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6.002x CIRCUITS AND ELECTRONICS
The Impedance Model !!!
Reading: Section 13.3 from text
2
n Sinusoidal Steady State (SSS) Reading 13.1, 13.2
+
–
vtVv iI cos+ –
RC
Review
n Focus on sinusoids. n Focus on steady state, only care about vP as vH dies away.
SSS
3
3
4
Hv
total
Review
Vp contains all the information we need:
p
p
V
V
Amplitude of output cosine
Phase
sneak in Vi e jωt
drive
complex algebra
take real part
The Sneaky Path
Vp
tVi cos Vp cos wt +ÐVpéë ùû
set up DE
usual circuit model
nightmare trig.
1
tj
p eV
RCj
Vi
1
2
Sinusoidal Steady State SSS Approach
4
transfer function
jHRCjV
V
i
p
1
1
Review Frequency response - magnitude RCj
VV i
p
1
pp VtVv cos
The Frequency View
5
RC
1
4
2
0
i
p
V
V
1
RCtan 1
The Frequency View
Review
CvOv
BIASV
+ –
+ –
GSC
R
SVFrequency response - phase
6
Is there an even simpler way to get Vp ?
RCj
VV i
p
1
t
agony
sneaky
approach
Eff
ort
easy
Usual
diff eqn.
approach
? Is there
any even
simpler
way
without
DEs?
RCj
VV i
p
1
7
The Impedance Model
Is there an even simpler way to get Vp ?
Consider resistor:
R
Ri+
– RvResistor
8
The Impedance Model -- Capacitor Consider capacitor:
C
Ci+
– CvCapacitor
9
The Impedance Model -- Inductor Consider Inductor:
L
Li+
– LvInductor
10
The Impedance Model -- Inductor Consider Inductor:
L
Li+
– Lv
ts
lL eIi
ts
lL eVv
dt
diLv L
L
Inductor
11
inductor resistor capacitor
The Impedance Model -- Summary
12
Back to RC example…
+
– Cv
Iv + –
R
C
capacitor
sCZC
1
cCc IZV
impedance
cI
+
– cV CZ
Is there an even simpler way to get Vp ?
13
Impedance model:
Back to RC example…
+
– Cv
Iv + –
R
C
+
– cViV +
–
RZR
sCZC
1
cI
capacitor
sCZC
1
cCc IZV
impedance
cI
+
– cV CZ
Is there an even simpler way to get Vp ?
14
t
sneaky
approach
Eff
ort
agony
easy
Usual
diff eqn.
approach
There you have it
super
sneaky… no DEs!
Impedance method Today
15
Signal Notation
16
Impedance Method Summary
17
Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)
18
Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)
19
The Big Picture… Vi coswt Vp cos wt +ÐVpéë ùû
set up DE
usual circuit model
nightmare trig.
Vi e st
drive complex algebra
take real part
complex algebra
impedance-based circuit model
No D.E.s, No trig!
t
agony
sneaky
approach
Eff
ort
easy
Usual
diff eqn.
approach
Super
sneaky…
no DEs
20
Back to
21
L
Rj
LC
L
Rj
V
V
i
r
+ –
L
rI
C +
– rV
iV RLet’s study this transfer function
Review complex algebra in Appendix C of textbook
21
Graphically
2221 RCLC
RC
V
V
i
r
:Low
:High
:1
LC
Find limit
+ –
L
rI
C +
– rV
iV R
22
Graphically
2221 RCLC
RC
V
V
i
r
:Low
:High
:1
LC
Observe
+ –
L
rI
C +
– rV
iV R
1
Is there an even simpler way to get Vp ?
RCj
VV i
p
1
t
agony
sneaky
approach
Eff
ort
easy
Usual
diff eqn.
approach
? Is there
any even
simpler
way
without
DEs?
RCj
VV i
p
1
1
The Impedance Model
Is there an even simpler way to get Vp ?
Consider resistor:
R
Ri+
– RvResistor
2
The Impedance Model -- Capacitor Consider capacitor:
C
Ci+
– CvCapacitor
3
The Impedance Model -- Inductor Consider Inductor:
L
Li+
– LvInductor
1
The Impedance Model -- Inductor Consider Inductor:
L
Li+
– Lv
ts
lL eIi
ts
lL eVv
dt
diLv L
L
Inductor
2
inductor resistor capacitor
The Impedance Model -- Summary
1
Back to RC example…
+
– Cv
Iv + –
R
C
capacitor
sCZC
1
cCc IZV
impedance
cI
+
– cV CZ
Is there an even simpler way to get Vp ?
1
Impedance model:
Back to RC example…
+
– Cv
Iv + –
R
C
+
– cViV +
–
RZR
sCZC
1
cI
capacitor
sCZC
1
cCc IZV
impedance
cI
+
– cV CZ
Is there an even simpler way to get Vp ?
2
t
sneaky
approach
Eff
ort
agony
easy
Usual
diff eqn.
approach
There you have it
super
sneaky… no DEs!
Impedance method Today
1
Signal Notation
2
Impedance Method Summary
1
Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)
1
Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)
1
The Big Picture… Vi coswt Vp cos wt +ÐVpéë ùû
set up DE
usual circuit model
nightmare trig.
Vi e st
drive complex algebra
take real part
complex algebra
impedance-based circuit model
No D.E.s, No trig!
t
agony
sneaky
approach
Eff
ort
easy
Usual
diff eqn.
approach
Super
sneaky…
no DEs
1
Back to
21
L
Rj
LC
L
Rj
V
V
i
r
+ –
L
rI
C +
– rV
iV RLet’s study this transfer function
Review complex algebra in Appendix C of textbook
2
Graphically
2221 RCLC
RC
V
V
i
r
:Low
:High
:1
LC
Find limit
+ –
L
rI
C +
– rV
iV R
1
Graphically
2221 RCLC
RC
V
V
i
r
:Low
:High
:1
LC
Observe
+ –
L
rI
C +
– rV
iV R
