1• CAPACITORS AND INDUCTORS• KIRCHOFF'S LAWS• RC CIRCUITS• LR CIRCUITS• RC INTEGRATOR AND DIFFERENTIATORChapter 2 - CAPACITORS AND INDUCTORS2A dielectric is polarizeable material that strengthens the capacitance by attracting more charge to the plates. κ = dielectric strength CAPACITORS- Charge Storage devicesC=εοA/dC=κεοA/dκεc= -q/C or q=CVE+++++------+-+-+-+-+++++++++++---------- !V = "!Eid!l !ab#!V = "E dlab#!V = "E !d3Electrolytic Capacitors+paper soakedelectrolyteInsulating Dieletric OxideLayer K=10-+++++Metal electrodes•High capacitance but unipolar!•When voltage is applied a thin oxidelayer (d~50µm) forms on the + electrode.The capacitance C=εA/d is thus verylarge.•If the polarity is reversed for a length oftime the insulating oxide layer isdamaged and the capacitor will conductcurrent and heat.•Either leaking or exploding may result.•Attach + side to more positive side ofcircuit.• Designer must keep +side at higher DCpotential.• Some electrolytics allow AC!+ions41/C =1/C1+1/C2Series and Parallel ConnectionsC = C1+C2+V -V +V -V ΔVΔV+V -V+V -V55F 5F50F10F2.5F60FCAB = (2.5+60) F =62.5 FExampleceABAB6INDUCTORS- Current Opposing DevicesεL= - L di/dtB(t)back emfopposingB(t)An iron core strengthens inductance. Magnetic domains react to the di/dt, thereby increasing the emf. (Similar to dielectric!) self inductancei7CIRCUIT EMFS- Kirchoff's LawsVR C LV(t) -iR - q/C - L di/dt = 0 general case2nd order DE and easily solveable.i+-8V(t) -iR - q/C = 0 Kirkoff’s lawdq/dt + q/RC = V(t)/Rτ = RC = capacitive time constantSolving V(t) = Vo charging and V(t)=0 dischargingq(t) = CVo(1-e-t/RC) charging V(t)=Vo, q(0)=0I(t) = dq/dt= Vo/R e-t/RC V(t) =I(t)R= Vo e-t/RCq(t) = CVoe-t/RC dis-harging V(t)=0, q(0)=CVoI(t) = dq/dt= -Vo/R e-t/RC V(t) =I(t)R = -Vo e-t/RCRC CircuitR CiqV(t)qC(t)tt=RC0.63qmaxqmax=CVchargingDis-chargingq(t)V(t)Voqmax=CVo9V(t) -iR - Ldi/dt = 0 Kirkoff’s Law di/dt + iR/L = V(t)/Lτ = L/R = inductive time constantsolvingi(t) = Vo/R(1-e-tR/L) V(t) on, i(0) =0 VR(t)= I(t)R = Vo (1-e-tR/L) VL(t)= L di/dt = Vo e-tR/Li(t) = Vo/R e-tR/L V(t) offI(t)tτ = L/R0.63 Vmaximax= Vo/RRL CircuitR LiV(t)VR=IRVR Back emf holdscurrent down fora short period τ ~L/R!V(t) onV(t) offV(t)=VR(t)+VL(t)10Example 2-#6τ = L/R=.025/1000=30µs imax =V/R=100mAWe can replace L/R =1/RC to obtain anequivalent circuit.C = L/R2=2.5e-8F=25nFR=1KL=25mHi10VAn engineer wishes to replace the inductor witha capacitor giving the same time constant.What is the value of C?The voltage characteristics of the circuit remains the same!25nF11Example 2-#7 RC CircuitDescribe the current which flows to point A when the switch is closed.qC(t) =CVo(1-e-t/RC)IC(t)=dq/dt =Vo/R e-t/RC = Imax e-t/RCPositive charges flow to the left plateand negative charges flow to point Auntil the capacitor is fully charged,according to IC(t) above.Negative charges flowing CCW areequivalent to + charges flowing CWto replenish + battery charges!RCiqVoA+-+12Transformer - Example of an InductorgroundCenter tapPrimarySecondaryTransformer Equations:I1 V1 = I2 V2 Conservation of energyV2 = ( N2/N1) V1 Balance of Induced EmfV1(t) V2(t)13DC and AC WaveformsV(t) = Vdc + Vac sin(ωt)VdcV=0TVdcV=0V(t) = Vdc + Vac(t) V(t) with DC blocked!TmaxTmin14Integration and DifferentiationDifferentiatorVout(t) = I(t) R = R dQ/dtQ = Qdc + QacVout(t) = R dQ/dt = R dQac/dtOnly the AC current is passedthrough a series capacitor !RC << ΔTminCapacitor quickly charges.Resistor current spikes and quicklyfalls to zero.IntegratorVout(t) = q(t)/C = 1/C i(t)dtThe capacitor integrates theincoming current (smoothing)!RC >> ΔTmax (Integration)Capacitor integrates over manynoise cycles adding to zero! .CR VoutRCV(t)VoutΔTV(t)ΔTIqI 0RC! RCV(t) = q/C + iR15Noise FilterOften electronic noise accompanies a signal we are trying to capture or amplify. An RC integrator may be the solution!If we Integrate the bipolarity noise signal the integral will vanishdue to the alternating sign of the voltage. RC > ΔT to average out the spikes! f=1/ΔT = frequency of noise. RCRC16RLC Analogue CircuitkF(t) = m dx2/dt2 + µ dx/dt + k xV(t) = L dq2/dt2 + R dq/dt + (1/C) qVimµ