RC Circuit System Investigation

advertisement
RC Circuit System Investigation
Dr. Kevin Craig
Greenheck Chair
in Engineering Design
&
Professor of Mechanical Engineering
Marquette University
RC Circuit System Investigation
K. Craig
1
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
RC Circuit System Investigation
K. Craig
2
Physical System for Investigation
RC Circuit
Electrical System
RC Circuit System Investigation
K. Craig
3
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
RC Circuit System Investigation
C
out
K. Craig
4
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
RC Circuit System Investigation
K. Craig
5
Resistors In Series
Resistors
in Parallel
Voltage
Divider
RC Circuit
Components
Ground
RC Circuit System Investigation
K. Craig
6
Resistance Measurement
RC Circuit System Investigation
K. Craig
7
Capacitance Measurement
RC Circuit System Investigation
K. Craig
8
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.
RC Circuit System Investigation
K. Craig
9
• 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.
RC Circuit System Investigation
K. Craig
10
• 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.
RC Circuit System Investigation
K. Craig
11
• 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.
RC Circuit System Investigation
K. Craig
12
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
K. Craig
13
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
RC Circuit System Investigation
p tank = ρgh
dh
RC + h = H
dt
deout
τ
+ eout = Kein
dt
K. Craig
14
RC Circuit System Investigation
K. Craig
15
• 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
⎦
RC Circuit System Investigation
K. Craig
16
• 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
RC Circuit System Investigation
K. Craig
17
• 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.
RC Circuit System Investigation
K. Craig
18
• 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
K. Craig
19
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
K. Craig
20
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
K. Craig
21
• 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.
RC Circuit System Investigation
K. Craig
22
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
RC Circuit System Investigation
log10
1.8
60
1.9
2.0
70 80 90 100
1.699 1.778 1.845 1.903
2.0
1.954
K. Craig 23
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)
RC Circuit System Investigation
Semi-log plots: Magnitude (dB) vs. log10 f
Phase (deg) vs. log10 f
K. Craig
24
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
RC Circuit System Investigation
1
1.5
2
time (sec)
2.5
3
3.5
4
x 10
-3
K. Craig
25
• 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.
RC Circuit System Investigation
K. Craig
26
Measurements
• RC Circuit
– Time Response (Step and Sine)
– Frequency Response
RC Circuit System Investigation
K. Craig
27
Time Response
Function Generator
FGEN
Analog Input Channel
AI 6+
DC Power
Ground
RC Circuit System Investigation
•
•
•
•
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
28
RC Circuit Time Response
RC Circuit System Investigation
K. Craig
29
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
RC Circuit System Investigation
K. Craig
30
RC Circuit System Investigation
K. Craig
31
63%
τ = 0.118ms
RC Circuit System Investigation
K. Craig
32
RC Circuit System Investigation
K. Craig
33
RC Circuit System Investigation
K. Craig
34
RC Circuit System Investigation
K. Craig
35
RC Circuit System Investigation
K. Craig
36
f = 1061 Hz
T = 0.943 ms
Magnitude Ratio: 0.798
Phase Angle: 40.5 deg
RC Circuit System Investigation
K. Craig
37
RC Circuit System Investigation
K. Craig
38
RC Circuit System Investigation
K. Craig
39
RC Circuit System Investigation
K. Craig
40
RC Circuit System Investigation
K. Craig
41
RC Circuit System Investigation
K. Craig
42
f = 10 KHz
T = 0.100 ms
Magnitude Ratio: 0.156
Phase Angle: 79.2 deg
RC Circuit System Investigation
K. Craig
43
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
44
RC Circuit Frequency Response
RC Circuit System Investigation
K. Craig
45
RC Circuit System Investigation
K. Craig
46
RC Circuit System Investigation
K. Craig
47
Download