EE/BME 374K,
EE 385J.31, BME 384J.1
Lecture 3
Sept. 2, 2021
Medical Device Costs?
FDA Goal: Provide assurance that medical devices on the market are safe and
effective
• Bringing a new medical device to market takes on average 3-7 years
– Preclinical: benchtop and animal testing: 2-3 years, $10-20 million
– Clinical: completely depends à $0 to $10+ million
– FDA application: up to $260k
• Total cost: up to several tens of millions. Depends a lot on the device.
• Average cost of low to medium risk devices, $31 million to bring to market
•
See papers by Van Norman, 2016.
Also: https://mdepinet.org/wp-content/uploads/S5_1_Martinsen.pdf
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Comment on Homework #1 Problem 1
1b) Write one possible problem statement,
related to the problems observed from the
video.
à i.e., a ”Needs Statement” (see slide 23 & 30
of Lecture 1)
3
Comment on Homework #1 Problem 2
Tips:
àCalculate +/-A% and +/-B% for EACH reading (each x value)
àIndependent nonlinearity = maximum deviation from straight line; will vary
with the reading value!
Today’s Plan
• Static vs. Dynamic Systems
• Review of Transfer Functions
• Dynamic Characteristics:
– 0 Order
– 1st Order
– 2nd Order
– Damping
5
Static vs Dynamic Characteristics
• Static: describes performance of instruments
for DC or very low frequencies
• But most biological signals are time-varying
à Want instrument behaviour itself to not
change over time
• Dynamic: requires differential / integral
equations to describe the quality of
measurements
– Transfer function, frequency response, time delay
6
Time-varying Signals (EE 313)
Input, output relations:
dny
dn−1y
dy
dmx
dx
an n + an−1 n−1 +...+ a1 + a0y(t) = bm m +...+b1 +b0x(t)+ C
dt
dt
dt
dt
dt
• C = constant offset (not time-varying)
• ai and bi depend on the physical/electrical parameters of
the system
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Operator Notation
dny
dn−1y
dy
dmx
dx
an n + an−1 n−1 +...+ a1 + a0y(t) = bm m +...+b1 +b0x(t)+ C
dt
dt
dt
dt
dt
Introduce D operator:
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“Ordinary Linear” Differential Equation
• The system is linear i.f.f.
( )
ai ≠ f t,y
( )
bi ≠ f t,x
ai and bi are constants, don’t depend on time
or output/input
• Ordinary differential equations because:
– 1 independent variable, x
– i.e., it’s not a partial derivative
9
“Ordinary Linear” Differential Equation
• We’re assuming:
The instrument’s method of measuring /
analyzing the input signal to obtain the output
signal does not change with time or the
magnitude of the input
• Thankfully many engineering instruments can
be described by ordinary linear differential
equations, with constant coefficients
10
Transfer Function
• Transfer function (TF) expresses relationship between
input and output signals
• With known TF, can predict output for any input
• Operational TF (function of operator D):
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Transfer Function
• Frequency TF (function of jw):
• Input periodic (assuming transient response has died out):
• Output (i.e., solution):
12
Zeroth Order Instrument
• Simplest form of differential equation (all coefficients = 0
except a0 and b0)
()
()
a0y t = b0x t
• Transfer function:
• Does not have an energy storage element (no d/dt)
• “Ideal” performance: output proportional to input for all
frequencies, no phase delay
13
First Order Instruments
• Single energy storage element (e.g. either L or
C, but not both):
• Rewrite using operator format:
where:
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First Order Instruments: Transfer Functions
• Differential equation that describes dynamic
response:
• Using operator:
àOperational TF =
àFrequency TF =
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First Order: Example
• RC Circuit (has only one storage element)
+
x(t)
-
+
x(t) = input voltage
y(t)
y(t) = output voltage, i.e., voltage
across capacitor
-
• Described by first order differential equation:
• Time constant:
• K=1
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Second Order Instruments
• Second-order differential equation describes
dynamic response:
Reduce/re-arrange to:
where
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Second Order: Transfer Functions
• Operational TF:
• Frequency TF:
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Conditions
• Overdamped: ζ > 1
• Underdamped:
ζ <1
• Critically damped:
ζ =1
(Fig. 1.7 Medical Instrumentation)
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Time Delay in Instruments
• Time-delay elements: Instruments where output is exactly
the same as input, except that it is delayed in time (by ):
• Time delay (time domain) ó Phase Shift (frequency domain)
• Transfer function:
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Review of dB Descriptions (EE411, BME 311)
• Decibel units (dB) are a standard logarithmic
description in signal analysis (applied to ratios).
• Definitions:
[Voltage gain]dB = 20 log10{Vout/Vin}
[Power gain]dB = 10 log10{Pout/Pin}
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