREQ DOMAIN

Extreme Frequency effects on Capacitors and Inductors  Using the reactances of an inductor and a capacitor you can show the effects of low and high frequencies on them.

Frequency effects  At low freqs (f=0):  an inductor acts like a short circuit.  a capacitor acts like an open circuit.  At high freqs (f=∞):  an inductor acts like an open circuit.  a capacitor acts like a short circuit.

Ex2.  Represent the below circuit in freq domain;

REVIEW QUIZ -What is the keyword use to remember the relationships between AC voltage and AC current in a capacitor and inductor What is the equation for capacitive reactance? Inductive reactance? -T/F A capacitor at high frequencies acts like a short circuit. -T/F An inductor at low frequencies acts like an open circuit.

IMPEDANCE

 The V-I relations for three passive elements;  The ratio of the phasor voltage to the phasor current:

 From that, we obtain Ohm’s law in phasor form for any type of element as:  Where Z is a frequency dependent quantity known as IMPEDANCE, measured in ohms.

IMPEDANCE  Impedance is a complex quantity: R = Real part of Z = Resistance X = Imaginary part of Z = Reactance

 Impedance in polar form: where;

IMPEDANCES SUMMARY ImpedancePhasor form:Rectangular form ZRZR R+j0 ZLZL 0+jX L ZCZC 0-jX C

ADMITTANCE

 The reciprocal of impedance.  Symbol is Y  Measured in siemens (S)

ADMITTANCE  Admittance is a complex quantity: G = Real part of Y = Conductance B = Imaginary part of Y = Susceptance

Z AND Y OF PASSIVE ELEMENTS ELEMENTIMPEDANCEADMITTANCE

TOTAL IMPEDANCE FOR AC CIRCUITS  To compute total circuit impedance in AC circuits, use the same techniques as in DC. The only difference is that instead of using resistors, you now have to use complex impedance, Z.

TOTAL IMPEDANCE FOR PARALLEL CIRCUIT

 As a conclusion, in parallel circuit, the impedance can be easily computed from the admittance:

Ex3: SERIES CIRCUIT R=20 Ω L = 0.2 mH C = 0.25μF