Download Engineering Science EAB_S_127_Ch3

Engineering Science EAB_S_127
Electricity Chapter 3 & 4
Overview
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Measurement of electrical resistance
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The Wheatstone Bridge
Capacitance
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Energy stored in a capacitor
Charging and Discharging through a resistor
Time constants
The Wheatstone Bridge
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We use an “Ohmmeter” to measure an unknown resistance
The heart of the simplest Ohmmeter is a so-called
“Wheatstone Bridge” circuit
If R1 was a variable resistor, we can adjust it until Vab = 0
The Balanced Wheatstone Bridge

When Vab = 0, a special condition occurs: the bridge is said
to be “balanced”, i.e. Va = Vb

This implies that ig = 0, hence from KCL, i4 = i3 and i2 = i1
Further, from Ohm’s Law; i4R4 = i2R2 and i3R3 = i1R1

The Wheatstone Bridge continued

Hence
i1 R1 i3 R3

i2 R2 i4 R4
R3
R1 
R2
R4
The Wheatstone Bridge: Example

Calculate R1 in a Wheatstone bridge when it is balanced
and when R2 = 300Ω, R3 = 200Ω, R4 = 100Ω .
R3
R1 
R2
R4

Answer:
R3
200
R1 
R2 
300  600
R4
100
Unbalanced Wheatstone Bridge
The following unbalanced Wheatstone Bridge is constructed.
Calculate the output voltage across points C and D
For the first series arm, ACB
The voltage across points C-D is
given as
For the second series arm, ADB
Capacitors
Type of Capacitors
Applications of capacitors
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Energy storage
Pulsed power and weapons
Power conditioning
Motor starters
Signal processing
Capacitance
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Capacitors are devices which store electrical charge
A capacitor consists of two plates separated by an
insulator, as shown in Figure 4.1
The negative plate is connected to a low potential and
the positive plate to a high potential
Insulator
Q
-
+
-
+
-
+
V
Negative plate
Figure 4.1
Positive plate
Capacitance continued
The total amount of the charge stored, is denoted by Q
and the voltage across the plates by V
Q
 The capacitance then is defined as
C
[F ]
V
 Where C is in Farads
 1 Farad = 1 Coulomb per Volt
 1mF, µF, nF (nano), pF(pico)
 0.1 pF is the smallest available in capacitors
Insulator
Q
for general use in electronic design

+
-
+
-
+
V
Negative
plate
Figure 4.1
Positive
plate
Energy Stored in a Capacitor
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When charged, a capacitor stores electrical energy
Recall the formula for electrical energy in a circuit, which
is W = VQ
However, we need to be careful as the voltage between
the plates in a capacitor varies from 0 to V
Hence, to be more accurate we should use the average
0  Vab Vab
voltage
Vm 
W  VmQ 

So

Hence W 
Vab
Q
2
2

2
and we know
Vab
1
CVab  CVab2
2
2
Q  CVab
Energy Stored in a Capacitor: Example


Question: A capacitor is supplied with 10 V in a circuit. If
its capacitance is 150µF, what is the electrical energy
stored in the capacitor?
Answer:
W
1
1
CVab2  150 10 6 10 2  75 10  4 J  7.5mJ
2
2
Charging and discharging a capacitor
Charging and discharging a capacitor
Charging and Discharging a Capacitor

Charging and discharging a capacitor from a DC (direct
current) source is shown below
V


We assume that the voltage source,V, has no internal
resistance
If the switch was held in position 2 for a long time, then the
capacitor would be completely discharged, Vc = 0V
Charging a Capacitor


If the switch is then moved to position 1, current will start
to flow through the resistor R, thereby charging the
capacitor, C
The voltage across the plates of the capacitor will rise in
time, until after a long time, the capacitor will have the same
voltage as the supply,V
V
VC
Discharging a Capacitor

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If the switch is then moved back to position 2, current will
start to flow through the resistor R, thereby discharging
the capacitor, C
The voltage across the plates of the capacitor will fall in
time, until after a long time, the capacitor will have no
charge at all and again, Vc = 0V
V
VC
Time Constant of an RC Circuit
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It can be shown mathematically, that the time for the
voltage to fall to 37% of its original voltage, t = RC
The charging and discharging curves have an exponential
nature
When discharging
V
C

When charging
V
C
RC Time Constant: Example
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Question: If R = 1000 and C = 0.1mF, what is the time
constant of the circuit?
Answer: t = RC = 1000x0.1x10-6 = 0.1 x10-3 = 100ms
Hence, when discharging, the following equation can be
used to calculate the voltage
When charging
Summary
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Learning Outcomes:
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Wheatstone Bridge
Balanced Condition
Capacitors and capacitance
Energy stored in a capacitor
Charging a capacitor
Time constants
Exponential charging and discharging curves