Digital Systems Capacitance and Inductance CMPE 6501 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6Step ResponseThe step response measurement is very useful in our analysis.From it, we can derive a curve of impedance versus frequency.Test setup:Rules of thumb in characterizing the DUT using the step response:• Resistors display a flat step response, i.e., at time t = 0, the output rises andremains at a fixed value.X(t)+-ZI(t)+-RsY(t)Step responseOutput impedance of step sourceDevice under test (DUT)Pulse generatorrepeatedly appliesstep input toDUT.Y(t)tX(t)tDigital Systems Capacitance and Inductance CMPE 6502 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6Step Response• Capacitors display a rising step response, i.e., at time t = 0, the output startsrising and later reaches its full value.• Inductors display a sinking step response, i.e., at time t = 0, the output risesinstantly to its full value and then later decays back toward 0.Capacitors and inductors subdivide into ordinary and mutual categories.Ordinary capacitance and inductance (two-terminal devices) can be ahelp or hindrance.Mutual capacitance and inductance usually creates unwanted crosstalk.Y(t)tX(t)tY(t)tX(t)tDigital Systems Capacitance and Inductance CMPE 6503 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6CapacitanceThe capacitor is a 2-terminal element in which the branch voltage and currentvariables are related by integral and differential equations:Charge and voltage are related by the linear relationship:Power is negative or positive depending on the value of the term v(t)dv/dt inthe following expressionv t( )1C----i τ( ) τdt0t∫v t0( )+=i t( ) Cdvdt------=and(Voltage oncap dependson history of i)q t( ) Cv t( )=slope = Cqvp t( ) Cv t( )dvdt------=Digital Systems Capacitance and Inductance CMPE 6504 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6CapacitanceBut energy (the integral of power) is always positive or zero:Therefore, it’s a passive element (like the resistor) but it is non-dissipa-tive (unlike the resistor).All the energy supplied to the cap. is stored in the electric field.Note that the voltage appearing across a capacitor must always be a continu-ous function (voltage steps not allowed -- require an infinite i).Current, on the other hand, is allowed to change instantenously.w t( )Cv2t( )2-----------------=i(t)v(t)tti(t)+-v(t)i(t)tv(t)tDigital Systems Capacitance and Inductance CMPE 6505 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6CapacitanceThe energy stored in the electric field of a capacitor is supplied by the drivingcircuit.Since the driving source is a limited source of power, the voltage takes afinite time to build up.The reluctance of voltage to build up quickly in response to injectedpower (or decay quickly) is called capacitance.X(t)I(t)1 ns+-Y(t)X(t)+-I(t)Rs = 30Ωmax value VCC/Rvoltage approaches VCCY(t)Y(t)I(t)open circuitimpedanceshort circuitDigital Systems Capacitance and Inductance CMPE 6506 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6CapacitanceBear in mind that a capacitor behaves like an inductor at high frequencies(unfortunately).This is due to the mounting leads on capacitors.This inductance causes the step response to have a tiny pulse (a couplehundred ps) at time 0, followed by a drop to 0 and then a capacitiveramp.Note that you will not be able to see this unless your step source rise time issharp.At Tr, you can characterize the circuit element for frequencies up to:Reactance on leading edge (to est. distortion in digital wfm by a cap.):FA0.5Tr-------=XCTrπC-------=Digital Systems Capacitance and Inductance CMPE 6507 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6Capacitance Test GigA measurement setup ideal for characterizing capacitors:Note that the resistances are known, so by measuring the rise time of theresulting waveform, the capacitance of the DUT can be computed.The test gig is dimensioned at 1 square inch to ensure it behaves in a lumpedfashion.The test gig should include a ground plane of 1 square inch.Pulsegenerator+-V(t)50 Ω50 Ω1 KΩ 1 KΩDUTScope+-50 ΩterminationDUT can be a circuittrace, a bypass cap,etc.1 square inch50Ω back terminationreduces output by 1/2Digital Systems Capacitance and Inductance CMPE 6508 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6InductanceThe inductor is a 2-terminal element in which the branch voltage and currentvariables are related by integral and differential equations:Flux linkages (in weber-turns) and current are related by the relationship:Power is negative or positive depending on the value of the term i(t)di/dt inthe following expressioni t( )1L---v τ( ) τdt0t∫i t0( )+=v t( ) Ldidt-----=andλ t( ) nφ t( ) Li t( )= =slope = LLn turns of wirefluxproducesλiv t( )dλdt------=+-vip t( ) i t( )v t( ) i t( )Ldidt-----= =Digital Systems Capacitance and Inductance CMPE 6509 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6InductanceBut energy (the integral of power) is always positive or zero:Therefore, it’s a passive element (like the resistor) but it is non-dissipa-tive (unlike the resistor).All the energy supplied to the ind. is stored in the magnetic field.Note that the current flowing through an inductor must always be a continu-ous function (current steps not allowed -- require an infinite v).Voltage, on the other hand, is allowed to change instantenously.w t( )Li2t( )2---------------=v(t)i(t)ttv(t)i(t)v(t)ti(t)t+-Digital Systems Capacitance and Inductance CMPE 65010 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6InductanceThe energy stored in the magnetic field of a inductor is supplied by the driv-ing circuit.Since the driving source is a limited source of power, the current takes afinite time to build up.The reluctance of current to build up quickly in response to injectedpower (or decay quickly) is called inductance.X(t)I(t)1 ns+-Y(t)X(t)+-I(t)Rs = 30 Ωvoltage output decays to 0.Y(t)Y(t)I(t)short circuitimpedanceopen circuitcurrent approaches VCC/RDigital Systems Capacitance and Inductance CMPE 65011 (2/5/08)UMBCU M B CUNIVERSITY OF MARYLAND BALTIMORE COUNTY1 9 6 6Inductance Test GigA measurement setup ideal for characterizing inductors:Reactance on leading edge (to est. distortion in digital wfm by an