PHYSICS AS A SYSTEMS SCIENCE
SIMPLE RC-SYSTEMS
(AN EXAMPLE OF SYSTEM BEHAVIOR)
Charging and discharging single tanks
5.0
0.40
0.20
0.10
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0.00
0
100
200 300
Time / s
400
0.40 bar
4.0
Oil level / m
Level / m
0.30
0.30 bar
3.0
0.20 bar
2.0
1.0
0.0
0.0E+0
5.0E+3
1.0E+4
1.5E+4
Time / s
A straight-walled tank filled with oil is discharged through a horizontal pipe at the bottom (center diagram)
or charged with the help of a pump (diagram on the right). The measured levels closely fit exponential
functions of time.
Hans U. Fuchs 2010
2
Charging and discharging capacitors
6.0
R2
V
4.0
2.0
UC
0.0
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0
100
200
Time / s
300
Voltage / V
C
Voltage / V
R1
+
6.0
4.0
2.0
0.0
400
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Time / s
Discharging and charging of a simple capacitor. Left: Diagram of a circuit that allows for charging and
subsequent discharging (R1 = 9.8 kOhm, R2 = 108 kOhm). Center: Voltage across the capacitor as a
function of time, as it discharges. Right: Voltage across capacitor during charging.
Hans U. Fuchs 2010
3
Equilibrating levels (of fluids)
Flow
Level / mm
150
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100
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0
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Time / s
Two tanks containing rape seed oil are connected by a hose at the bottom, the liquid flows from the one
having the higher fluid level to the one having a lower level (this is independent of the size of the tanks;
levels equilibrate, not quantities of liquid). Right: Data for the levels of rape seed oil.
Hans U. Fuchs 2010
4
Establishing electrical equilibrium
5
V
V
UC1
UC2
Voltage / V
4
3
2
1
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0
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Time / s
30
40
Left: Differently charged bodies in (electrical) contact: Charge flows from the one more strongly charged.
Flowing charge can light a glow lamp. The phenomenon can be observed under controlled conditions with
two connected capacitors (second from left; the parallel lines in the circuit diagram are the symbol for
capacitors, a rectangle is a symbol for a resistor, V stands for volt-meter, U for voltage). The diagram on
the right shows the voltages of the capacitors as functions of time.
Hans U. Fuchs 2010
5
Driving charge apart with a battery
Voltage / V
4
2
0
-2
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[[[[[[[[[[[[[[[[[[[
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[[[[[[[[[[[[[[[
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[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
-4
0
10
20
30
Time /s
Two capacitors in a single circuit with a battery between them. The diagram shows how the voltage of the
capacitors changes with time.
Remarks. In a single circuit, the behavior is very simple. Solutions of models of single circuits having constant resistance
and capacitance are simple exponential functions.
Hans U. Fuchs 2010
6
Flow through chains of tanks
1.00
Level / m
0.80
0.60
0.40
0.20
0.00
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0
100
200
Time / s
300
400
A chain of tanks filled with water. Water stands at different levels at first (the four tanks on the right are
virtually empty at first). When the pipes are opened, water levels change with time.
Remarks. If there are two or more circuits, the behavior is more complex. Solutions are combinations of exponential
functions.
Hans U. Fuchs 2010
7
Chains and networks of capacitors and resistors
Voltage / V
6
4
2
0
0
200
400
Time / s
600
A (short) chain of capacitors with resistors between them (photograph at left) forms a physical model for
diffusion of electric charge. In a first experiment, all capacitors have the same capacitance, and all but the
leftmost are uncharged. The graph at the center shows the voltages across the capacitors as functions of
time. Note the similarity of behavior with that observed in a chain of water tanks (Fig.1.14).
Hans U. Fuchs 2010
8
Time constants
h
R
C
UR
UC
τC
t
τC
t
R
h
+
Pump
UR
US
C
UC
The (capacitive) time constant of charging or discharging of a simple (single circuit) RC-system is the time
it would take for the storage element to be charged (discharged) if the process ran constantly at its initial
speed.
The time constant can be found graphically by constructing the tangent to the charging (discharging)
function at the beginning and extending it to the maximum (minimum) value.
Hans U. Fuchs 2010
9
Exponential function and time constants
The models of the examples shown lead to exponential functions if the capacitance and the resistance
are constants. The expression of the exponentially decaying function is
h ( t ) = h 0e − t τ
The expression for the rate of change of this function is given by
1
ḣ( t ) = − h0e− t τ
τ
Therefore, the slope at the beginning is – h0/τ which means that τ is the time constant.
The value of the function h(t) at t = τ is
1
h(τ ) = h0e−τ τ = h0 = 0.37h0
e
Therefore, we can find the time constant graphically by determining the 37%-point on the curve.
Remarks. The concept of time constant if related directly to the concept of half-life (of exponential decay).
Hans U. Fuchs 2010
10
Graphical determination (1)
1.0
Pressure / bar
0.8
0.6
0.4
37%
0.2
0.0
1
1.5
2
Time constant
2.5
3
Time / s
3.5
4
Remarks. To find the time constant of the decay, determine 37% of the maximum value of the pressure. Determine the
associated time, determine the time span for the decay from the maximum to 37%.
Hans U. Fuchs 2010
11
Graphical determination (2)
1.6
1.4
Level / m
1.2
Final level
Initial tangent
for h(t)
1
0.8
63% of
total change
0.6
0.4
0.2
Time constant
0
0
100
200
300
Time / s
400
500
600
Remarks. Determine the final level and then find the total change since the beginning. Calculate 63% of this total change
(63% = 100% – 37%). The value of the function for which 63% of the total change has been reached gives the time
constant.
Hans U. Fuchs 2010
12
Graphical determination (3)
5.0
4.0
Voltage / V
37% of 3 V
3.0
2.0
63% of 2 V
1.0
0.0
0
τC = 6.0 s
10
20
Time / s
30
40
Remarks. In equilibration of two levels, find the common level. Determine the change of the upper function to 37% of its
total change. Determine 63% of the total change of the lower curve. The time constants determined in this manner should
be the same.
Hans U. Fuchs 2010
13
Graphical determination (4)
Temperature / K
360
340
320
315
300
293
0
5000
11500
10000
Time / s
15000
20000
Remarks. To find the time constant, always find the total change of the function even if it is not shown. In the case shown
here we know that the temperature must decrease to the temperature of the environment (293 K).
Hans U. Fuchs 2010
14
Graphical determination (5): Logarithmic scale
Remarks. The time constant of the simple decaying exponential function h0exp(–t/τ) can be found by representing the
natural logarithm of the values (ln(h(t)) as a function of time. The slope of the resulting linear function equals the inverse of
the time constant.
Hans U. Fuchs 2010
15
Dependence of the time constant upon resistance and capacitance
The time constant depends in a simple manner upon the resistance and the capacitance of the resistor
and the capacitor of the simple circuit:
If we double the resistance, discharging takes twice as long.
if we double the capacitance, discharging takes twice as long.
This suggests that
τ = RC
Remarks. This can be proved easily by calculating the initial rate of change of discharging of a capacitor or of a tank, and
the relating the result to the time constant.
Hans U. Fuchs 2010
16