TRANSIENT SIGNALS IN COMPUTERS
Ideal World:
• Only 1’s and 0’s
• Instantaneous links
Reality:
•
•
•
•
Voltages exhibit propagation delay, decay, reflections
Spurious transients can superimpose to flip bits erroneously
RFI generated and picked up by wires can flip bits
Ground loops matter
vs.
good
risky
L12-1
TEM LINE THEVENIN EQUIVALENT
Example: Given Vs(t):
z
z
v(z,t) = v + (t - ) + v - (t + )
c
c
1
z
z
i(z,t) = [v + (t - ) - v - (t + )
Zo
c
c
Rs
+
Vs
-
i(t, z=0)
+
v(t, z=0)
-
Zo
Assume v- = 0 (no other sources)
Then v(t, z=0) = Zo i(t, z=0) yields equivalent circuit
Solution at terminals:
v+(t, z=0) = Vs(t)
Zo
(voltage divider equation)
Zo + R s
Solution for all t,z:
Zo
v + (t,z) = Vs (t − z )
c Z o + R s (forward propagating wave only)
L12-2
TEM LINE THEVENIN EQUIVALENT
Voltages at an Open Circuit:
v+(t – z/c)
i(t,D) = Yo[v+(t,D) – v-(t,D)] = 0 (open circuit)
⇒
v-(t,D) = v+(t,D)
⇒
v(t,D) = v+(t,D) + v-(t,D) = 2v+(t,D)
+
v(t,D)
-
Thevenin Equivalent for TEM source:
vopen circuit(t) = 2v+(t) = VTh(t)
z
D
0
+
VTh(t) = 2v+(t)
-
+
Zo v(t,D)
-
Example—Resistive Load:
Zo
At z = D: v-(t) = vL(t) – v+(t)
+
+
R
But:
v(t) = 2v+(t)
VTh = 2v+(t)
vL(t)
R + Zo
R − Zo
= v+(t) Γ
Thus:
v-(t) = v+(t)
D
R + Zo
Rn − 1
Define:
Γ = “reflection coefficient” = v-/v+ at load = R + 1
R
z
n
(= 0 if R = Z0, = -1 if R = 0, and = +1 if R = ∞)
L12-3
CAPACITIVELY TERMINATED TEM LINE
Example: Capacitive Load
1
Zo
u(t)
+
Zo,c
-
C
VTh = 2v+(t,D) +
D VTh
= u(t - ) -
Zo vL(t)
+
C
z
v+(t,D)
0.5
t
0
0 z D
c
vL(t), τ = ZoC sec
t =D/c
v-(t,D)
vL = v- + v+ at the load ⇒
v-(t,D) = vL(t,D) – v+(t,D)
v(t,z)
0.5
z
z
i(z,t) = Yo [v + (t - ) - v - (t + )]
c
c
0
t = 0+, short-circuit response
t → ∞, open-circuit response
1/2Zo
D
z
i(t,z)
0
D
z
L12-4
DIODE-TERMINATED TEM LINE
Example -- Logic Circuit:
Zo
+
i
-
ideal
diode
0
0.5
1
1
vs(t)
0.5
τ
v(z,t)
vTh(t) 3
v+(t,D) 2
0
D/c sec vL(t) 4
Zo,c
Vs(t)
D
Zo
D +
v Th = Vs (t - )
c -
physical
diode
v
Let vs(t) = 1-volt ramped step
v(t)
+
vL(t) 0.5v
+
- -
0 v+(t – z/c)
+
vL(t) 0.5 V
+
-
6
v(z,t)
0.5
t
z
τ
0.25
5
v-(t,z)
D
z
vL(t) = v+(t,D) + v-(t,D)
L12-5
INITIAL CONDITIONS
Example – charged TEM line:
t = 0 Charged to V, t<0
+
Zo,c Γ = +1
-
v(z,t) = v+(z – ct) + v-(z + ct)
1
i(z,t) = [v+(z – ct) - v-(z + ct)]
Zo
z D
Initial voltage and currents:
1
v+(z,t) = [v(z,t) + Zo i(z,t)]
2
v-(z,t) = 1 [v(z,t) - Zo i(z,t)]
2
v(z, t<0)
⇒
V
vi(t < 0) = 0
v+
Γ= +1
at ends 0
D
z
Subsequent voltages and currents:
v(z, 0 < t < D/c)
V
v+
Γ = -1 vv- Γ = +1
z
0 v+
D
V
Γ = -1
0
v(z, D/c < t < 2D/c)
vv+
Γ = +1
z
D
v-
Lossless
system,
rattles
forever
L12-6
INITIAL CONDITIONS
Arbitrary initial conditions:
v,i
v,i
v-
v(t=0)
i(t=0)
0
D
v+
z
0
v+
z
v- D
Arbitrary circuits:
Put switches anywhere
+
Zo,c
z D
L12-7
LIGHTNING STRIKE EXAMPLE
Lightning strikes the midpoint
What happens?
+
-
Zo,c
z
D
Figure by MIT OpenCourseWare.
L12-8
LIGHTNING STRIKE EXAMPLE
Revenge?
+
-
Zo,c
z
D
Figure by MIT OpenCourseWare.
L12-9
LIGHTNING STRIKE EXAMPLE
One possibility:
Temporary short through ionized channel
+
-
Zo,c
D
z
v(z,to)
Γ = -1
v+
0
v-
v+
v+
z
v-
v2cto
L12-10
LOSSY TRANSMISSION LINES
Equivalent Circuit:
i(z) RΔz LΔz
v(t,z)
CΔz
RΔz LΔz
GΔz
CΔz
GΔz
L [Hy m-1], C [F m-1],
R[Ω m-1], G[S m-1]
Δz
Equations for Δz → 0 :
dV(z)
= −(R + jωL) I (z)
dz
d I (z)
= −(G + jωC)V(z)
dz
V(z) = V+e-jk’z – k”z
I(z) = Yo V+e-jk’z – k”z
d2 V(z)
dz
2
= (R + jωL)(G + jωC)V(z)
⇒ V(z) = V e− jkz + V e+ jkz
+
−
k = −(R + jωL)(G + jωC) = k '− jk "
decay
for forward wave: Yo = (G + jωC) /(R + jωL)
Propagation e-jkz : decay rate (e-k”z) and phase velocity (vp) = f(ω)
(Exception: “distortionless line” for which R/L = G/C)
L12-11
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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