CHEM*3440Chemical Instrumentation

Topic 3

Rudimentary Electronics

Current ConventionCONVENTION: Electrical current flows from a region of positive potential energy to a region of more negative (or less positive) potential energy.

Truth is that electron flow is electrical current. They are negative and so flow

from negative to positive.

Convention is historical.

All circuit behaviour can be correctly analyzed based on this convention.

Even works when true circuit may consist of mixed carriers (positive ioins

and negative electrons flowing in opposite directions.

ChargeElectrical charge is a fundamental property of nature. Comes in two forms

we identify as “positive” and “negative”.

Measured in units of coulombs (C) and is commonly represented by Q or q.

Charge of an electron is 1.6022 x 10-19 C.

There are 6.2414 x 1018 electrons in a coulomb of charge.

In a mole of electrons there is 96485.31 C of charge. This is called the

Faraday constants. It is used to convert electrical measurements (like

current) into chemical measurements (like moles).

Potential EnergyMatter likes to be electrically neutral. Separating a positive charge from a

negative charge requires work (energy). Bringing together two like charges

requires energy also.

The potential difference between two such particles is called the electrical

potential difference or the electromotive force (emf).

Potential energy difference is measured in joules of energy needed per

coulomb of charge separation. It has units of volts (V).

1 V = 1 J/C

The space between two separated charges is said to be filled by an electric

field. The emf is said to change continuously when moving from one charge

towards the other.

Mobile ChargesAn ion is formed when an electron is removed (cation) from or added (anion)

to a neutral atom or molecule.

Matter is held together by the attractive force between oppositely charged

particles. This is done by the sharing of the outermost (valence) electrons

between atoms.

In metals, the valence electrons are shared by so many other atoms, they

are essentially free to move through the whole sample.

Other materials share their electrons very specifically and they cannot move

through the whole material. They are insulators.

Other materials donate electrons completely to a neighbour and the material

consists of ions. Charges move through this material by the displacement of

ions rather than electrons.

Current DensityCurrent flow per unit area: J = I/A = Q/tA

Positive charge carriers which are mobile move in the direction of the

imposed electric field E. Mobile negative carriers move in the opposite

direction.

Both contribute to the current density J, the rate of charge motion per unit

area.

Electric Field, E

Current I = Q/t

Area A

+

--

+

MobilityCurrent density depends on number density of carriers n, charge on each

particle q, and the velocity of the carriers, v.

J = n q v

The velocity depends upon the particle!s charge q, the strength of the

electric field E driving it, and various physical aspects peculiar for each

material, collected under the variable m.

v = q E m

Generally one collects the particle!s properties together to a single variable

called mobility µ.

µ = q m

Current density then is

J = n q E µ

ConductivityCurrent density arises from the sum of the current density of all possible

types of charge carriers. Each carrier will have a particular charge q, a

certain mobility µ, and a given number density n.

J = E (q1 n1 µ1 + q2 n2 µ2 + q3 n3 µ3 + …)

The terms in parentheses are unique to a certain material under specific

conditions (temperature, preparation history, exposure to light, etc.). This

property is called the conductivity of the material and is given the symbol ".

! = q1 n1 µ1 + q2 n2 µ2 + q3 n3 µ3 + …

Current density is thus dependent on the applied electric field E and the

material specific property of conductivity.

J = E !

ResistivityIt is often easier to measure the inverse property of conductivity, called

resistivity.

t= v1

In this case, the current density is written as

J= tE

Like conductivity, resistivity is a property of matter that includes information

about carrier number density, charge, and mobility.

Ohm’s LawTotal current that flows through a conductor of cross section A is

I = JA=EvA

The electric field is related to the potential difference V by the length of

material over which the field is applied.

I = LVvA

= VG G= LvA

For a specific piece of material, the physical properties and the piece!s

geometry are combined to give a new property called conductance G. It has

units of siemens.

The reciprocal of conductance is resistance R. Hence we have the relation

I= GV= RV

V= IR

This is Ohm!s Law.

PowerIt takes energy to push a charge carrier through a resistor (anything which

has a resistivity > 0). Current flow is resisted because the carrier bumps into

the atoms and defects of the substance. The energy shows up as heat in

the resistor. The rate at which energy is dissipated in the resister is power P.

P = I V

Power is measured in units of Watts which is Joules/second. As energy is

dissipated in a resistor, it will heat up. The ability of a resistor to shed this

energy determines its power rating.

Alternate forms of the expression for power is

P= I2R= RV2

A Complete CircuitConsists of at least a voltage source and a load.Current is the same at every point in the circuit.If +ive current leaves a device through its +ive terminal, energy is added to the circuit.If +ive current enters a device through its +ive terminal, energy is removed from the circuit.

+ +

– –

Current flow, from positive to negative

Load (resistor)

Energy removed

from circuit

(gives off heat).

Voltage source

(battery, …)

Energy added

to circuit.

I

V R

Kirchhoff’s Current LawTwo laws guide our analysis of fundamental DC circuits.

First is Kirchhoff!s Current Law - really conservation of charge.

At any junction in a circuit, the sum of the currents must equal o.

A

BI0 I1

I2

I3At point A: Io + I1 = 0

I0 = -I1

At point B: I1 + I2 + I3 = 0

I1 = -I2 - I3

Kirchhoff’s Voltage LawSecond law is Kirchhoff!s Voltage Law - really conservation of energy.

Around any complete loop in a circuit, the sum of the potential differences must be zero.

+ +

– –

I

V R

I

– +R

V1 V2

V3

V1 is a voltage source. Positive

current passes through it from

negative to positive, entering

the circuit through its positive

terminal. It increases the

energy of positive charges.

V2 and V3 are resistors.

Current enters their positive

terminals and exits their

negative terminals; potential

energy of positive charges

decreases.

V1 + V2 + V3 = 0 V1 = - V2 - V3

Passive Circuit ElementsPassive circuit elements do not produce energy (though they can store energy).

Resistors dissipate

energy in a circuit.

Capacitors store

energy in an

electric field.

Inductors store

energy in a

magnetic field.

Active Circuit ElementsActive circuit elements add energy to a circuit.

An AC or DC

power supply.

Batteries.

Transistors.

Photodiodes.

Analyzing Resistive CircuitsApply Ohm!s Law, Kirchhoff!s Laws, definitions of power, current, etc. to find unknowns.

+

–

10 V14 !

49 !

(a) In which direction does the

current flow?

(b) What is the magnitude of the

current?

(c) How much power is

dissipated in each resistor?

(d) How much power does the

battery supply to the circuit?

ARC 2

(a) In which direction does the

current flow?

(b) What is the magnitude of the

current?

(c) How much power is

dissipated in each resistor?

(d) How much power does the

battery supply to the circuit?

+

–

10 V14 !49 !

Resistors: Series and ParallelRecall that resistor networks can be simplified for analysis by two simple

observations:

Resistors in series behave as if they were a single resistor whose resistance value is the sum of the individual resistors.

R total = Ri/

Resistors in parallel behave as if they were a single resistor whose resistance value is the inverse of the inverse sum of the individual resistors.

R total1= Ri

1/ R total =

Ri1/

1

In a series network, the total resistance is larger than any one resistor in it; in

a parallel network, the total resistance is smaller than any one resistor in it.

Voltage Divider CircuitThe battery raises the

potential energy of an

electron at its positive

terminal 10 V above that

at its negative terminal.

Point A is 10 V positive

of point D.

(a) What is the potential

difference between B

and D?

(b) Between C and D?

(c) Between A and C?

+

–

10 V

1 k!

2 k!

3 k!

A

B

C

D

V j= VtotalRi/

R jj! i

/> H

Current Divider CircuitPotential at points A, B, and

C are identical and are held

at 10V positive by the

battery. Similarly the

potential at points D, E, and

F are identical.

Hence, the same potential

drop is across all three

resistors; Ohm!s Law gives

the current flowing in each

one.

The current from the battery

is divided into the three legs

of the circuit.

+

–

10 V1 k!2 k!3 k!

A B C

D E F

(a) What is the current through each resistor?

(b) What is the current through the voltage source?

(c) What is the power in each circuit element?

VoltmeterMost common instrument. Measures voltage difference. Really sets up a

voltage divider network: every voltage source has some internal resistance

and every voltmeter has some internal resistance.

+

–

Vsource

Rsource

Rmeter

Vmeter

Vmeter = Vsource Rmeter + RsourceRmeterc m

Accuracy of meter reading depends upon

how close the resistor ratio term in

parentheses is to 1. Must know the

relative magnitude of the two internal

resistances.

Loading ErrorIf the source resistance is an appreciable fraction of the meter resistance,

the voltage measurement will be in error. Meter resistance must be very

large compared to source resistance. If it is too small, to draws current from

the source - it “loads” the source.

Meter resistance must

be several orders of

magnitude greater

than source

resistance.

Most meters have a

high input resistance

1012 to 1014 ! to

minimize such errors.

Rmeter Rmeter/Rsource %Error

10 1 50

100 10 9.1

1000 100 1

10000 1000 0.1

106 105 10-5

for a source resistance of 10 ohms

Wheatstone Bridge CircuitEasier to distinguish 0.0000 V from 0.0001 V (requires precision of 1 part in

10) than 2.0000 V from 2.0001 V (requires precision of 1 part in 105).

Nulling circuits have inherently high precision.

V(A) = V(B)

V(E) = V(F)

If R1 = R3 and R2 = R4 then

V(C) = V(D)

and therefore I(C-D) = 0

+

–

10 V

R4R2

R1 R3

A B

C D

EF

WBC 2Same bridge, drawn differently to emphasize points of common potential.

+

–

R4R2

R1 R3

C DI

Wheatstone bridge measures

difference in potential between

points C and D by measuring

the current that flows between

them. When current is 0 (null

condition) then potentials are

equal.

Can easily find the null condition

for the Wheatstone bridge with

the equations for a voltage

divider. Can show that

R2R1 = R4

R3

Instrumentation Using Null Measurement

Choose R2 and R4 to be

precision, known, fixed

resistors. Let R1 be a

precision, adjustable resistor.

Let R3 be a transducer whose

resistance changes with a

physical property.

Transducer can be strain

gauge, resistive temperature

device (RTD), gas sensor,

conductivity sensor, etc.

+

–

R4R2

R1 R3

C DI

Expose R3 to sample environment. Adjust R1 until current I(C-D) is 0 (the null

condition). Knowing R2 and R4 and reading the value for R1 we can find R3.

Calibrate system property to R3 resistance and the measurement is complete.

Instrumentation Using Error SignalFinding the null condition requires

careful adjustments. It is time

consuming. A more modern

approach is to get close but

measure the resulting error

signal.

Expose R3 to a blank or a

reference sample.Adjust R1 to

achieve the null condition.

Expose R3 to unknown samples,

amplify and measure the current

that flows because of being no

longer in the null condition.

Relate this current to the sample

property.

+

–

R4R2

R1 R3

C DI

to current amplifier

Time Varying CircuitsCapacitors and inductors store and release energy; leads to time varying

potentials and currents in circuits.

Two conductive sheets, separated by

an insulator.

Stores energy in an electric field.

Current depends on rate of change of

potential.

Capacitance (in units of farad) relates

changing potential to current.

Open circuit at DC condition.

“Resistance” decreases as frequency

increases.

A wire wrapped into a coil, often around

a ferrite core.

Stores energy in a magnetic field.

Potential depends on rate of change of

current.

Inductance (in units of henry) relates

changing current to potential.

Closed circuit at DC condition.

“Resistance” increases as frequency

increases.

i t] g= C dtdv t] g

v t] g= L dtdi t] g

RC CircuitClose switch to start

charging the capacitor.

Resistor limits the current

and hence the rate at which

the capacitor can

accumulate charge.

Potential across capacitor

increases until it reaches

potential of voltage source.

Move switch to discharge

position; current flows out of

capacitor, limited by the

resistor, until no voltage

appears across resistor.

+

–

V

R

C

charge

discharge

switch

Vinput

Voutput

Vin

pu

t

Vinput

Vou

tpu

tRC Time Constant

Integration leads to

equation for time

evolution of RC circuit.

Exponential growth or

decay.

Equation for discharge:

Voutput = Vmax e-tx

x= RC

R in ohms and C in

farads, then " is in

seconds.

Time

Ou

tpu

t V

olta

ge

"

V(t= ") = 0.368 Vmax

For example, V(t= 5") = 0.0067 Vmax

RC Time Constant con’tThe growth equation is

Voutput = Vmax 1 - e-tx^ h

Time

Ou

tpu

t V

olta

ge

"

V(t= ") = 0.632 Vmax

For example, V(t= 5") = 0.9933 Vmax

The time constant

indicates the time

necessary for the

output voltage to reach

63.2% of its final value.

RL CircuitDifferent behaviour

from the RC circuit. At

end of charging

process, the current

through the device is

at a maximum but the

potential across the

inductor is 0, and the

stored energy in the

magnetic field is

maximum. Upon

discharging, this

energy is released as

a decreasing current

flow in the circuit.

+

–

V

R

L

charge

discharge

switch

VinputVoutput

Vin

pu

tV

ou

tpu

t R in ohms and L in

henrys, then " is in

seconds.

x= LR

Circuit Response TimeAll circuits have some resistance, some capacitance, and some inductance.

Therefore, all circuits have a non-zero response time; when there is a change in

the input voltage, there is a response time before the output voltage reflects the

change.

All measuring instruments are governed by this effect, known by different names:

time constant

response time

rise time

This governs how quickly an instrument can make a measurement.

ReactanceResistors, capacitors, and inductors tend to impede the current flow in a

circuit.

Resistors dissipate energy: capacitors and inductors store energy.

Resistance (R) measures a resistor!s ability to impede current flow.

Resistance is independent of frequency.

Reactance (X) measures a capacitor!s or inductor!s ability to impede current

flow. Reactance is dependent on frequency.

Equations which define reactance for a capacitor or and inductor are:

XC = 2rfC1

XL = 2rfL

ImpedanceThe overall impedance in a circuit is a result of the cumulative effect of the

circuit!s resistance and reactance. This impedance has the symbol Z.

Z= R2 +X2

Important phase relationships arise because the reactance is out of phase

with the driving potential while the resistance is in phase. This needs to be

understood for a thorough circuit analysis, but we will go no further in this

class.