urrent onvention HEM*3440 hemical nstrumentation Topic 3 udimentary Electronics ONENTON: 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. onvention 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. harge Electrical charge is a fundamental property of nature. omes in two forms we identify as positive and negative. Measured in units of coulombs () and is commonly represented by Q or q. harge of an electron is.6022 x 0-9. There are 6.244 x 0 8 electrons in a coulomb of charge. n a mole of electrons there is 96485.3 of charge. This is called the Faraday constants. t is used to convert electrical measurements (like current) into chemical measurements (like moles). Potential Energy Matter 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. t has units of volts (). = J/ 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 harges An 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. n 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. harges move through this material by the displacement of ions rather than electrons. urrent ensity urrent flow per unit area: J = /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. urrent = Q/t Area A - - Electric Field, E Mobility urrent 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. urrent density then is µ = q m J = n q E µ onductivity urrent 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 (q n µ 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 ".! = q n µ q2 n2 µ2 q3 n3 µ3 urrent density is thus dependent on the applied electric field E and the material specific property of conductivity. J = E!
esistivity t is often easier to measure the inverse property of conductivity, called resistivity. t = v n this case, the current density is written as J = t E Like conductivity, resistivity is a property of matter that includes information about carrier number density, charge, and mobility. Ohm s Law Total current that flows through a conductor of cross section A is = JA= EvA The electric field is related to the potential difference by the length of material over which the field is applied. va = L = G va G = L For a specific piece of material, the physical properties and the piece!s geometry are combined to give a new property called conductance G. t has units of siemens. The reciprocal of conductance is resistance. Hence we have the relation = G = = This is Ohm!s Law. Power t takes energy to push a charge carrier through a resistor (anything which has a resistivity > 0). urrent 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. A omplete ircuit onsists of at least a voltage source and a load. urrent is the same at every point in the circuit. f ive current leaves a device through its ive terminal, energy is added to the circuit. f ive current enters a device through its ive terminal, energy is removed from the circuit. P = 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 oltage source (battery, ) Energy added to circuit. Load (resistor) Energy removed from circuit (gives off heat). P = 2 = 2 urrent flow, from positive to negative
Kirchhoff s urrent Law Two laws guide our analysis of fundamental circuits. First is Kirchhoff!s urrent Law - really conservation of charge. At any junction in a circuit, the sum of the currents must equal o. 2 A 0 At point B: 2 3 = 0 At point A: o = 0 B = -2-3 3 0 = - Kirchhoff s oltage Law Second law is Kirchhoff!s oltage Law - really conservation of energy. Around any complete loop in a circuit, the sum of the potential differences must be zero. is a voltage source. Positive current passes through it from negative to positive, entering the circuit through its positive terminal. t increases the energy of positive charges. 2 2 and 3 are resistors. urrent enters their positive terminals and exits their negative terminals; potential energy of positive charges decreases. 3 2 3 = 0 Passive ircuit Elements Passive circuit elements do not produce energy (though they can store energy). = - 2-3 Active ircuit Elements Active circuit elements add energy to a circuit. An A or power supply. esistors dissipate energy in a circuit. Batteries. apacitors store energy in an electric field. nductors store energy in a magnetic field. Photodiodes. Transistors.
Analyzing esistive ircuits Apply Ohm!s Law, Kirchhoff!s Laws, definitions of power, current, etc. to find unknowns. A 2 (a) n which direction does the current flow? (a) n which direction does the current flow? 0 49! 4! (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? 0 49! 4! (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? esistors: Series and Parallel ecall that resistor networks can be simplified for analysis by two simple observations: esistors in series behave as if they were a single resistor whose resistance value is the sum of the individual resistors. total = esistors in parallel behave as if they were a single resistor whose resistance value is the inverse of the inverse sum of the individual resistors. total / = / i total = / i n 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. i 0 oltage ivider ircuit k! 2 k! 3 k! A B / / j j! i j = total > H i The battery raises the potential energy of an electron at its positive terminal 0 above that at its negative terminal. Point A is 0 positive of point. (a) What is the potential difference between B and? (b) Between and? (c) Between A and?
0 urrent ivider ircuit 3 k! A B 2 k! k! 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? Potential at points A, B, and are identical and are held at 0 positive by the battery. Similarly the potential at points, 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. source oltmeter Most common instrument. Measures voltage difference. eally sets up a voltage divider network: every voltage source has some internal resistance and every voltmeter has some internal resistance. source meter meter meter meter = source c m meter source Accuracy of meter reading depends upon how close the resistor ratio term in parentheses is to. Must know the relative magnitude of the two internal resistances. Loading Error f 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. f it is too small, to draws current from the source - it loads the source. meter meter/source %Error 0 50 00 0 9. 000 00 0000 000 0. 0 6 0 5 0-5 Meter resistance must be several orders of magnitude greater than source resistance. Most meters have a high input resistance 0 2 to 0 4! to minimize such errors. 0 Wheatstone Bridge ircuit Easier to distinguish 0.0000 from 0.000 (requires precision of part in 0) than 2.0000 from 2.000 (requires precision of part in 0 5 ). Nulling circuits have inherently high precision. 2 A 3 4 B (A) = (B) (E) = (F) f = 3 and 2 = 4 then () = () and therefore (-) = 0 for a source resistance of 0 ohms E F
WB 2 Same bridge, drawn differently to emphasize points of common potential. 2 Wheatstone bridge measures difference in potential between points and by measuring the current that flows between them. When current is 0 (null condition) then potentials are equal. 3 4 an easily find the null condition for the Wheatstone bridge with the equations for a voltage divider. an show that = 3 2 4 nstrumentation Using Null Measurement 2 3 4 hoose 2 and 4 to be precision, known, fixed resistors. Let be a precision, adjustable resistor. Let 3 be a transducer whose resistance changes with a physical property. Transducer can be strain gauge, resistive temperature device (T), gas sensor, conductivity sensor, etc. Expose 3 to sample environment. Adjust until current (-) is 0 (the null condition). Knowing 2 and 4 and reading the value for we can find 3. alibrate system property to 3 resistance and the measurement is complete. nstrumentation Using Error Signal Finding the null condition requires careful adjustments. t is time consuming. A more modern approach is to get close but measure the resulting error signal. Expose 3 to a blank or a reference sample.adjust to achieve the null condition. Expose 3 to unknown samples, amplify and measure the current that flows because of being no longer in the null condition. elate this current to the sample property. 2 3 4 to current amplifier Time arying ircuits apacitors 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. urrent depends on rate of change of potential. apacitance (in units of farad) relates changing potential to current. Open circuit at condition. esistance decreases as frequency increases. i] t g dv] t g = dt 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. nductance (in units of henry) relates changing current to potential. losed circuit at condition. esistance increases as frequency increases. v] t g di] t g = L dt
ircuit Time onstant input output charge discharge switch input output lose switch to start charging the capacitor. esistor 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. Output oltage " (t= ") = 0.368 max Time For example, (t= 5") = 0.0067 max ntegration leads to equation for time evolution of circuit. Exponential growth or decay. Equation for discharge: output = max e - t x x = in ohms and in farads, then " is in seconds. input Time onstant con t L ircuit Output oltage " (t= ") = 0.632 max Time For example, (t= 5") = 0.9933 max The growth equation is output = max ^ - e - t x h The time constant indicates the time necessary for the output voltage to reach 63.2% of its final value. input output charge discharge switch input L output x = L in ohms and L in henrys, then " is in seconds. ifferent behaviour from the 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.
ircuit esponse Time All 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 eactance esistors, capacitors, and inductors tend to impede the current flow in a circuit. esistors dissipate energy: capacitors and inductors store energy. esistance () measures a resistor!s ability to impede current flow. esistance is independent of frequency. eactance (X) measures a capacitor!s or inductor!s ability to impede current flow. eactance is dependent on frequency. Equations which define reactance for a capacitor or and inductor are: This governs how quickly an instrument can make a measurement. X = 2rf X L = 2rfL mpedance The 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 = 2 X 2 mportant 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.