Appendix A
Flip-book animations
This appendix demonstrates certain principles through the use of “flip-book” animation. Each page of these appendix sections forms one frame of the “animation,” viewed by rapidly flipping pages (if the book is printed), or rapidly clicking the “next page” button (if the book is viewed on a computer). While crude, this animation technique enjoys the benefits of low technology (it even works in paper form!) and convenient pausing at critical frames.
Enjoy!
A.1 Polyphase light bulbs animated
The key to understanding how three-phase electric motors work is to have an accurate mental picture of the rotating magnetic field created by the stator windings of a polyphase motor. One of the best ways to visualize this phenomenon is to observe a string of “chaser” lights blinking in a polyphase sequence. Just in case you don’t happen to have a string of polyphase lights at your viewing convenience, I have provided a simulation here to demonstrate how the illusion of motion is created by the sequential energization of light bulbs.
Note how the lights appear to “move” from left to right as the energization sequences moves from A to C. After a few cycles of ABC, two of the wires are crossed to reverse phase sequence. This has the result of reversing the apparent direction of motion! It matters not which two phases are reversed. In this animation, I reverse phases A and B, but I could have just as well swapped phases B and C, or phases A and C, and created the exact same e ect.
2829
2830 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C
B
A
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2831 |
C
B
A
2832 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C
B
A
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2833 |
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B
A
2834 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2835 |
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B
A
2836 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C
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A
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2837 |
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A
2838 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2839 |
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2840 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2841 |
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2842 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2843 |
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2844 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2845 |
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2846 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2847 |
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2848 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2849 |
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2850 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C |
B |
A |
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2851 |
C |
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A |
2852 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C |
B |
A |
A.1. POLYPHASE LIGHT BULBS ANIMATED |
2853 |
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A |
2854 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
C |
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A.1. POLYPHASE LIGHT BULBS ANIMATED |
2855 |
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A |
2856 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
A.2 Polyphase induction motor animated
The following animation shows how the “rotating” magnetic field of a three-phase AC induction motor is produced by the interaction of three stator winding sets energized with di erent phases (A, B, and C) of a three-phase AC power source. A red arrow shows the direction of the resultant magnetic field created by the interaction of the three winding sets.
A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2857 |
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2858 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2859 |
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2860 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2861 |
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2862 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2863 |
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2864 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2865 |
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2866 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2867 |
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2868 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2869 |
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2870 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2871 |
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2872 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2873 |
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2874 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A
A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2875 |
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A
S N
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2876 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2877 |
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2878 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.2. POLYPHASE INDUCTION MOTOR ANIMATED |
2879 |
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2880 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A
A.3. ROTATING PHASOR ANIMATED |
2881 |
A.3 Rotating phasor animated
The following animation shows a rotating phasor in three-dimensional form. The phasor rotates in a complex plane (with real and imaginary axes), but travels linearly along the time axis. In doing so it traces a path that looks like a circle when viewed along the centerline (time axis) but looks like a sinusoidal wave when viewed from above or along the side.
Euler’s Relation describes the phasor’s position in the complex plane:
ejωt = cos ωt + j sin ωt
Where,
e = Euler’s number (approximately equal to 2.718281828) ω = Angular velocity, in radians per second
t = Time, in seconds
cos ωt = Horizontal projection of phasor (along a real number line) at time t
j sin ωt = Vertical projection of phasor (along an imaginary number line) at time t
If you imagine the phasor’s length either growing or decaying exponentially over time, the result will be a spiral that either widens like a horn or shrinks like a funnel. This would be a visualization of a complex exponential, where the s variable defines both the rate of growth/decay (the envelope of the spiral) and the angular velocity (the pitch of the spiral):
est = e(σ+jω)t = eσtejωt
Where,
s = Complex growth/decay rate and frequency (sec−1)
σ = τ1 = Real growth/decay rate (time constants per second, or sec−1) jω = Imaginary frequency (radians per second, or sec−1)
t = Time (seconds)
The example of a unit-length rotating phasor is nothing more than a special case of the complex exponential, where σ = 0 (i.e. there is no growth or decay over time):
est = e(0+jω)t = e0ejωt = ejωt
2882 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2883 |
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2884 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2885 |
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2886 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2887 |
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2888 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2889 |
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2890 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2891 |
time
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2892 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2893 |
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2894 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2895 |
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2896 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2897 |
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2898 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2899 |
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2900 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2901 |
time
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2902 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
time
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A.3. ROTATING PHASOR ANIMATED |
2903 |
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2904 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2905 |
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2906 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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2907 |
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2908 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2909 |
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2910 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2911 |
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2912 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2913 |
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2914 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2915 |
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2916 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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2917 |
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2918 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2919 |
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2920 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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A.3. ROTATING PHASOR ANIMATED |
2921 |
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2922 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
A.4 Di erentiation and integration animated
The following animation shows the calculus concepts of di erentiation and integration (with respect to time) applied to the filling and draining of a water tank.
The animation shows two graphs relating to the water storage tank: one showing the volume of stored water in the tank (V ) and the other showing volumetric flow rate in and out of the tank (Q). We know from calculus that volumetric flow rate is the time-derivative of volume:
Q =
dV
dt
We also know that change in volume is the time-integral of volumetric flow rate:
Z t1
V = Q dt
t0
Thus, the example of a water storage tank filling and draining serves to neatly illustrate both concepts in relation to each other.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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2923
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V = ò Q dt |
2924 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Q 0
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2925
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V = ò Q dt |
2926 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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V = ò Q dt |
2928 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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dt |
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Q |
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Volume meter
Q 0 
-Max.
2929
Integration |
Volume (V) is the |
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integral of flow rate |
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with respect to time |
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V = ò Q dt |
2930 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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Q |
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dt |
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Volume meter
Q 0 
-Max.
2931
Integration |
Volume (V) is the |
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integral of flow rate |
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with respect to time |
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V = ò Q dt |
2932 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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Q |
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dt |
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+Max. |
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Volume meter
Q 0 
-Max.
2933
Integration |
Volume (V) is the |
|
integral of flow rate |
||
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|
with respect to time |
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|
V = ò Q dt |
2934 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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Q |
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dt |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
2935
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2936 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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|||||
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with respect to time |
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|||||
Flow meter |
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Q = |
dV |
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Q |
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dt |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
2937
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2938 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
|
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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||||||
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derivative of volume |
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|||||
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with respect to time |
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|||||
Flow meter |
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Q = |
dV |
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dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
2939
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2940 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
|
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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|||||
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with respect to time |
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|||||
Flow meter |
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Q = |
dV |
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Q |
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+Max. |
dt |
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V |
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Volume meter
Q 0 
-Max.
2941
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2942 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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||||||
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derivative of volume |
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with respect to time |
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|||||
Flow meter |
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Q = |
dV |
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Q |
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+Max. |
dt |
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V |
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Volume meter
Q 0 
-Max.
2943
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2944 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
|
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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||||||
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derivative of volume |
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|||||
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with respect to time |
|
|||||
Flow meter |
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Q = |
dV |
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||||||||
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Q |
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+Max. |
dt |
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V |
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Volume meter
Q 0 
-Max.
2945
Integration |
Volume (V) is the |
|
integral of flow rate |
||
|
||
|
with respect to time |
|
|
V = ò Q dt |
2946 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
|
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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|
Flow rate (Q) is the |
||||||
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||||||
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derivative of volume |
|
|||||
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|
with respect to time |
|
|||||
Flow meter |
|
|
Q = |
dV |
|
||||||||
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||||||||||
|
Q |
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+Max. |
dt |
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||||||
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V |
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Volume meter
Q 0 
-Max.
2947
Integration |
Volume (V) is the |
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integral of flow rate |
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with respect to time |
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V = ò Q dt |
2948 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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dV |
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dt |
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Volume meter
Q 0 
-Max.
2949
Integration |
Volume (V) is the |
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integral of flow rate |
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with respect to time |
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V = ò Q dt |
2950 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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Q |
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dt |
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V |
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Volume meter
Q 0 
-Max.
2951
Integration |
Volume (V) is the |
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integral of flow rate |
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with respect to time |
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V = ò Q dt |
2952 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED
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Max. |
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V |
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Liquid storage tank |
0 |
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Differentiation |
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Flow rate (Q) is the |
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derivative of volume |
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with respect to time |
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Flow meter |
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Q = |
dV |
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Q |
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dt |
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V |
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Volume meter
Q 0 
-Max.
2953
Integration |
Volume (V) is the |
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integral of flow rate |
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|
with respect to time |
|
|
V = ò Q dt |
2954 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Flow meter
Q
Max.
V
Liquid storage tank
0
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
V +Max.
Volume meter
Q 0 
-Max.
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2955 |
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Max. |
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Liquid storage tank |
V |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
2956 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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Max. |
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Liquid storage tank |
V |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2957 |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
2958 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
|
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2959 |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
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Flow rate (Q) is the |
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Volume (V) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
|||||||||||
Flow meter |
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Q = dt |
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V = ò Q dt |
||||||||||||||
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Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
2960 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
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|||||||
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|||||||||
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Flow rate (Q) is the |
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Volume (V) is the |
|||||||||||
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derivative of volume |
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integral of flow rate |
|||||||||||
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with respect to time |
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with respect to time |
|||||||||||
Flow meter |
|
|
Q = dt |
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V = ò Q dt |
||||||||||||||
|
Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2961 |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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|
Integration |
Volume (V) is the |
||||||||||||
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|||||||||||||||
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|||||||||||||||
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Flow rate (Q) is the |
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||||||||||||
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derivative of volume |
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|
integral of flow rate |
|||||||||||
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with respect to time |
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with respect to time |
|||||||||||
Flow meter |
|
|
Q = dt |
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|
|
V = ò Q dt |
||||||||||||||
|
Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
2962 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
|
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2963 |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
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Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
2964 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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|||||||||||||
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
||||||||||||
Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2965 |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
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Flow rate (Q) is the |
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||||||||||||
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derivative of volume |
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integral of flow rate |
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with respect to time |
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with respect to time |
|||||||||||
Flow meter |
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Q = dt |
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V = ò Q dt |
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Q |
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+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
2966 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
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Max. |
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Liquid storage tank |
V |
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0 |
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dV |
Differentiation |
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Integration |
Volume (V) is the |
||||||||||||
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|||||||||||||||
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|||||||||||||||
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Flow rate (Q) is the |
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||||||||||||
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derivative of volume |
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|
integral of flow rate |
|||||||||||
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with respect to time |
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with respect to time |
|||||||||||
Flow meter |
|
|
Q = dt |
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|
V = ò Q dt |
||||||||||||||
|
Q |
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|
+Max. |
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V |
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||
Volume meter
Q 0 
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2967 |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
2968 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2969 |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
2970 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2971 |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
2972 |
APPENDIX A. FLIP-BOOK ANIMATIONS |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.
A.4. DIFFERENTIATION AND INTEGRATION ANIMATED |
2973 |
Note how the height of the flow graph directly relates to the slope of the volume graph . . .
Max. |
V |
0 |
Flow rate (Q) is the derivative of volume with respect to time
Q =
dV dt
+Max.
Q 0 
Differentiation
Integration
Volume (V) is the integral of flow rate with respect to time
V = ò Q dt
-Max.