Lecture 6 (III): AC RESPONSE

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.

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

AC CAPACITOR

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

PHASE RELATIONSHIP 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.

Note: Phase relationship of I and V in a capacitor Graph vC(t) and iC(t) Note: Phase relationship of I and V in a capacitor vc(t) 90° ic (t)

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

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

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

Ex. Ex: f = 500 Hz, C = 50 µF, XC = ?

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

PHASE ANGLE FOR XC If V is our reference wave: I.C.E

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

AC INDUCTOR

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

Graph vL(t) and iL(t) vL(t) Note the phase relationship 90° iL(t)

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

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

INDUCTIVE REACTANCE XL will have units of Ohms (W). Note direct proportionality to f and L. Magnitude of XL

Ex. f = 500 Hz, C = 500 mH, XL = ?

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

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

COMPARISON OF XL & XC XL is directly proportional to frequency and inductance. XC is inversely proportional to frequency and capacitance.

SUMMARY OF V-I RELATIONSHIPS ELEMENT TIME DOMAIN FREQ DOMAIN

Extreme Frequency effects on Capacitors and Inductors Using the reactance 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): At high freqs (f=∞): 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.

Ex. Represent the below circuit in freq domain; CTH LITAR RLC

Solution =2 rad/s: Time domain Freq domain R = 2 Ω C = 0.25 F XC = -j(1/C) = -j2 Ω L = 1 H XL = jL = j2 Ω Vs = 5 cos 2t Vs = 5<0

Solution

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. ELI and ICE True False

IMPEDANCE

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 ZR R+j0 ZL 0+jXL ZC 0-jXC Impedance Phasor form: Rectangular form ZR R+j0 ZL 0+jXL ZC 0-jXC

ADMITTANCE

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 IMPEDANCE ADMITTANCE

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.

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

Solution: Convert to Freq Domain

Circuit in Freq domain R=20Ω jL = j20 Ω -j(1/C) = -j40 Ω

(a): Find Total Impedance

(b): Draw Impedance Triangle j q q XC XC ZT ZT - j

(c): Find is, vR, vC, vL RMS value for power calculation

(c): Find is, vR, vC, vL Remember to convert to RMS values when converting from sinusoid to phasors.

(d):Using Voltage Divider Voltage divider still works, too.

TOTAL IMPEDANCE FOR PARALLEL CIRCUIT

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

Ex: Parallel Circuits

(a) Find total impedance The circuit given is already in freq domain:

Circuit simplification

POSITIVE/NEGATIVE QUADRANT +j -,+ +,+ SUGAR ADD 180-j60 -, - +,- TO COFFEE -j

(b) Draw Impedance Triangle j q q XC ZT ZT - j

(c) Find is, iC, iRL

(d) Using Current Divider

Ex: Find i and vc

Elements in Frequency domain =4 rad/s Time domain Freq domain

Circuit in Freq Domain

(a) The current, i The current in series circuit, Total impedance is;

(b) The voltage, VC