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ST09 – Capacitor

Capacitor

Lecturer:Smilen Dimitrov

Sensors Technology – MED4

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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:• Describing AC:

DCUtU

ftUtUtU 2sinsin maxmax

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.

• 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.

U

QC

dt

dQ

t

QI

dt

tdUC

dt

tdQI

)()(

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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.

d

SC

vacuum

materials

0

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• When the switch opens:

00 tVc

000 tVtV cc

dt

tdVC

R

tVtVtI ccs )(

RC

dt

tVV

tdV

cs

c

)(->

RC

dt

VtV

tdV

sc

c )(

sc VtVtZ RC

dt

tZ

tdZ

)(

)(

Cannot change instantly

replacement

Integrate both sides RC

dt

tZ

tdZ

)(

)(-> const

RC

tVtVtZ sc lnln

Exponent both sides constRC

ttZ ee

ln

RC

t

RC

tconst

constRC

t

sc eceeeVtVtZ

0obtaining sVc 0For t=0

RC

t

sRC

t

ssc eVeVVtV 1

Capacitor voltage Resistor voltage

RC

t

scsR eVtVVtU

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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

• Use of imaginary number Euler formula• Shorthand notation for sinusoidal signals only

• Complex representation

• Differentiation is now• Similarity to Ohms law:

• No more resistance (R) , but instead complex impedance Z

1j sincos je j

tUtU m cos)( f 2T

f1

tjte tj sincos

tjm eUU tjm eUUtU ReRe)(

UCjeUCjedt

dCU

dt

UdC

dt

QdI tj

mtj

m

I

UR

R

UI

Cj

CjUCj

U

I

UXZ cc

11

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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

C

jR

C

j

jTV

V

i

o

)(

21

1)(

RCV

V

V

VT

i

o

i

o

complex transfer characteristic (amplitude and phase)real transfer characteristic (only amplitude)

The cutoff frequency for the circuits is the frequency for which Vo is 3 dB lesser than Vi

RCf c 2

1

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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,

– 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.

RCf c 2

1

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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.

Nparallel CCCC ...21Nseries CCCC

1...

111

21

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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

• 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.

)(Pfd )(Pf

S

d

SC

)(

)(

PfS

Q

Pf

SQ

C

QU

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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 circuithas 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 Wien-Bridge 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).