objective of lecture describe the mathematical relationships between ac voltage and ac current for a...

Objective of LectureDescribe the mathematical relationships

between ac voltage and ac current for a resistor, capacitor, and inductor.Discuss the phase relationship between the ac

voltage and current.Chapter 9.4 Fundamentals of Electric Circuits

Explain how Ohm’s Law can be adapted for inductors and capacitors when an ac signal is applied to these components.Derive the mathematical formulas for the

impedance and admittance of a resistor, inductor, and capacitor.

Chapter 9.5 Fundamentals of Electric Circuits

ResistorsOhm’s Law

if i(t) = Im cos(t + )

then v(t) = Ri(t) = R Im cos(t + )

V = RIm RI where

The voltage and current through a resistor are in phase as there is no change in the phase angle between them.

Capacitorsi(t) = C dv(t)/dt where v(t) = Vm cos(t)

i(t) = -C Vm sin(t)

i(t) = CVm sin(t )

i(t) = CVm cos(t - )

i(t) = CVm cos(t )

CapacitorsV = Vm

I = CVm cos(t + )

Vm cos(t ) = V ej= V = jV

I = jCV = CV

or

V = (1/jC) I = - (j/C) I = (1/C) I -

Capacitors90o phase difference

between the voltage and current through a capacitor.Current needs to flow first

to place charge on the electrodes of a capacitor, which then induce a voltage across the capacitor

Current leads the voltage (or the voltage lags the current) in a capacitor.

Inductorsv(t) = L d i(t)/dt where i(t) = Im cos(t)

v(t) = - L Im sin(t) = LIm cos(t )

V = LIm

I = Im cos(t)

Im cos(t ) = I ej= I = jI

V = jLI = LI

or

I = (1/jL) V = - (j/L) V = (1/L) V -

Inductors90o phase difference

between the voltage and current through an inductor.

The voltage leads the current (or the current lags the voltage).

ImpedanceIf we try to force all components to following Ohm’s Law, V = Z I, where Z is the impedance of the component.

Resistor: ZR =

Capacitor: ZC =

Inductor: ZL =

o

o

o

LLj

CCj

RR

90

901

0

AdmittanceIf we rewrite Ohm’s Law:I = Y V (Y = 1/Z), where Y is admittance of the component

Resistor: YR =

Capacitor: YC =

Inductor: YL = o

o

o

LLj

CCj

GGR

901

90

0 /1

Impedances

Value at = Admittances

Value at =

0 rad/s

∞ rad/s

0 rad/s

∞ rad/s

ZR = R = 1/G

R R YR = 1/R = G

G G

ZL = jL 0 ∞ YL =-j/(L) ∞ 0

ZC = -j/(C)

∞ 0 YC = jC 0 ∞ Inductors act like short circuits under d.c. conditions and like open circuits at very high frequencies.

Capacitors act like open circuits under d.c. conditions and like short circuits at very high frequencies.

ImpedanceGeneric component that represents a resistor, inductor, or capacitor.

sin

cos

tan 1

22

ZX

ZR

RX

XRZ

jXR

Z

Z

Z

Admittance

22

22

1

XR

XB

XR

RG

jXR

Z1Y

sin

cos

tan 1

22

YB

YG

GB

BGY

jBG

Y

Y

Y

SummaryOhm’s Law can be used to determine the ac

voltages and currents in a circuit when impedance or admittance are used.A resistor’s voltage and current are in phase. Voltage leads current through an inductor by

90o.Current leads voltage through a capacitor by

90o.Component

Impedance Admittance

Resistor ZR

Capacitor ZC

Inductor ZL 09 1 09

09 09 1

0 0

o

o

oo

LLjLLj

CCjCCj

GGRR