AERO
ASTRO
16.682 - Prototyping Avionics
Spring 2006
LECTURE 3
February 15, 2006
DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS
Alvar Saenz-Otero
Outline
•
More on Components
– Resistors, Capacitors, Inductors: ideal vs. real
– First and second order systems
– Diodes
•
Amplifiers
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16.682 - Prototyping Avionics
Last time...
•
Four component laws
–
v = iR
–
dv
i=C
dt
•
Two network laws
– KCL - Kirchoff’s Current Law
∑i
n
In/out of node
=0
R1
R2
R3
node
–
v=L
i1 + i 2 + i 3 + i 4 = 0
di
dt
– KVL - Kirchoff’s Voltage Law
∑v
2
–
R4
n
Around a loop
v
P = iv = i R =
R
2
=0
+
+
V3
V2
+
V4
+
v1 + v 2 = v 3 + v 4
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V1
3
-
-
16.682 - Prototyping Avionics
Ideal vs. Real
•
Ideal
•
C
Real
– Wire
R
• R ≠ 0, C ≠ 0, L ~ 0
• R=0, C=0, L=0
R
– Resistor
R
• C ~ 0,L ~ 0
• C=0, L=0
C
C
– Capacitor
• R ≠ 0, L ~ 0
• R=0, L=0
R
– Inductor
L
• R ≠0, C ~ 0
• R=0, C=0
L
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R
16.682 - Prototyping Avionics
Review of Resistors
•
Serial
•
Parallel
R1
R1
R2
•
R3
=
Voltage divider
R4
R1
Bat
R2
VR
R2
=
R4
1
1
1
1
+
+
=
R1 R 2 R 3 R 4
R3
R1 + R 2 + R 3 = R 4
•
For two resistors
R 1R 2
R4 =
R1 + R 2
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GND
VR = VBat
R2
R1 + R 2
16.682 - Prototyping Avionics
First Order Systems
•
Relation of different inputs
i(t)
q0
t
Vout
t
R
C
I0
Vout
vc(t)
i(t)
t
or
I0
t
T
Vout
R
t
Bat
C
vc(t)
VBat = i(t)R
q −
= 0 e RC
C
t
−
⎛
⎞
RC
= I 0 R ⎜⎜1- e ⎟⎟
⎝
⎠
⎛ − RCt
⎞⎞
I 0 R ⎜⎛
=
t + RC⎜⎜ e
−1⎟⎟ ⎟
⎟
T ⎝⎜
⎝
⎠⎠
t
−
⎛
⎞
0 < t < τ → Vout = I 0 R ⎜⎜1- e RC ⎟⎟
⎝
⎠
I0
τ
i(t) = I o sin (ωt )
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t
t
t
−
⎛
⎞ − ( t-τ )
t > τ → Vout = I 0 R ⎜⎜1- e RC ⎟⎟e RC
⎝
⎠
Vout =
I0 R
1+ (ωRC )
2
6
(
)
sin ωt − tan −1 (ωRC )
16.682 - Prototyping Avionics
Second Order Systems
•
Circuits that combine capacitors and inductors are higher order
L
Vout = V0 (1- cos(ω0t ))
V(t)
C
ω0 =
vc(t)
t
LC
V0
t
L
R
C
V(t)
vc(t)
(
Vout ≈ V0 1 - cos(ω0t ) ⋅ e −ξ ⋅t
ω0 =
)
t
R
,ξ =
2L
LC
Resistor adds dampening
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16.682 - Prototyping Avionics
Diodes
•
Ideal
•
Real
I
I
D
+
V
V
– Does not allow current f low when
– Voltage drop: minimum voltage
voltage is reversed
before current can go through
• Stops all current
– Current leak: small amount of
– Allows infinite current flow when
current goes through in reverse
positive voltage is applied
– Maximum/Minimum voltage in both
forward and reverse
– Maximum current in forward
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16.682 - Prototyping Avionics
Introduction to Operational Amplifiers
• Utilize an “external” power source to amplify/modify an input signal
– Allow the use of feedback to closely track the signal
vpwr+
i+=0 v+
i-=0 v-
+
-
vout
vpwr– Adjusts the output voltag Vout to try make v+ and v- be the same
• The users adds elements (wires, resistors, capacitors, etc) which create
current loops between the output and inputs to create feedback loops
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16.682 - Prototyping Avionics