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Sensors Technology – MED4
ST09 – Capacitor
Capacitor
Lecturer:
Smilen Dimitrov
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ST09 – Capacitor
Introduction
•
The model that we introduced for ST
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ST09 – Capacitor
Introduction
•
We have discussed
– The units of voltage, current and resistance, from both a microscopic
and macroscopic (electric circuits) perspective
– The definition of an elementary electric circuit, Ohm’s law and Kirschoff
Laws
– Solving and measurement of voltage divider circuit and more
complicated circuits - and applications in sensors
– Resistive based sensors
•
This time we discuss the capacitor as an electronic element, and as a basis
for different types of sensors
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ST09 – Capacitor
Alternate and direct current (AC/DC)
•
Difference between direct and alternating current
– Direct current – potential difference (voltage) doesn’t change in time –
current goes always in the same direction
– Alternate current – voltage changes periodically in time – and so does the
direction of current
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ST09 – Capacitor
Alternate and direct current (AC/DC)
•
•
Enough to describe DC: U t   U DC
Describing AC: U t   U max sin t   U max sin 2ft 
DC - constant
AC – periodically alternating - sinusoidal
Signal – general (random) change in time
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ST09 – Capacitor
Alternate and direct current (AC/DC) - resistance
•
Resistance in context of AC – can be seen through an UI characteristic
Whereas the resistor behaves all the “same”
in AC or DC regimes, the capacitor doesn't - a
capacitor will in general not conduct DC
current, so it's influence becomes obvious
only in AC and signal regime.
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ST09 – Capacitor
Capacitor (condenser)
•
•
•
A capacitor, also known as condenser, is any system of two conducting
bodies separated by an insulator.
A capacitor is a device that stores energy in the electric field created
between a pair of conductors, on which equal but opposite electric charges
have been placed.
Electrons cannot directly pass across the dielectric from one plate of the
capacitor to the other. When there is a current through a capacitor, electrons
accumulate on one plate and electrons are removed from the other plate.
This process is commonly called 'charging' the capacitor even though the
capacitor is at all times electrically neutral.
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ST09 – Capacitor
Capacitor (condenser)
•
In fact, the current through the capacitor results in the separation rather than
the accumulation of electric charge. This separation of charge causes an
electric field to develop between the plates of the capacitor giving rise to
voltage across the plates. This voltage V is directly proportional to the
amount of charge separated Q.
C
•
Q
U
I
Q dQ

t
dt
I
dQ(t )
dU (t )
C
dt
dt
Here we can see that the capacitor is not an Ohmic element, which means
that it doesn’t follow Ohm’s law – since in Ohm’s law we have a linear
relationship between current and voltage.
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ST09 – Capacitor
Capacitor (condenser)
•
In a capacitor, the current through it is proportional to the change (first
derivative over time) of the voltage across it, which is not a linear
relationship. So, if the voltage changes, there is current - is the voltage is
constant, there is no current
– which is why a capacitor blocks the flow of DC current.
•
The charge on a capacitor cannot change instantaneously. The current is given by I
= DQ / Dt. Hence the change in charge DQ = I Dt goes to zero as the time interval Dt goes to zero .
•
The current flowing into a capacitor in the steady state that is reached after
a long time interval is zero. Since charge builds up on a capacitor rather than flowing through it,
charge can build up until the point that the voltage V=Q/C balances out the external voltage pushing charge onto
the capacitor.
•
“The current through the capacitor is said to lead the applied voltage”, or in a
capacitor, the voltage 'lags' behind the current: "in a capacitor the current
must first pile up electrons on the plates, for voltage to develop across
them."
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ST09 – Capacitor
Capacitor (condenser)
•
The capacitance is proportional to the surface area of the conducting plate
and inversely proportional to the distance between the plates. It is also
proportional to the permittivity of the dielectric (that is, non-conducting)
substance that separates the plates.
S
C 
d
vacuum
  0
materials
  0
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ST09 – Capacitor
Hydraulic analogy of a capacitor
•
Analogy to a water system
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ST09 – Capacitor
Capacitor as an electronic element
•
Different shapes and sizes
Electrolytic capacitors, are different
from the usual ones, by having a
polarity of their terminals - that is, a
certain way in which they have to
be placed in an electric circuit.
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ST09 – Capacitor
Measuring capacitors
•
Note that most common multimeters have functionality for measurement of
voltage (voltmeter), current (ampermeter) and resistance (ohmmeter).
Unfortunately, measurement tools that have a possibility to measure
capacitance of capacitors are a bit more rare.
•
With a normal multimeter, one can ensure that the capacitor gives infinite
resistance between its terminals - if this is not the case, then the capacitor is
malfunctioning.
•
Another possibility is to charge a capacitor (by connecting its terminals to a
terminal of a battery), then take the capacitor and measure if it still keeps the
voltage given by the generator; this however will be dependent on the
internal resistance of the instrument
•
With an oscilloscope - by using a known resistor and capturing the time
constant during charging
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ST09 – Capacitor
Capacitor - construction
•
We could approximate any capacitor to a system of parallel conducting
plates, separated by an insulator/dielectric.
–
•
To control the capacitance, one can then either control the cross-sectional area, the distance between the
plates, or the dielectric material placed between conducting plates that form the capacitor system.
Plain parallel plate design is sometimes followed in laboratory displays,
or in some certain capacitor designs
•
The concept of stacking of several plates on top of each other,
which effectively increases the total cross-sectional area
•
Use very thin foil to implement the conducting
plates and the dielectric,
and then to roll (wind) a wider sheet
into a compact capacitor
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ST09 – Capacitor
Capacitor (condenser)
•
Sometimes, the dielectric material can be recognized by the packaging
•
Variable capacitors –
also known as trimmers
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ST09 – Capacitor
Electrolytic (polarized) capacitors
•
There are also capacitors which are polarized - meaning that one of their terminals is
constantly meant to be at more positive potential than the other (that is, there are [+] and [-] terminals which cannot
switch roles)
•
Practical capacitors are often classified according to the material used as
the dielectric: bulk insulators and metal-oxide films (so-called electrolytic
capacitors)
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ST09 – Capacitor
Electrolytic (polarized) capacitors
•
Aluminum electrolytic capacitor... contain corrosive liquid and can burst if
the device is connected backwards. The aluminum oxide layer is held in place by the electric
field, and when reverse-biased, it dissolves into the electrolyte. This allows a short circuit between the electrolyte
and the aluminum. The liquid heats up and the capacitor may explode.
•
Tantalum capacitors ... are intolerant of voltage spikes and are destroyed
(often exploding violently) if connected backwards or exposed to spikes
above their voltage rating.
•
Since an electrolytic capacitor is polarity sensitive, its use is ordinarily
restricted to a dc circuit or to a circuit where a small ac voltage is
superimposed on a dc voltage.
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ST09 – Capacitor
Basic circuits
•
•
No linear relationship between voltage and current
The systems of equations that stem from the Kirchoff laws for a given circuit,
increase greatly in complexity
– cannot discuss proper design
•
Analysis goes through differential equation systems for general signals; or
complex number analysis for AC regime – both with own difficulties
•
Will just look briefly at basic circuits
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ST09 – Capacitor
Capacitor charging - RC series connection
•
•
Before time 0 – switch off – no current, no Vc Vc t 0   0
When the switch opens:
Cannot change instantly Vc t 0   Vc t 0   0
V t   Vc t 
dV (t )
I t   s
C c
R
dt
replacement
dVc (t )
dt

Vs  Vc t  RC
Exponent both sides
dZ (t )
dt


 Z (t )  RC
e ln Z t   e
obtaining Z t   Vc t   Vs  e


t
 const
RC
Capacitor voltage
t
t




RC
RC


Vc t   Vs  Vs  e
 V s 1  e



dVc (t )
dt

Vc t   Vs
RC
dZ (t )
dt

Z (t )
RC
Z t   Vc t   Vs
Integrate both sides
->
ln Z t   ln Vc t   Vs   
->
t
 const
RC
t
 const
RC
e
const
e

t
RC
 c0 e

t
RC
For t=0 c0  Vs
Resistor voltage
U R t   Vs  Vc t   Vs  e

t
RC
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ST09 – Capacitor
Capacitor (condenser)
•
The change of these voltages, Vc(t) and Ur(t), after the switch is applied,
can be usually visualised on time diagrams
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ST09 – Capacitor
AC complex number analysis
•
•
j
Use of imaginary number j   1 Euler formula e  cos   j sin 
Shorthand notation for sinusoidal signals only
f 
1
T
  2f
U (t )  U m  cost 
e jt  cost   j sin t 

U (t )  ReU   Re U m  e jt

•
jt
Complex representation U  U m  e
•
•
dQ
dU
d
C
 CU m e jt   jC U m e jt  jC U
Differentiation is now I 
dt
dt
dt
Similarity to Ohms law:
I
•
U
U
R
R
I
Zc  X c 
U
U
1
1


j
I
jC  U
jC
C
No more resistance (R) , but instead complex impedance Z
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ST09 – Capacitor
RC filters
•
RC connection has filtering properties:
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ST09 – Capacitor
Low-pass RC filter
•
Simplest RC low pass filter - a series RC connection, where the voltage
output is the capacitor voltage.
Input and
output signals
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ST09 – Capacitor
Low-pass RC filter
•
Filter characteristics – can be seen from transfer characteristics
j

complex transfer
Vo
C
 T ( j ) 
characteristic
j
Vi
R

(amplitude and phase)
C
real transfer
characteristic (only
amplitude)
T ( ) 
Vo
Vi

Vo

Vi
1
1  RC  
The cutoff frequency for
1
the circuits is the
fc 
frequency for which Vo
2RC
is 3 dB lesser than Vi
2
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ST09 – Capacitor
High pass RC filter - AC coupling
•
Simplest RC high pass filter - a series RC connection, where the voltage
output is the resistor voltage.
Input and
output signals
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ST09 – Capacitor
High pass RC filter - AC coupling
•
The cutoff frequency formula is the same as in the low pass filter,
1
fc 
2RC
– but the meaning (and the transfer characteristic) is different
•
•
This role of a high-pass filter has a special function when linking circuits this is known as AC capacitor coupling
The precise behaviour of a capacitor coupling is determined by its time
constant (RC). Note that the resistance (R) may be inside the next circuit
section rather than a separate resistor.
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ST09 – Capacitor
High pass RC filter - AC coupling
•
A coupling capacitor with a resistor voltage divider:
– For raising the “working point” when biasing a transistor
•
Adds a DC value (from the voltage divider) to an AC signal (that might have
a DC value of 0)
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ST09 – Capacitor
Series and parallel connection
•
Simplest RC high pass filter - a series RC connection, where the voltage
output is the resistor voltage.
C parallel  C1  C2  ...  C N
1
Cseries
1
1
1
 
 ... 
C1 C2
CN
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ST09 – Capacitor
Sensing application - Microphone and interfacing
•
When a sound wave falls on the microphone, force (pressure) is exerted on
it – hence the distance between the microphone capacitors plates will change
d  f (P)
C 
S
S

d
f (P)
•
we need to obtain voltage out of this
– hence the microphone capacitor
needs to be charged (commonly with
48 Volts, known as phantom power)
•
If Q = const.
U
Q

C
Q
Q

f ( P)
S

S

f ( P)
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ST09 – Capacitor
Electret microphone
•
An electret microphone is a relatively new type of condenser microphone,
which eliminates the need for a high-voltage bias supply by using a
permanently-charged material
•
Typical electret microphone preamp circuit
has a built in FET for amplification –
needs only a resistor (for biasing)
and a capacitor (for DC filtering)
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ST09 – Capacitor
Capacitive sensing
•
Touchpads – “the surface of a TouchPad sensor is an array of conductive
metal electrodes, covered by a protective insulating layer … when you place your
finger on a TouchPad, a tiny capacitance forms between your finger and the metal electrodes in the TouchPad.”
Usually needs a high
frequency signal to
operate
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ST09 – Capacitor
Capacitive sensing
•
Also the whole body can be seen as participating in capacitive sensing
Schematic of a WienBridge oscillator circuit can be used as a practical
capacitive sensor
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ST09 – Capacitor
Safety
•
Capacitors may retain a charge long after power is removed from a circuit;
this charge can cause shocks (sometimes fatal) or damage to connected
equipment.
•
Capacitors of any kind can explode or arc over if subjected to voltages
beyond their rating. This permanently ruins the capacitor.
•
If any of the above signs [of imminent failure] are noticed, turn off power
immediately and short capacitor from a distance - If it is not possible to
short from a distance, wait one full minute before going in to short it the old
way (with a heavy wire on a stick).
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