Jan 09, 2016
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