How Does Dell Compete with a Retailer Who Already Has a Stock

I

3.4 Analysis of RL and RC Networks
3.4.1 Series RL Network

+

+
VR
R-

Im
Z

jXL = jωL

V
+
L VL
-

Re
0

R

Z = ZR + ZL = R + jXL XL = ωL > 0
2
2
Z = R + XL
−1 ⎛ X L ⎞ φZ = tan ⎜ ⎟
⎝R⎠
V = VR + VL = IR + IjXL = IZ

Im

V

VL
0

Re
I

VR

I

3.4.2 Series RC Network
Im

+

+
R VR
+
C VC
-

R

0 jXC = -j/ωC

Z = ZR + ZC = R + jXC

Re

V
-

Z

XC = -1/ωC < 0

2
Z = R2 + X C
−1 ⎛ X C ⎞ φ Z = tan ⎜
⎟
⎝R⎠

V = VR + VC = IR + IjXC = IZ

Im
0

I

VR

VC
V

Re

3.4.3 Parallel RL Network
I
11
1
=+
Z R jX L jRX L
Z=
R + jX L
RX L
Z=
2
R2 + X L
⎛ XL ⎞ φ Z = 90 − tan ⎜ ⎟
⎝R⎠

+
R

L

V
-

IL = V/jXL
= -jV/ωL

IR = V/R

−1

I = IR + IL

V
V
⎛1
1⎞ V
I= +
=V⎜ +
⎟=
R jX L
⎝ R jX L ⎠ Z

Im

V

IR

0
IL

Re
I

3.4.4 Parallel RC Network
I
11
1
=+
Z R jX C jRX C
Z=
R + jX C
RX C
Z=
2
R2 + X C

+
R

C

V
-

IR = V/R

⎛ XC ⎞ φ Z = −90 − tan ⎜
⎟
⎝R⎠

IC = V/jXC
= jVωC

−1

IC

Im

I

I = I R + IC

⎛1
1⎞V
V
V
=V⎜ +
I= +
⎟=
R jX C
⎝ R jX C ⎠ Z

0

Re
V

IR

3.5 Resonance
3.5.1 Series RLC Network j Z (ω ) = R + jωL −
= R + jX (ω ) ωC 1
X (ω ) = ωL − ωC When reactance = 0 → resonance
1
X (ω0 ) = ω0 L −
=0
ω0 C
1
Resonant frequency ω0 =
LC

I
+

V

-

+
VR
R+
L VL
+
C VC
-

ωL
X=ωL-1/ωC

0

ω0

-1/ωC

ω

A two-terminal network undergoes resonance when the imaginary portion of its impedance (or admittance) becomes zero; or equivalently, that the impedance (or admittance) becomes real.
For series RLC circuit, j Z (ω ) = R + jωL −
= R + jX (ω ) ωC ⎛ ω 2 LC − 1 ⎞
Z (ω ) = R + j ⎜
⎟
⎝ ωC ⎠
At resonance,
Z (ω 0 ) = R

⎛ ω LC − 1 ⎞
2
Z (ω ) = R + ⎜
⎟
⎝ ωC ⎠
2

2

⎛ ω 2 LC − 1 ⎞
−1
∠Z (ω ) = tan ⎜
⎟
⎝ ωRC ⎠

|Z |

R
0

∠Z

ω0

ω

VL

ω < ω0
I

VL

ω > ω0

VR

V

0
VL + VC

VL + VC