We will also understand how to model drug delivery
If the input duration is much shorter than system time constant, then shape or duration of input does not matter, only total amount of drug delivered. Model input as a short pulse à Impulse
More on this soon…
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First, let’s understand a step input
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Rising step response of RC circuit
Sec. 10.1.1 of A&L
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Falling step response of RC circuit
)(ti R )(tvCC−
+
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Pulse input: Rising step followed by falling step (much later)!
)(ti R )(tvCC−
+
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Short pulse input: Rising step followed by falling step (shortly thereafter)!
)(ti R )(tvCC−
+
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Let’s Stare at Current Pulse
Becomes “impulse” as Lim T à 0 Denoted by Impulse is an important signal type
)(tδ
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As the pulse gets narrower à impulse
Becomes “impulse” as Lim T à 0 Denoted by Impulse is an important signal type
)(tδ
)(ti
0 tT
TQI =
)(tQδ)(tvC
t0 T
)(tiR )(tvCC
−
+
T2 T3 T1 T2 T3 T1
3TQ
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Response to impulse – limit case
)(ti R )(tvCC−
+
0)0( =−Cv
)(ti
t0 T
QCharge Q
t: 0 – T
)(tvC
t0
t > T
Lim T à 0
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Response to impulse
)(ti R )(tvCC−
+
0)0( =−Cv
)(ti
t0
Q
)(tvC
t0
remember drug delivery
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Current and voltage impulses
R )(tvCC−
+
0)0( =−Cv)(ti
t0
Q
)(tvC
t0
CQ
RCt
eCQ −
)()( tQti δ=
Current impulse delivers charge
Voltage impulse delivers flux linkage Λ
Q
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Superposition with initial conditions and sources
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Impulses, steps and ramps
For linear systems (remember, no sources and initial conditions)