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Revision Notes Electron Devices Electron Drift Velocity Vd n Where mu-sub-n is the mobility of electrons (defined for the material) and epsilon is the electric field. Contact Potential Equation: kT N a N d ln q ni2 Where V0 is contact potential kT/q is 25mV (at 300K) Na is concentration of acceptors (holes) on p-side Nd is concentration of donors (electrons) on n-side ni is the intrinsic carrier concentration. V0 Fermi-Dirac Distribution function (first term’s work) f (E) 1 E Ef 1 exp kT Diode Equation qV I D I 0 exp D 1 kT Diode Capacitance Cj W Junction capacitance, where W is the width of the depletion/transition region and epsilon is the permittivity. This is dominant during reverse bias. qV C S e kT Storage capacitance, varies with forward bias voltage. Normally dominant in forward bias, and used in voltage-variable capacitors (varicap diodes). Page 1 of 16 Communications (Tim Tozer) LCR parallel circuit: 0 Q 1 LC f R 0 L BW Amplitude Modulation e A0 (1 m sin( mt )) sin( c t ) e = modulated signal m = modulation index (between 0 and 1, usually 0.3) Nyquist’s sampling theorem Sampling frequency = 2 x maximum frequency Quantisation noise power (v) 2 Pqn 12 Delta-v is the step size. Thermal noise power N kTB N= noise power in watts k = Boltzmann’s Constant = 1.38 x 10-23 J/K = -229 dB J/K T = noise temperature in Kelvin B = bandwidth in Hz This equation does assume a precisely matched load. Page 2 of 16 Maths (Ken Todd- term 2) Total Differential dz z dx z dy dt x dt y dt Where z is a function of x and y and x and y are functions of t. Change of Variables If z is a function of x and y, and x and y are both functions of u and v, then: z z x z y u x u y u And, similarly z z x z y v x v y v Taylor’s Series in two variables f ( x, y ) f (a, b) Df (a, b) D 2 f ( a, b) D 3 f ( a, b) ... 2! 3! k x y h xa Dh k y b Stephenson’s method Treat h and k as constants, to get the series in terms of h and k Substitute h and k for a,b,x & y. Page 3 of 16 Finding Stationary points At a stationary point: f 0 x f 0 y To check for a saddlepoint, use the Del operator: 2 2 f 2 f 2 f 2 2 x y xy If >0 then you have a saddlepoint. Otherwise, check the partials for a maximum or minimum. Double Integrals A Double Integral is the volume under a surface, as defined by four limits. x b xa dx y d y c f ( x, y )dy Page 4 of 16 Static Fields Electrostatics: Coulomb’s Law F Q1Q2 4r 2 Electric Field due to a point charge E Q 4r 2 Electric Field due to a line charge 2r Rho is charge density in Cm-1 E Electric Field due to an infinite sheet of charge 2 Rho is charge density in Cm-2 E P.D. between two points V Ed Where E is electric field strength and d is distance Capacitance of two sheets C A d where A is the area of the sheets and d is their separation. Energy stored in a capacitor: E 12 CV 2 Electric Flux Density: D E Page 5 of 16 Displacement current density: JD dD dt Conduction current density: J dQ dt Generalised form of Ohm’s law: J E where sigma is conductivity. Magnetic fields: Biot-Savart Law: Il rˆ B 4r 2 Faraday’s Law: V d Bd A dt Inductors Total flux in an inductor (in Webers) Bds kNI where k is a geometric constant. Voltage induced in inductor: V N d dI kN 2 dt dt Self-inductance (in Henrys) L kN 2 V L dI dt Page 6 of 16 Transistors Diode equation (approximate form) qV I D I S exp D kT Small-signal diode resistance 25mV ID ID is the DC bias current. rD Transistor large-signal behaviour: IC 0 I B Transistor small-signal behaviour: IE 25mV 1 re gm gm r 1 re Common-emitter amplifier: Voltage gain Av g m RC Input resistance rin RB1 // RB 2 // r Output resistance rout RC Overall gain of CE amplifier with source and load resistance: RL rin Av VL VS (rs Rin )( ro RL ) Page 7 of 16 Common-collector amplifier Voltage gain (just less than unity) g m RE Av 1 1 g m RE Input resistance (very high) rin rT // RB1 // RB 2 rT (1 g m RE )r Output resistance (very low) ro RE // r // re re Bear in mind that, in a CC amplifier, changing the load resistance changes the input resistance of the amplifier. Changing the source resistance changes the output resistance. Common-base amplifier: AV g m RC Very low input impedance and moderate output impedance rin r // re // RE rout Rc Amplifier characteristic summary (ignoring source and load resistances): Configuration Voltage gain Current gain CE CC Av -gmRC 1 AI CB gm R C 1 Input resistance Output resistance rin rout RC r // RBias (re+RE) // re RBias RC re Onset of clipping in single transistor amplifiers For CE and CB amplifiers: Clipping on the positive half-cycle occurs when the device switches off completely and the voltage across RC is reduced to zero. Clipping on the negative half-cycle occurs when VCE is less than 0.7V and the transistor saturates. To get maximum output swing, bias should make VC halfway between VB and VCC. Page 8 of 16 For CC amplifiers: The output always clips first on the negative half-cycle. Distortion occurs when the bias current is equal to the current into RE // RL The bias current must be large enough to handle the largest desired voltage swing. Miller’s Effect This limits the high-frequency behaviour of transistor amplifiers. Effective capacitance of transistor: Ceff C (1 g m RC )CC RC RC // RL High-frequency cut-off point (-3dB point): 1 fC 2RS Ceff Myles’s Transistor Amplifier design guidelines: Maximum gain from a single-stage amplifier (split-supply): gain MAX 20VCC VCC then you need a CC output stage 2I E 25 Source impedance: if RS where IE is in mA, then use a CC input stage. IE Load impedance: if RL Field Effect Transistors FETs are transconductance devices- the gate voltage controls the source-drain current. MOSFETs (Metal-oxide-semiconductor field effect transistors) come in two types, each of which comes in two polarities: N-channel MOSFETs conduct solely by electrons. P-channel MOSFETs (less common) conduct solely by holes. Enhancement-mode MOSFETs have no built-in channel. The channel is induced by the presence of a positive gate voltage. The threshold voltage VT is positive. With no gate voltage, the device is switched off (normally-off). Depletion-mode MOSFETs are the reverse. The channel is built into the silicon and is controlled by the presence of a negative gate voltage. The threshold voltage is negative. With no gate voltage, the device is switched on (normally-on). To turn the device off, apply a negative gate voltage. Page 9 of 16 MOS Capacitors Assuming p-type silicon substrate: Making the metal negative relative to the silicon causes accumulation: Holes are attracted to region under gate Bands in silicon bend closer to valence band (gets more p-type) Making the metal positive relative to the silicon causes depletion Holes at surface of semiconductor are repelled, causing a depletion region to form. Bands in silicon bend closer to conduction band (gets less p-type) Making the metal more postive causes inversion: Energy bands bend further down Eventually the bands bend enough to make the surface appear locally n-type. Strong inversion occurs when the surface is as n-type as the bulk is p-type. When strong inversion occurs, the device has enough charge carriers to conduct usefully. This point is termed the threshold voltage (VT) MOS FET DC behaviour Drain Current 1 2 I D VGS VT VDS VDS 2 W C L Where: ID is the drain current is the device constant (or gain constant), normally specified in questions. VGS is the gate-source voltage VT is the threshold voltage (normally defined by manufacturer). VDS is the drain-source voltage. W is the channel width L is the channel length (source-drain distance) is the mobility C is the gate capacitance (varies with oxide thickness) Drain Current at saturation 1 (VGS VT ) 2 2 This is the region used for small-signal amplifiers. I D ( SAT ) Page 10 of 16 MOS FET small-signal AC behaviour Small signal transconductance g m (VGS VT ) This equation holds when the device is saturated i.e. VDS>VGS-VT When biasing FETs, remember that the gate current is zero. Common-source amplifier Used as a gain stage where very high input impedance is required. Frequently used as differential input stages in opamps. Small-signal gain: AV g m RD Input impedance: Z in RG RG may be one resistor (depletion-mode) or two in parallel (enhancement mode) Output impedance: Z out RD Common-drain amplifier Used as a voltage buffer. Near unity gain with very high input impedance and very low output impedance. Usually found at input stages to instrumentation (voltmeters, scopes) Small-signal gain: AV 1 Input impedance: Z in RG Output impedance: Z out RS 1 g m RS Common-gate amplifier Gain stage with low input impedance- usually used for video or RF systems requiring 50 or 75 ohm input impedance. Also used as current amplifiers. Small-signal gain: AV g m RD Page 11 of 16 Input impedance: Z in RS 1 g m RS Output impedance: Z out RD NMOS Logic circuits NMOS inverter threshold equation Z pu Z pd 2 VTd (Vinv VTe ) 2 Zpu is the aspect ratio (length-to-width ratio) of the pull-up device. Zpd is the aspect ratio of the pull-down device. VTd is the threshold voltage of the depletion mode device (pull-up) VTe is the threshold voltage of the enhancement mode device (pull-down) Vinv is the cross-over voltage of the inverter (the point at which it changes state). The equation gives the ratio of the aspect ratios of the devices. Transmission Lines Velocity of wave on line velocity 1 LC Characteristic impedance Z0 L C Voltage Reflection coefficient V Z L Z0 Z L Z0 Voltage Transmission coefficient 1 V Standing waves: If you have sinusoidal excitation and v = 1, then the result is a standing wave: V 2V0 cos( x)e jt Page 12 of 16 Reactive behaviour of mismatched transmission line: Short-circuited line of length l: l Z in jZ 0 tan velocity 1 velocity LC (quarter-wave system) 4 Zero reactance (short circuit) occurs when l (half-wave system) 2 Infinite reactance occurs when l Input impedance of a Tx line with arbitrary termination: Z cos( l ) jZ0 sin( l ) Z in Z 0 L Z 0 cos( l ) jZ L sin( l ) (half wavelength) then Zin = ZL, so if the tx line is an integer number of half2 wavelengths then Zin is independent of Z0. If l If l Z2 (quarter wavelength) then Z in 0 4 ZL Rearranging: Z 0 Z in Z L This is used for quarter-wave impedance matching: a quarter-wave length of transmission line of a specified characteristic impedance can act as a matching piece: If you have a 50 source and a 100 load, then Z 0 50 100 72 . Thus a quarterwave length of transmission line of 72 will match a 50 system to a 100 system, at the specified frequency. Page 13 of 16 Transmission Lines & Static Fields (term 3) Gauss’s Law: Q E ds This boils down to the fact that the total field radiated by a number of point charges inside an enclosing surface is equal to the sum of all the charges enclosed divided by the permittivity. Ampère’s Law loop H dl I A number of currents are surrounded by a loop. The total magnetic field on the whole loop is equal to the total current passing through it. Field Analysis of Coaxial cables Capacitance per unit length 2 b ln a b is the radius of the cable outer. a is the radius of the cable inner. is the permittivity of the dielectric. C Inductance per unit length L b ln 2 a Propagation Velocity V 1 Characteristic Impedance Z0 1 2 b ln a Page 14 of 16 Intrinsic impedance This is the ratio of electric to magnetic fields in the cable: Zi Field Analysis of Parallel Plate Transmission Line Capacitance per unit length C w d w is the width of the plates d is the plate separation is the permittivity of the dielectric. Inductance per unit length L d w Propagation Velocity V 1 Characteristic Impedance Z0 d w Page 15 of 16 Circuits LC Resonant circuits Resonant frequency 0 1 LC Q-factor (Series LC) Q 0L R 1 0 CR Q-factor (Parallel LC) Q R 0 CR 0 L Real, Reactive and Apparent Power Apparent Power PApparent Vrms I rms Apparent Power is measured in Volt-Amps (VA) Real Power PRe al Vrms I rms cos Reactive Power PRe active Vrms I rms sin Page 16 of 16