RC Circuit System Investigation
Dr. Kevin Craig
Greenheck Chair
in Engineering Design
&
Professor of Mechanical Engineering
Marquette University
RC Circuit System Investigation
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Engineering
System
Investigation
Process
RC Circuit
Electrical
System
Engineering System Investigation Process
START HERE
Physical
System
System
Measurement
Parameter
Identification
Physical
Model
Mathematical
Model
Measurement
Analysis
Mathematical
Analysis
Comparison:
Predicted vs.
Measured
Design
Changes
YES
Is The
Comparison
Adequate ?
NO
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Physical System for Investigation
RC Circuit
Electrical System
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Physical Modeling
• Simplifying Assumptions
– Resistors and Capacitors are pure and ideal
– Voltage sources are ideal and supply the intended
voltage to the circuit no matter how much current (and
thus power) this might require
– Measuring devices are ideal and do not load the
circuits by drawing any current
out
R
in
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C
out
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Model Parameter Identification
• Measure component values using the DMM by
connecting the components as shown.
• RC Circuit
– 15 kΩ nominal (5% tolerance), 14.986 kΩ measured
– 0.01 μF nominal, 9.85 nF = 0.00985 μF measured
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Resistors In Series
Resistors
in Parallel
Voltage
Divider
RC Circuit
Components
Ground
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Resistance Measurement
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Capacitance Measurement
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Mathematical Modeling:
General Circuit Laws
• We focus on basic analysis techniques which apply to all
electrical circuits.
• Kirchhoff’s Laws are basic to electrical circuits.
• One needs to know how to use these laws and combine
this with knowledge of the current/voltage behavior of the
basic circuit elements to analyze a circuit model.
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• Kirchhoff’s Voltage Loop Law (KVL)
– It is merely a statement of an intuitive truth; it
requires no mathematical or physical proof.
– This law can be stated in several forms:
• The summation of voltage drops around a
closed loop must be zero at every instant.
• The summation of voltage rises around a
closed loop must be zero at every instant.
• The summation of the voltage drops around a
closed loop must equal the summation of the
voltage rises at every instant.
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• Kirchhoff’s Current Node Law (KCL)
– It is based on the physical fact that at any point
(node) in a circuit there can be no accumulation of
electric charge. In circuit diagrams we connect
elements (R, L, C, etc.) with wires which are
considered perfect conductors.
– This law can be stated in several forms:
• The summation of currents into a node must be
zero.
• The summation of currents out of a node must
be zero.
• The summation of currents into a node must
equal the summation of currents out.
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• Sign Conventions
– In electrical systems we need sign conventions for
voltages and currents.
– If the assumed positive direction of a current has
not been specified at the beginning of a problem,
an orderly analysis is quite impossible.
– For voltages, the sign conventions consist of +
and – signs at the terminals where the voltage
exists.
– Once sign conventions for all the voltages and
currents have been chosen, combination of
Kirchhoff’s Laws with the known voltage/current
relations which describe the circuit elements leads
us directly to the system mathematical model.
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Mathematical Modeling of System
out
R
in
C
out
i R = i C + i out
KCL
iR = iC + 0
iR = iC
ein − eout
deout
=C
R
dt
RC Circuit System Investigation
ein − eout = iR
Basic Component
Equations
deout
(Constitutive Equations) i = C dt
deout
RC
+ eout = ein
dt
deout
τ
+ eout = Kein
dt
K =1
τ = RC
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pS = constant = ρgH (bottom of reservoir)
Analogies
A
C=
ρg
ρ = fluid density
g = acceleration due to gravity
B df o
+ fo = fi
K dt
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p tank = ρgh
dh
RC + h = H
dt
deout
τ
+ eout = Kein
dt
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• Numerical Solution
– How do we solve this differential equation
numerically?
deout
τ
+ eout = Kein
dt
– We use a numerical approximation:
deout
τ
+ eout = Kein
dt
Δeout
τ
+ eout = Kein
Δt
1
Δeout = ⎡⎢ ( Kein − eout ) ⎤⎥ Δt
⎣τ
⎦
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• Algorithm for Solving this Equation
– Step 1: Initialize Variables
τ ein t start = 0 t end
Δt
( eout )initial
– Step 2: Increment time and stop when done
t = t + Δt
If t = t end then stop
– Step 3: Compute ein(t)
– Step 4: Solve
1
Δeout = ⎡⎢ ( Kein − eout ) ⎤⎥ Δt
⎣τ
⎦
– Step 5: Determine new eout
( eout )new = ( eout )old + Δeout
– Step 6: Go back to Step 2
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• Plotting Numerical Data
– Engineers are well known for their ability to plot many curves
of experimental data and to extract all sorts of significant facts
from these curves.
– The better one understands the physical phenomena involved
in a certain experiment, the better is one able to extract a wide
variety of information from graphical displays of experimental
data.
– Understand The Physical Processes Behind The Data!
– When data may be approximated by a straight line, the
analytical relation is easy to obtain; but when almost any other
functional variation (e.g., exponential, polynomial, complex
logarithmic) is present, difficulties are usually encountered.
– It is convenient to try to plot data in such a form that a straight
line will be obtained for certain types of functional
relationships.
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• Analytical Solution to a Step Input
eout
t
− ⎞
⎛
= Kein ⎜1 − e τ ⎟
⎝
⎠
t
−
eout
= 1− e τ
Kein
t
−
eout
=e τ
1−
Kein
t
−
Kein − eout
=e τ
Kein
t
Kein
= eτ
Kein − eout
RC Circuit System Investigation
t
Kein
= eτ
Kein − eout
deout
+ eout = Kein
dt
t
− ⎞
⎛
eout = Kein ⎜1 − e τ ⎟
⎝
⎠
τ
⎡ Kein ⎤ t
log10 ⎢
= log10 ( e )
⎥
⎣ Kein − eout ⎦ τ
⎡ Kein ⎤ ⎡ 1
⎤t
log10 ⎢
log
e
=
(
)
⎥ ⎢ τ 10 ⎥
Ke
e
−
⎦
⎣ in out ⎦ ⎣
⎡ Kein ⎤
log10 ⎢
⎥
Ke
e
−
⎣ in out ⎦
1
slope = log10 ( e )
τ
t
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Another Approach: Impedance + Voltage Divider
Impedance
e/i
Allows us to
use Algebra
in
1
CD
out
e = iR
e
=R
i
e
1
=
i CD
de
i = C = CDe
dt
d
D=
Differential Operator
dt
1
eout
1
Transfer
CD
=
=
ein R + 1
RCD + 1 Function
CD
( RCD + 1) eout = (1) ein
( RCD ) eout + eout = ein
RC Circuit System Investigation
deout
RC
+ eout = ein
dt
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Mathematical Analysis and Prediction
RC Circuit Unit Step Response
de
RC out + eout = ein
dt
RC ( Deout ) + eout = ein
1
0.9
0.8
( RCD + 1) eout = ein
eout
1
K
=
=
ein RCD + 1 τD + 1
Amplitude
0.63
0.7
0.6
R = 15 KΩ
C = 0.01 μF
0.5
0.4
0.3
0.2
K = 1 = Steady-State Gain
τ = RC = Time Constant
MatLab Commands
RC Circuit System Investigation
0.1
0
0
1
K = 1; tau = 0.15E − 3;
RC _ System = tf (K,[tau 1])
step(RC _ System)
2
3
4
5
Time (sec)
τ = RC = 0.15 m sec
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• Time Constant τ
– Time it takes the step response to reach 63% of the
steady-state value, Kein.
• Rise Time Tr = 2.2 τ
– Time it takes the step response to go from 10% to
90% of the steady-state value, Kein.
• Delay Time Td = 0.69 τ
– Time it takes the step response to reach 50% of the
steady-state value, Kein.
• Steady-State Value
– The steady-state value of the response is Kein and at
4τ seconds (4 time constants), the response has
reached 98% of the steady-state value; for all
practical purposes, this is steady state.
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Semi-Log Paper
1.0
1.1
1.2
1.3
10
1.0
1.4
1.5
1.6
1.7
50
20
30
40
1.301
1.477
1.602
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log10
1.8
60
1.9
2.0
70 80 90 100
1.699 1.778 1.845 1.903
2.0
1.954
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RC Circuit
Frequency Response
DC Value = K
( x ) dB = 20log10 ( x )
Slope = -20 dB/decade
-3 dB
1 1
rad
=
= 6666.7
τ RC
sec
= 1061 Hz = Bandwidth
MatLab Commands
K = 1; tau = 0.15E − 3;
RC _ System = tf (K,[tau 1])
bode(RC _ System)
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Semi-log plots: Magnitude (dB) vs. log10 f
Phase (deg) vs. log10 f
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RC Circuit
Amplitude Ratio = 0.707 = -3 dB
Response to Input 1061 Hz Sine Wave
1
Input
1061 Hz
Sine Wave
Phase Angle = -45°
0.8
0.6
0.4
amplitude
0.2
0
-0.2
R = 15 KΩ
C = 0.01 μF
Output
-0.4
-0.6
-0.8
-1
0
0.5
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1
1.5
2
time (sec)
2.5
3
3.5
4
x 10
-3
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• Bandwidth
– The bandwidth is the frequency where the amplitude
ratio drops by a factor of 0.707 = -3dB of its gain at
zero or low-frequency.
– For a 1st-order system, the bandwidth is equal to 1/ τ.
– The larger (smaller) the bandwidth, the faster (slower)
the step response.
– Bandwidth is a direct measure of system susceptibility
to noise, as well as an indicator of the system speed
of response.
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Measurements
• RC Circuit
– Time Response (Step and Sine)
– Frequency Response
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Time Response
Function Generator
FGEN
Analog Input Channel
AI 6+
DC Power
Ground
RC Circuit System Investigation
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•
•
•
ELVIS Connections
Circuit Input to FGEN
Circuit Input to Channel AI 6+
Measured Signal to Channel AI 7+
Channels AI 6- and AI 7- and Circuit
Ground to DC Power Ground
Analog Input Channel
AI 7+
Analog Input Channels
AI 6- and AI 7K. Craig
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RC Circuit Time Response
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Why 500 Hz ?
5τ = ( 5 )( 0.15ms ) = 0.75ms
Let’s Use 1.00 ms
Square Wave Period = 2.00 ms
Square Wave Frequency = 500 Hz
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63%
τ = 0.118ms
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f = 1061 Hz
T = 0.943 ms
Magnitude Ratio: 0.798
Phase Angle: 40.5 deg
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f = 10 KHz
T = 0.100 ms
Magnitude Ratio: 0.156
Phase Angle: 79.2 deg
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Frequency Response
Function Generator
FGEN
Analog Input Channel
AI 1+
Power Ground
RC Circuit System Investigation
•
•
•
•
ELVIS Connections
FGEN to Circuit Input
Circuit Input to Channel AI 1+
Measured Signal to Channel AI 0+
DC Power Ground to Channels AI 0- and
AI1- and Circuit Ground
Analog Input Channel
AI 0+
Analog Input Channels
AI 0- and AI 1K. Craig
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RC Circuit Frequency Response
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