Chapter 8. Natural and Step Responses of RLC Circuitswp.kntu.ac.ir/faradji/EC1/EC1_Ch8.pdf ·...

Chapter 8. Natural and Step Responsesof RLC CircuitsBy: FARHAD FARADJI, Ph.D.

Assistant Professor,Electrical Engineering,

K.N. Toosi University of Technology

http://wp.kntu.ac.ir/faradji/ElectricCircuits1.htm

Reference: ELECTRIC CIRCUITS, J.W. Nilsson, S.A. Riedel, 10th edition, 2015.

Chapter Contents8.0. Introduction

8.1. Introduction to the Natural Response of a Parallel RLC Circuit

8.2. The Forms of the Natural Response of a Parallel RLC Circuit

8.3. The Step Response of a Parallel RLC Circuit

8.4. The Natural and Step Response of a Series RLC Circuit

8.5. A Circuit with Two Integrating Amplifiers

8.6. Summary

Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 2

8.0. Introduction

8.1. Introduction to the Natural Resppponnsse of a Parallel RLC Circuit

8.2. The Forms of the Natural Ressssppppppooooooonnnnnnnsssssseeeeee oooooooffffff aaaa Parallel RLC Circuit

8.3. The Step Response of a Parallllllleeeeeellll RRRRRRLLLLLLCCCCCCCC CCCCCCCCiiiiiirrrrrccccuit

8.4. The Natural and Step Responsee oooooooooooooffffffffffffff aaaaaaaaaaa SSSSeeries RLC Circuit

8.5. A Circuit witthhhh TTTTwwwwooo IIInnnntttttteeeegggrrrraaaatttttttiiiiiiinnnngggggg AAAAAAAAmmmmppppllllllliiiiiiiifffffffffffiiiiiieeeerrrrrssss

8.6. Summary


8.0. Introductiono Discussion of natural response and step response of circuits

containing both inductors and capacitorsis limited to 2 simple structures:

parallel RLC circuit and series RLC circuit.


o Discussion of natural response and step response of circuitscontaining both inductors and capacitors

is limited to 2 simple structttttttuuuuuuurrrrrrreeeeeesssssss:::::parallel RLC circuit anddddd series RLC circuit.


8.0. Introduction

o Finding natural response of a parallel RLC circuitconsists of finding voltage across parallel branches

by release of energy stored in L or C or both.

o Initial voltage on C, V0, represents initial energy stored in C.

o Initial current in L, I0, represents initial energy stored in L.

o If individual branch currents are of interest,you can find them after determining terminal voltage.


o Finding natural response of a paraaallllllllllllllllllleeeeeeeeeeeelllllll RRRRRRRRRLLLLC circuitconsists of findinggg vvvvoooooooooolllllllltttttttttttttaaaaaaaaaaggggggggggeeeeeeeeeee aaaaaaaaccccccccccrrrrrrrrrroooooooooosssssssssssss ppppppppppppaaaaaaaaaaaaarrrrrrrrraaaaaaaaaaaaallllllllllllllllllleeeeeeeeeeeelllll bbbbbbbbbbbbrrrrrrrrrraaaaaaaaaaaannnnnnccches

by releaaaassssssssssseeeeeeeeeee ooooooooffffffffffff eeeeennnnnnnnnneeeeeeeeerrrrrrrrrrrggggggggggggyyyyyyyy sssttttttttttooooooooorrrrrrrrrreeeeeeeeedddddddddddddd iiiiiiiinnnnnnnn LLLLLLLLLLLL oooooooooorrrrr CCCCCCCCCCCCC oooooooooor bbbbbbbbbbboooooooootttttttttthhhhhhhhh.

o Initial voltage on C, C VVV0VV , rreeeepppppppppprrreeeesssseeeennnnttttsss iiinnnniiiitttttiiiiaaaallll eeeennnnnnneeeerrrrggggyyyy sssstttooorrrred in C.

o Initial current in L, I0II , represents iniitiall energy stored in L.

o If individual branch currents are of interest,you can find them after determining terminal voltage.


8.0. Introduction

o In step response of a parallel RLC circuit,we are interested in voltage across parallel branches

as a result of sudden application of a dc current source.

o Energy may or may not be stored in circuitwhen current source is applied.


o In step response of a parallel RLCC cccccccccciiiiiiiiiirrrrrrrrrrrcccccccccuuuuuuuuuiiiitttt,we are interested innnn vvvvvvvvvvoooooooooooolllllllllltttttttttttaaaaaaaaaaagggggggggggeeeeeeeee aaaaaaaaaaccccccccccrrrrrooooooooosssssssssssssssss pppppppppppppaaaaaaaaaaaarrrrrrrrrraaaaaaaaaaaallllllllllleeeeeeelllllll bbbbbbbbbbbbbrrrrrrrrraaaanches

as a resssuuuuuulllllllltttttttttt oooooooooffffffffffffff sssssuuuuudddddddddddddddddddddddddeeeeeeeeeeennnnnnnn aaaaaaaaappppppppppppppppppppllllliiiiiiiiiiccccccccccccaaaaaaaaaaattttttiiiiiiiioooooooooooonnnnnnn oooooofffffffff aaaaaaaaa ddcccccccccc cccccccccuuuuuuuuuuuurrrrrrrrrrrrrrrrrrreeeeeeeeeennnnnnnntttttttttttt sssssssssssoooooooooooouuuuuuurrrce.

o Energy may or may nnoooooootttttt bbbbeeeee sssttttoooorrrreeeedddd iinnnn ccccciiiiirrrrccccuuuuiiiitttttttwhen current sourceee iiiiss aaaapppppplllliiieedddd.


8.0. Introduction

o Finding natural response of a series RLC circuitconsists of finding current generated in series connected elements

by release of initially stored energy in L, C, or both.

o As before, initial L current, I0 and initial C voltage, V0

represent initially stored energy.

o If any of individual element voltages are of interest,you can find them after determining current.


o Finding natural response of a seriieeeeeeeeesssssssssssss RRRRRRRRRRRRLLLLLLLLLCCCC circuitconsists of findinggg ccccuuuuuuuuuurrrrrrrrrrrrrrrrrrrrreeeeeeeeeeeennnnnnnnnnttttttttttt ggggggggeeeeeeeeeennnnnnnnnnneeeeeeeeeeerrrrrrrraaaaaatttttttttteeeeeeeeeeedddddddddddddd iiiiinnnnnnnnnn ssssssseeeeeeerrrrrrrrrrriiiiiiiieeeeeeeeeeessssssssss ccconnected elements

by releaaaassssssssssseeeeeeeeeee ooooooooffffffffffff iiiinnnniiiiiiiittttttttiiiiiiiaaaaaaaaaaallllllllllllllllyyyyyyyy sssstttttttttttoooooooooorrrrrrrrrrreeeeeeeeeedddddddddddddd eeeeeeeeeennnnnnnnnnneeeeeeeeeeerrrrrggggggggggyyyyy iiiiiiiiinnnnnnnnnn LLLL,,, CCCCCCCCCCCC,, oooooooooooorrrrrrrr bbbbbbbbbbbooooooooottttttttttthhhhhhhhhhhhhh.....CCCC

o As before, initial L cuurrrrrrrrreeeennnnttttt,,,,, IIIII0000 aaaannnnddddd iiiinnnniiittttiiiiiaaaaallll CCCCC vvvvvvvvoooollllttttaaaggggeeee, VVVV0VVrepresent initially stooorrrreedddd eennneeerrggggyyy.

o If any of individual element voltages are of interest,you can find them after determining current.


8.0. Introduction

o In step response of a series RLC circuit,we are interested in current

resulting from sudden application of dc voltage source.

o Energy may or may not be stored in circuitwhen switch is closed.


o In step response of a series RLC ciiirrrrrrrrrcccccccccccccuuuuuuuuuuuuuiiiiiiiiittttttttt,,,,we are interested innnn cccccccccuuuuuuuuuuuurrrrrrrrrrrrrrrrrrrrreeeeeeeeennnnnnnnntttttttttt

resultinnngggggg ffffffffffrrrrrrrrrroooooooommmmmmmmmmmm sssssssuuuuuuuuddddddddddddddddddddddddddeeeeeeeennnnn aaaaaaaappppppppppppppppppppppplllllllliiiiiiiiiccccccccccaaaaaaaattttttttttttiiiiiiiiiiooooooonnnnnnnnnn oooooooooofffffffff dddcccccccc vvvvvvvvvoooooooooollllllttttttttttttaaaaaaaagggggggggggeeeeeee ssssssssssooooooooooouuuuuuuuuuurrrcce.

o Energy may or may nnoooooootttttt bbbbeeeee sssttttoooorrrreeeedddd iinnnn ccccciiiiirrrrccccuuuuiiiitttttttwhen switch is closeedddd.








8.6. Summary


8.0. Introduction






8.6. Summary


8.1. Natural Response of a Parallel RLCo First step in finding

natural response of circuit isto derive differential equation

that voltage v must satisfy.

o We choose to find voltage first,because it is same for each component.

o A branch current can be found by usingcurrent-voltage relationship for component.

o We easily obtain differential equation for voltage bysumming currents away from top node:


o First step in findingnatural response of circuit is

to derive differential equaaaatttttttiiiiiioooooooonnnnnnthat voltage v must satttttiiiiiiisssssssffffffyyyyyy....

o We choose to find voltage first,because it is same fffooooooooorrrrrrrr eeeaaaacccccchhhh cccccoooommmmmmpppppppooooonnneeeeeennnnttt...

o A branch currrreeeeeeennnnnnnnnnnnnnnttt ccccccccaaaaaaaannnn bbbbbbbbbbbeeeeeeeeeee ffffffffffffoooooooooouuuuuuuuuuunnnnnnnnndddddddddddd bbbbbbbbbbyyyyyyyyyyyyyyy uuuuuuuuuuusssssssssiiiinnnnnnnnnnnggggggggggggggcurrent-voltage rrrrrreeeeeeeeeeeeeellllllllllaaaaaaaaaaatttttttiiiooooooooonnnnnnnnnnnnsssssssshhhhhhhhhhhhhiiiiiiiiipppppppppppppp fffffffffffooooorrrrrrrrr ccccccccccccooooooooooommmmmmmmmmmmppppppppppppooooooooooonnnnnnnnneeeennnnnnnnnnnnntttttttt.......

o We easily obtain differennnnnnnnnnnnttttttttttttiiiiiiiiiiaaaaaaaaaaaalllllll eeeeeeeeeeqqqqqqqqqquuuuuuuuuuuaaaaaaaaattttttttttttttiiiiiiiioooonnnnnnn fffffffffffffooooooooooooorrrrrrrrr vvvvvvvvvvvvooooooooooolllllllttttttttttttaaaaaaaaaaaagggggggggeee bysumming currents away from top node:


8.1. Natural Response of a Parallel RLC

o Equation is anordinary, 2nd order differential equation with constant coefficients.

o Circuits here contain both L and C.

o Differential equation describing these circuits is of the 2nd order.

o We sometimes call such circuitsthe 2nd order circuits.


o Equation is annnordinary, 2nd ordeerrrr ddddiiifffffffffffffeeerrrreeeeeennnnnttttiiiiaaaalll eeeeqqqqqqqqquuuuuuaaaattttiiiiooooonnnn wwwwiiitttthhhh cccoooonstant coefficients.

o Circuits here contain bothhh L anddd C.

o Differential equation describing these circuits is of the 2nd order.

o We sometimes call such circuitsthe 2nd order circuits.



o Classical approach is to assume thatsolution is of exponential form:

A and s are unknown constants.

o Equation is calledcharacteristic equation of differential equation.

o Roots of this quadratic equation determinemathematical character of v(t).


o Classical approach is to assume ttttthhhhhhhaaaaaaatttttttsolution is of exponential fooooorrrrrrrmmmmmmm::::

A and s are unknown connnsssstttttttaaaaaaannnnnnntttttttssssssss...

o Equation is calledcharacteristic equation of differential equation.

o Roots of this quadratic equation determinemathematical character of v(t)t .



o We can show that sum of v1 and v2 is also a solution:


o We can show tttthhhhhhhhhhaaaattttttttt ssssssuuuummmmmmmmmmm ooofff vvvvvvv11111111 aaaaaaaaannnnnnnnnndddddddd vvvv222222 iiiissss aaaaaaalllllllllssssooo aaaa ssssssoooooooooolllllluuuuuuuttttttttiiiiooooooooonnnn:::



o Roots of characteristic equation (s1 and s2) are determined bycircuit parameters R, L, and C.

o Initial conditions determineconstants A1 and A2.

o Form of v must be modified if 2 roots s1 and s2 are equal.Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 13

o Roots of characteristttttttiiiiiccccccccc eeeeqqqqqqquuuuaaaattttttiiiiiioooonnnn (((((sss1111 aaaannnnddddddd ssss22222))))))))))) aaaaarrreeee dddddddddddeeeetttttteeeermined bycircuit parameters R, LLLLL, aaaannndddddd CCCC.


o Form of v must be modified if 2 roots s1 and s2 are equal.Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 13


o is the neper frequency.

o is the resonant radian frequency.


o is the neper frequency.

o is the resonant radian frequency.



o s1 and s2 are referred to as complex frequencies.

o is neper frequency.

o is resonant radian frequency.

o Exponent of e must be dimensionless.

o s1 and s2 ( and ) must have dimension of 1/time or frequency.

o All these 4 frequencies have dimension ofangular frequency per time (rad/s).

o Nature of roots s1 and s2 depends on and .


o s1 and s2 are referred to as commmmpppppplllleeeeeeexxxxxx fffffffrrrrrrrreeeeeeeqqqqqqquuuueeencies.

o is neper frequency.

o is resonanttt rrrraaaadddiiiiaaannnn ffffrrrreeeeqqqquuuueeeennnnccccyyyyyyy....

o Exponent of e musttt bbbbbbbbbbbbeeeeeee ddddddiiimmmmmmmmmeeeeeennnnnnnssssssssiiiiooooonnnnnnnnnnlllleeeeeeeeesssssssss..

o s1 and s2 ( and ) musssstttttttttttt hhhhhhhhhhhaaaaaaaaavvvvvvvvvvvvveeeeeeeeee dddddddddddiiiiiimmmmmmmmmmmmmmeeeeeennnnnnnnnsssssssssiiiiiiiiooooooooooooonnnnnnnnnnnn oooooooooooffffffff 111111111111///////////////tttttttiiime or frequency.

o All these 4 frequencies have dimension ofangular frequency per time (rad/s).

o Nature of roots s1 and s2 depends on and .



o There are 3 possible outcomes:If < 2, both roots will be real and distinct.

Voltage response is said to be overdamped.If > 2, both s1 and s2 will be complex and conjugates of each other.

Voltage response is said to be underdamped.If = 2, s1 and s2 will be real and equal.

Voltage response is said to be critically damped.

o Damping affects the wayvoltage response reaches its final (or steady-state) value.


o There are 3 possible outcomes:::If < 2, both roots will be rrreeeeeeeeeaaaaaaaaaalllllll aaaaaaannnd distinct.

Voltagee rreeesssppppooonnnsseeeee iiiissssssss sssssaaaaaaiiiiiiiddddddddddd ttttttttooooooo bbbbbbbbbbbeeeeee oooooooovvvvvvvvvveeeeeeeeeerrrrrrrrrrdddddddddddaaaaaaaaammmmmmmpppppppppeeeeeeeeeddddddddddd..If > 2, bbbbbbbbbbbbooootttttttthhhhhhhhhhhh ssss111111 aaaannnnnddddddddddd sss2222 wwwwiiiiiiiillllllllll bbbbbbbbbbbbbeeeee cccccoooommmmmmmmmmppppplllllllleeeeeeeeexxx aaaannnnndddddddddd ccccccooooonnnnnnnjjjjjjjuuuuugggggggggggaaaaaaaaaattttttees of each other.

Voltage responnnsssssssseeeeee iiissss ssssaaaaiiiidddddddd ttttttttoooo bbbeeee uuuunnnnddddddddddeeeeeerrrrrrrrddddddddddddaaaammmmppppeeedddddd.If = 2, s1 and s2 wwwwiiiillllllllll bbbbbbeee rrreeeaaaalllllll aaannndddddd eeeeqqquuuaaaallll.

Voltage response is said to be critically damped.

o Damping affects the wayvoltage response reaches its final (or steady-state) value.








8.6. Summary


8.0. Introduction






8.6. Summary


8.2. Forms of Parallel RLC Natural Response

o Behavior of a 2nd order RLC circuit depends on s1 and s2.

o s1 and s2 depend on circuit parameters R, L, and C.

o The 1st step in finding natural response is tocalculate s1 and s2, anddetermine whether response is over-, under-, or critically damped.


o Behavior of a 2nd order RLC circcccuuuuuuiiiittttttt ddddddeeeeeeeppppppppeeeeeeeennnnnnddddddds on s1 and s2.

o s1 and s2 depend on circuit parammeeeeeeeeettttttttttttteeeeeeeeeeeeerrrrrrrrrrsssssssss RRRR, L, and C.

o The 1st step inn ffffiiiinnnndddiiiinngggggg nnnnaaaattttttttuuurrrraaaalllll rrrreeeeessssppppoooonnnnsssseeeeeeeee iiiiiissss tttttttoooocalculate sss11 aaaannnndddddddddd ssss22222222222,,,, aaannndddddddetermine whethhhhheeerrrrr rrreeeeessssppppoooonnnnsssseeee iiiissss oooovvvvvveeeerrrr---,,,,,,,, uuuuuunnnnddddeeeerrr---, ooorrrr critically damped.



o The 2nd step is tofind 2 unknown coefficients, such as A1 and A2.


o Natural response should be matched toinitial conditions imposed by the circuit.

o Initial conditions are:initial value of current (or voltage) andinitial value of the 1st derivative of current (or voltage).


o The 2nd step is tofind 2 unknown coefficients, sssuuuuuuuuuccccccccccccccchhhhhhhhhhhhhh aaaaaaaassss A1 and A2.

o Initial conditions deterrrrmmmmmmmmmmiiiiiiiiinnnnnnnnnnnneeeeeeeeeeeeconstants AAAAAAAAAA111 aaaaaaaaaaannnnnnnnnnnddddddddd AAAAAAAAAA2222..

o Natural response shoouuuuuuuullldddd bbbbbeeee mmmmaaaattttcccchhhheeeeddddd ttttooooinitial conditions imppppooossseeeeddd bbbbyyy tttthhhhee cciiirrrcccuuuiittt.

o Initial conditions are:initial value of current (or voltage) andinitial value of the 1st derivative of current (or voltage).



o Response equations,as well as equations for evaluating unknown coefficients,

are slightly different for each of 3 damping configurations.

o This is why first we want to determinewhether response is

overdamped,underdamped, orcritically damped.


o Response equations,as well as equations for evaluuaaaatttttttttttiiiiiiiiiinnnnnnnnnnnnnnngggggggggg uuuunknown coefficients,

are slightly diffeerrreeeeeeeeeeennnnnnnnnnnttttttttttttt ffffffffffffffooooooooooooorrrrrrrrrrrr eeeeeeeeeeeeeeaaaaaaaaaaaccccccccchhhhhhhh ooooooffffffffffff 33333333333 dddddddddddddaaaaaaaaaaaaammmmmmmmmmmmpppppppppppiiiiiiiiiiinnnnnnnnnnnggggggggggggg cccconfigurations.

o This is why firrrssssstttttttttttt wwwwwwwwwwwwweeeeeeeeeee wwwwwwaaaaaaaaannnnnnnnnttttttttttt ttttttttttttoooooooo ddddddddddeeeeeeetttttttttteeeeeeeeerrrrrrrrrrrmmmmmmmmmmmmmiiiiiinnnnnnnnneeeeeeeeeeewhether responsseeeeeee iiiiiiiiissssss

overdamped,underdamped, orcritically damped.


8.2. Forms of Parallel RLC Natural ResponseOverdamped Voltage Response

o Roots of characteristic equation,s1 and s2, are real and distinct.

o Voltage response is said overdamped.

o Form of voltage is:

o A1 and A2 are determined by initial conditions: v(0+) and dv(0+)/dt.

o v(0+) and dv(0+)/dt are determined from:initial voltage on C, V0, andinitial current in L, I0:


Overdamped Voltage Response

o Roots of characteristic equation,s1 and s2, are real and distinnncccctttttt...

o Voltage response is said overdaaaaaaammmmmmmpppppeeeeeedddddddd...


o A1 and A2 are determinnneeeeeeeeeddddddddd bbbbbbbbbbbbyyyyyyyyyy iiiiiinnnnnnnnniiiiiiiiittttttttttiiiiiiiiaaaaaaaaalllllllll ccccccccoooooooonnnnnnnnnnnddddddddddddddiiiiiiiiiiitttttttttiiiiiiiiiiioooooooooooonnnnnnnsssss::::::: vvvvvvvvvvvv((((((((((((0000+) and dv(0+)/dt.

o v(0+) and dv(000+++++)))))/////dddddtttt aaaarrrreeee dddddddeeeetttteeeerrrrmmmmiiiiinnnneeeeddddddddd ffffffrrroooommmmmmm:::::initial voltage on CCCC, C VVVV000000VVVV ,,,,,,, aaannnnddddinitial current in L, I0II :::



o v(0+) is initial voltage on C, V0.

o We get dv(0+)/dt byfirst finding current in C at t = 0+:

o KCL at top node is:



o v(0+) is initial vvvvoooolttttaaaaggggeeee oooonnnn CCCCCCCCCCC,,, CC VVVVVVVV0000VVVV ...

o We get dv(0+)///dddttt bbbyfirst finding currenttt iiinnnnnnnn CCCCCCCC aaaaaaattttttttt tttttttt ==== 00000000++++::::

o KCL at top node is:



o Finding overdamped response, v(t):1. Find s1 and s2

using values of R, L, and C:

2. Find v(0+) and dv(0+)/dt using circuit analysis.3. Find A1 and A2 by solving:

4. To determine v(t) for t > 0,substitute s1, s2, A1, and A2 into:



o Finding overdamped response, v(t)))t :1. Find s1 and s2

using values of R, L, and CCCCCCCC:::::

2. Find v(0+) aaaannndddd ddddvvv((((0000+++++++)))))))///////////ddddddddddttttttt uuuusssssiiiiiinnnnnggggg cccciiiirrrccccccuuuuuuuuuiiiiiiiiiiitttttttt aaaaannnnnaaaalllllyyyyssssiiiiiiisssssss..3. Find A1 anndddddd AAAAAAAAAA22 bbbbbbbbbbbbyyyy ssssoooollllllllvvviiiiinnnnggggg:::

4. To determine v(t)t for t > 0,substitute s1, s2, A1, and A2 into:



o Finding currents using v(t):



o Finding currents using v(t)t :



o Roots are real and distinct.

o Response is overdamped.


o Roots are real and distinct.

o Response is overdamped.



o C holds initial voltage across parallel elements to 12 V.

iR(0+) = 12/200 = 60 mAElectric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 26

o C holds initial voltage acrossssssssss pppppppaaaaaarrrrraaaaalllllllllllleeeeellllll eeeeeeellllllleeeeeeeeemmmmeeeeennnntttttttsssssssss tttttttttooooo 11111111112222222222 VVVVVV.

iR(0+) = 12/200 = 60 mAElectric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 26


Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 27Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 27

8.2. Forms of Parallel RLC Natural ResponseUnderdamped Voltage Response

o When > 2, both s1 and s2 will becomplex and conjugates of each other.

o Voltage response is underdamped.

o d is called damped radian frequency.



Underdamped Voltage Response

o When > 2, both s1 and s2 will becomplex and conjugates of eeeeaaaaaccccccchhhhhhhh oooooootttttthhhhhhhhhhhheeeerrr.

o Voltage response is underdampppppppeeeeeeedddddd....

o d is called damped radiiian ffffrequency.




o Euler identity is:



o Euler identity is:



o Constants B1 and B2 are real, not complex.Because voltage is a real function.

o Don't be misled by the fact that B2 = j(A1 - A2).In this underdamped case, A1 and A2 are complex conjugates.

o Using B1 and B2 yields a simpler expression for voltage.

o Like A1 and A2, we determine B1 and B2

by initial energy stored in circuit:

o As with s1 and s2,and d are fixed by circuit parameters R, L, and C.



o Constants B1 and B2 are real, noooottttt cccccccoooooommmmmmppppppplllllleeeeeeexxxx...Because voltage is a real funnnnnnnccccccttttttiiiioooooonnnnnnn...

o Don't be misled by the fact that BBBB2222 ============= jjjjjjjjjjjj((((((((AAAAAAA((((((((((( 11 - A2).In this underdampeeeeddddddddddd ccccccccccccccaaaaaaaaaaaassssssssssseeeeeeeeeeee,,,,,,,,, AAAAAAAAAAAAAA11111111 aaaaaaaaaannnndddddd AAAAAAAAAAAAA2222222222 aaaaaaaaaarrrrrrrrrrreeeeeeeeeeeeeee ccccccccoooooooooooooommmmmmmmmmmmmpppppppllllex conjugates.

o Using B1 and BBBBBBB222 yyyyyyyyyyyyyiiiiieeeeeeeeeeelllllllldddddsssss aaaaaaaaa ssssssssssiiiiiiiiimmmmmmmmppppppppppllllllleeeeeeeeeerrrrrrrr eeeeeeeeeeexxxxxxxxxppppppppprrrrrrrreeeeeeeeeeesssssssssssssssssiiiiiiooooooonnnnnnnnn fffooorrrrrrrr vvvvvvvoooooooooooollllllltttttttttttaaaaaaaaaaggggggggggeeeeeeeee..

o Like A1 and A2, we deettteeeeeeeerrmmmmmiiiinnneeee BBBB11111111 aaaannndddd BBBBBB22222

by initial energy storrreeeeddd iiiinnn ccciiirrrccuuuuiiitt:

o As with s1 and s2,and d are fixed by circuit parameters R, L, and C.



o General nature of underdamped response:First, trigonometric functions indicate that response is oscillatory.

Voltage alternates between positive and negative values.The rate at which voltage oscillates is fixed by d.

Second, amplitude of oscillation decreases exponentially.The rate at which amplitude falls off is determined by .

is also referred to asdamping factor or damping coefficient.

d is calleddamped radian frequency.



o General nature of underdampeeeedddddd rrrrrrreeeeeessssppppppooooooonnnnnnnsssse:First, trigonometric functionnnnnnnssssss iiiinnnnddddddiiiiicccccccaaaaaattttteeeeeee that response is oscillatory.

Voltage alternates betweeeennnn pppppppppppppooooooooossssssssiiiiiiitttttiiiivvee and negative values.The rate at whicchhhhhhhh vvvvvvvvvvoooooooooooollllltttttttttttaaaaaaaaaaagggggggggggeeeeeeeeeeeee ooooooooosssssssssscccciiiilllllllllaaaaaaaaaaaattttttttteeeeeeeeeeeeesssssssssss iiiiiisssssssss fffffffiiiixxxxxxxxxxeeeeeeeeeeeeddddddddd bbbby d.

Second, ammmmmmmmmppppppppppplllliiiiiittttttttuuuuuuuuuddddeeeeeeeeeeee oooooooooooffffffffffff ooooooooooosssssssssssccccccciiiiilllllllllllllllaaaaaaaaaattttttttttttiiiiiiiiioooooooooooonnnnnnnnnnn dddddddddddeeeeeeeeeeeeeeccccccccccrrrrrrrrreeeeeeeeeeaaaaaaaaaaassssssssseeeeeeeeeessssssssss eeeeeeeeeexxxxxxxxxxpppppppppppppppooooooooooooonnnnnnnnnneeeeeeeeeeeennnnnnnnnnnnnttttttttttiiiiiiiiiiiaaaaaaaaallllllly.The rate at whhhhhhiiiiiiiiiiiccccccccccchhhhhhhhhhh aaaammmmmmmmmmpppppppppppplllllllliiiiiiitttttttttttuuuuuuuuuuuuuudddddddddddddeeeee fffffffaaaaaaaaaaaalllllllllllllllllssssssss ooooooooooooooffffffffffffffffff iiissss ddddddddddddddeeeeeeeeeettttttteeeeeeeeeerrrrrrrrrrrrmmmmmmmined by .

is also referreeeedddddddddddd ttttttttttoooooooooo aaaaaaaaaaaasssssssssssdamping factor or damping coefficient.

d is calleddamped radian frequency.



o General nature of underdamped response:If there is no damping:

= 0.Frequency of oscillation is d = 0.

If there is an R in circuit:is not zero.

Frequency of oscillation is d < 0.

d is said to be damped.



o General nature of underdampeeeedddddd rrrrrrreeeeeessssppppppooooooonnnnnnnsssse:If there is no damping:

= 0.Frequency of ossccciiiiiiiillllllllllllllaaaaaaaaaaaatttttttttttttiiiiiiiioooooooooooonnnnnnnnnnn iiiiiiissssssss dddd ===== 0000000000000.....

If there is aaaannnnnnnnnnnnnnn RRRRRRRRRRRR iiiiiinnn ccccccccccciiiiiirrrrrrrrrrrccccccccccuuuuuuuuuuuuiiiiitttttttttt::::is not zero.

Frequency of osciiiilllllllllllllaaaaaaaaaaaatttttttttttiiiiiiiiooooooooooooonnnnnnnnnnn iiiiiiisssssssssss dddddddd <<<<<< 00000000000....

d is said to be damped.



o General nature of underdamped response:Oscillatory behavior is possible

because of 2 types of energy storage elements in circuit:L and C.

A mechanical analogy of this electric circuit is that ofa mass suspended on a spring.

Oscillation is possiblebecause energy can be stored in both spring and moving mass.



o General nature of underdampeeeedddddd rrrrrrreeeeeessssppppppooooooonnnnnnnsssse:Oscillatory behavior is possiiiibbbbbbblllleeeeee

because of 2 types of eneeerrrggggyyyyyyyyyyyy sssssssssstttttttoooooooorrrraaaagge elements in circuit:L and C.

A mechannniiiccccccccccccaaaaaaaaaalllll aaaaaaaaaaannnnaaaaaaaaallllllllllooooooooooogggggggggggggyyyyyyyyyyyyyy oooooooooooffff tttttthhhhhhhhiiiiiissssssssssss eeeeeeeeeeeelllllllleeeeeeeeeccccccctttttttttttrrrrrrrrrrriiiiiiiiccccccccccc ccccccccccciiiiiiiirrrrrrrrrrccccccccuuuuuuuuuuiiiiiiittttttttttt iiiiiisssssssss ttttttttttttthhhhhhhhhhhhhaaaaaaaaatttttttttt ooooooooooooffffffffa mass suspennnnnnddddddddddddddeeeeeeeeeeedddddddd ooooooooonnnnnnnnnnnn aaaaaaaaaa ssssssssssppppppppppppprrrrrrrrrrrriiiiinnnnnnnnngggggggggggg.....

Oscillation is possiblllleeeeeeeeeeeebecause energy can be stored in both spring and moving mass.




iR(0+) = 0 mA


iR(0+) = 0 mA


o v(t) approaches its final value,

alternating between values that are

greater and less than final value.

o Swings about final value

decrease exponentially with time.


o v(t)t approacheeessss iiiiiiiiiiitttttttssssssssss ffffffffffiiiiiiiiiinnnnnnnnnnaaaallll vvvvvvvvvvvvaaaaaaaaalllllllluuuuuuuuuuuuueeeeeeeeee,,,,,

alternating betweeeeeeeeennnnnnn vvvvvvvvaaaaaallluuuuuuueeeeeeeesssssssssss tttthhhhhhhhhhhhaaaaaaaaaattttttttttt aaaaaarrreeeeeeeee

greater and less thannnn fffffffffffffiiiiiiinnnnnaaaalllllll vvvvaaaallllluuuuueeeeee.

o Swings about final value

decrease exponentially with time.


8.2. Forms of Parallel RLC Natural ResponseCritically Damped Voltage Response

o When = 2,s1 and s2 will be real and equal:


o Two simultaneous equationsneeded to determine D1 and D2 are:


Critically Damped Voltage Response

o When = 2,s1 and s2 will be real and equuuuaaaaallllllll:::::::


o Two simultaneeoouuusss eeeqqqqqqqqqqqqqquuuuuuaaattttiiioooooooonnnnnssssneeded to determiiiineeee DDDDDDDD11111111 aaaaaaannnnnddddddd DDDDD222222222 aaaaaaarrrrrrrrrrreeeeeee:::


8.2. Forms of Parallel RLC Natural ResponseCritically Damped Voltage Response

o You will rarely encounter critically damped systems in practice.

o Largely because 0 must equal exactly.

o Both 0 and depend on circuit parameters:

o In a real circuit,it is very difficult to choose component values

that satisfy an exact equality relationship.


Critically Damped Voltage Response

o You will rarely encounter critically daammpped systems in practice.

o Largely because 0 must equal eeeeeeexxxxxaaaaacccccctttttttllllllyyyyyyy...

o Both 0 and depend on circuiiiittttttt pppppppaaaaarrrrrraaaaaaammmmmmmeeeeeetttttteeeers:

o In a real circuit,it is very difficult to choose component values

that satisfy an exact equality relationship.








8.6. Summary


8.0. Introduction






8.6. Summary


8.3. Step Response of a Parallel RLC Circuito Finding step response of a parallel RLC circuit involves finding

v(t) across parallel branches ori(t) in individual branches

as a result of sudden application of a dc current source.

o There may or may not be energy stored in circuitwhen current source is applied.


o Finding step response of a parallel RLC circuit involves findingv(t)t across parallel branches ori(t)t in individual branches

as a result of sudden appppppllllllliiiiiiicccccccaaaaaattttiiiioooonnnnnn oooooooffff a dc current source.

o There may or may not be energgggyyyy sssssssstttttttooooooorrrrrreeeeeeeddddddd iiiinnn circuitwhen current sourcceeeeeeeee iiiiiiisssss aaaapppppppppppppppplllllliiiieeeedddddd...


8.3. Step Response of a Parallel RLC Circuito We focus on finding iL(t).

o iL(t) does not approach 0 as t increases.

o After switch has been open for a long time,iL(t) equals dc source current I.

o We assume thatinitial energy stored in circuit is 0.

o This assumptionsimplifies calculations anddoesn't alter basic process.


o We focus on finding iLi (t)t .

o iL(t)t does not approach 0 as t increassess.

o After switch has been open for aaaaa lllllooooooonnnnnngggggg tttttttiiiiiimmmmmmmeee,iL(t)t equals dc source currennnnnnnttttttt IIIIII...III

o We assume thatinitial energy storeeeeddddddddd iiiiinnnnnnnnnnnnnn ccccccccccccciiiiiiiirrrrrrrrrccccccccccccuuuuuuuuuuuuiiiiiittttttttttt iiiiiissssssss 00000000....

o This assumptiiioooooonnnnnnnnnnsimplifies calculaaaaaattttttttttiiiiiiiiooooooooonnnnnssssss aaaaaaaannnnnnnndddddddddddoesn't alter basic prrrrooooooooooccccccccccceeeeeeesssssssssssssss.


8.3. Step Response of a Parallel RLC Circuit



o The solution fora 2nd order differential equation with a constant forcing function equals

the forced response plusa response function identical in form to the natural response.

o The solution for the step response is in form:

o If and Vf representfinal value of response function.


o The solution fora 2nd order differential equaaaaattttttiiiiiiooooooonnnnnn wwwwwiiiiiitttttthhhhhhh aaaaaa constant forcing function equals

the forced response plusssa response function identiccaaaaaalllll iiiiiiiiiiinnnnnnnnn ffffffffoooorm to the natural response.

o The solution ffoooorrrrrr tttthhhheeee sssstttteeeepppp rrreeeessspppppppppoooonnnnsssseee iiiissss iiiiiinnnnnnnnn fffffffoooorrrrmmmm::::

o IfI and VfVV representfinal value of response function.



o No energy is stored in circuit prior to application of dc current source.o Initial current in L is 0.o L prohibits an instantaneous change in iL.o iL(0) = 0 immediately after switch has been opened.o Initial voltage on C is 0 before switch has been opened.o It will be 0 immediately after.o Since v = LdiL/dt:


o No energy is stoooorrrrrrrrreeeeeeeeeeedddddddd iiinnnnnn cccciiiiiiirrrrrrrrcccccccuuuiiitttt pppppppppprriiiioooooooooorrrrrr ttttttttooooo aaaappppppppppppppppplllliiiiiiiccccccccaaaatttiiioooonnnnnnn oooooooofffffffff dddddddcccc cccccccccuuuurrrrrrrreeeeeennnntttt sssooouuuurce.o Initial current in LLL iiis 00.o L prohibits an instantaneouuuusssssss ccccchhhhaaaaannnngggggggggeeee iiiiinnnnn iiiiiLLLL...o iL(0) = 0 immediately after swiiittchh hhhas bbeen openedddd.o Initial voltage on C is 0 before switch has been opened.o It will be 0 immediately after.o Since v = LdiL/dt:



o iL will be overdamped:


o iL will be overdamped:



o Increasing R to 625 decreases to 3.2×104 rad/s.

o Current response is underdamped, since:


o Increasing R to 625 decreases tttttttoooooo 33333...222222×××××××111111100000004 4444 rrrad/s.

o Current response is underdamped, siinnnnccccccccccccceeeeeeeeeeeeeee:::::::



o Current response is critically damped.


o Current response is critically damped..


o Underdamped responsereaches 90% of the final value in the fastest time,is desired response type when the speed is most important.



o Underdamped responsereaches 90% of the final value in the fastest time,is desired response type when the speed is most important.



o Underdamped responseovershoots final value.

o Neither critically damped nor overdamped responseproduces currents in excess of 24 mA.

o It is the best to use overdamped response.Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 50

o Underdamped responseovershoots final value.

o Neither critically damped nor overdamped responseproduces currents in excess of 24 mA.

o It is the best to use overdamped response.Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 50


o It would be impractical to requirea design to achieve

exact component values that ensure a critically damped response.


o It would be impractical to rrrreeeeeeeeeeqqqqqqqqqqqqqquuuuuuuuuiiiiiirrrrrrrrrrrreeeeeeeeeea design to achieve

exact component values that ensure a critically damped response.


o There cannot be an instantaneous change in iL.

o Initial value of iL in first instant after dc current source is applied must be 29 mA.

o C holds initial voltage across L to 50 V:

o Current response is critically damped:

o Effect of nonzero initial stored energy is on calculations for D'1 and D'2.



o There cannot be an instantaneous ccccchhhhhaaaaaannnnnngggggggeeeeeee iiiiiinnnnnn iiiiiLLL...

o Initial value of iL in first instant after ddddcccccccccc cccccccccccccccuuuuuuuuuurrrrrrrrrrrrreeeent source is applied must be 29 mA.

o C holds initial voltage acrrroooooossssssssssssssssss LLLLLLLLLLL tttttttttttooooooooooo 5555555500000000000 VVVVVVVVVVVV::::::::

o Current response is critically damped:

o Effect of nonzero initial stored energy is on calculations for D'1 and D'2.Electric Circuits 1 Chapter 8. Natural and Step Responses of RLC Circuits 52







8.6. Summary


8.0. Introduction






8.6. Summary


8.4. Natural & Step Response of a Series RLCNatural Response

o Finding natural or step responses ofa series RLC circuit are

same as those for a parallel RLC circuit.

o Both circuits are described by differential equations that have same form.


Natural Response

o Finding natural or step responses ofa series RLC circuit are

same as those for a paralllllllllleeeeeeellllll RRRRLLLLCCCC ccccciiiirrrrrcccccuit.

o Both circuits are described by ddiiiifffffffffffeeeeeeeerrrrrrrrreeeeeeeeeennnnnnnnttttttiiiiiaaaalll equations that have same form.


8.4. Natural & Step Response of a Series RLCNatural Response

o When you have obtained natural current response,you can find natural voltage response across any circuit element.


Natural Response

o When you have obtained natural current response,you can find natural voltage response across any circuit element.


8.4. Natural & Step Response of a Series RLCStep Response

o Vf is final value of vc, i.e., Vf = V.


Step Response

o VfVV is final value of vc, i.e., VfVV = V.VV


8.4. Natural & Step Response of a Series RLC



o iL is 0 before switch has been closed.

o It is 0 immediately after:

o There will be no voltage drop across resistor.

o Initial voltage on C appears across L:


o iL is 0 before swwwwiiiiiittttttttttttttcccccchhhhhhhhhhh hhhhhhhaaasssssss bbbbbbbbbeeeeeeeeeeeeeeeeeeeeeennnnnnnnn cccccccccllooooooooosssssssssseeeeeeeeeedddddddddddd......

o It is 0 immediately afteeeeeerrrrrrrrrrrr::::::::

o There will be no voltage drrroooooooooooooppppppppppppp aaaaaaaaaaccccccccccccrrrrrrrrrrooooooooooossssssssssssssssssss rrrrrrrrreeeeeeeeeeeessssssssiiiisssssstttttttoooooorrrrrrrrrrr...

o Initial voltage on C appears across L:



o Roots are complex, so voltage response is underdamped:


o Roots are comppplllleeeeeeeeeeeeeexxxxxx,,,,,,,, sssssssssooooo vvvvvvvvvooooooooooolllllllttttttttaaaaaaaaaggggggggggggeeeeeeeee rreeeeeeeeeesssssssssppppppppppooooooooooonnnnnnnnnnnnssssssssseeeeeeeeeee iiiissssssssss uuuuuuuuuuunnnnnnnnnnndddddeeeeeeeeeeerrrrrrrrrddddddddaaaaaaaaaammmmmmmmmmmppppppppppeeeeeeeeeeeddddddddddddd::::::::








8.6. Summary


8.0. Introduction






8.6. Summary


8.5. A Circuit with 2 Integrating Amplifierso A circuit with

2 integrating amplifiersconnected in cascade

is also a 2nd order circuit.

o Output voltage of the 2nd integratoris related to input voltage of the 1st

by a 2nd order differential equation.

o In a cascade connection,output signal of the 1st amplifier is input signal for the 2nd amplifier.

o We assume that op amps are ideal.

o Task is to derivedifferential equation that relates vo to vg.


o A circuit with2 integrating amplifiers

connected in cascadeis also a 2nd order circuuuuuuiiiiiiittttttt....

o Output voltage of the 2nd integrrrraaaattttttooooooorrroooorrrrrrris related to input vvoooooooooolltttttttttaaaagggggeeeeeeeeee oooooooooffffff tttthhhhhheeee 11111ssstttt

by a 2ndd oooooooooorrrddddddddddeeeeeerrr ddddddddddiiiiiiffffffffffeeeerrrreeeeeeeeennnntttttttttiiiiiiiiiaaaaalllll eeeeqqqqqqquuuuaaaatttttttttiiiiiiiioooonnn...

o In a cascade connectttttttiiiiiiiiiiiooooooooooonnnnnnnnn,,,,output signal of the 11111111111ssssssssttttttt aaaaaaaaaaammmmmmmmmmmmmmpppppppppppplllllliiiiiiiffffffffffiiiiiiieeeeeeeeeeeeerrrrrr iiiiiisssssssssss iiiiiiiiiinnnnnnnnnnnnppppppppppppuuuuuuuuuuutttttttt sssssssssssiiiiiiiiggggggggggggggnnnnnnnaaal for the 2nd amplifier.

o We assume that op amps are ideal.

o Task is to derivedifferential equation that relates vo to vgv .


8.5. A Circuit with 2 Integrating Amplifiers


o Energy stored in circuit initially is 0, and op amps are ideal:



o Energy stored in circuit initially is 0, and op amps are ideal:


o The 2nd integrating amplifier saturateswhen vo reaches 9 V or t = 3 s.

o But it is possible that the 1st integrating amplifier saturates before t = 3 s:

o At t = 3 s, vo1 = -3 V.

o Power supply voltage on the 1st integrating amplifier is ±5 V.

o Circuit reaches saturationwhen the 2nd amplifier saturates.



o The 2nd integrating amplifier saturatesssswhen vo reaches 9 V oooooorrrrrrr ttttttttttt ============= 3333333333333 sssssss......

o But it is possibllleeeee tttttttttthhhhhhhhhhhhaaaaaaaattttttttttt ttttttthhhheeeeeeeeeee 1111111111ststststststststtsstts iiiiiiiiiinnnnnnnntttteeeeeeeeeeggggggggggrrrrrrrrrraaaaaaaaaaattttttttttiiiiiiinnnnnnnnnnnnngggggggggg aaaaaaaaaaaammmmmmmmmmmmppppppppppllllllliiiifffffffffiiiiiiiiieeeeeeeeerrrrrrr sssaaaaaaaaaaattttttttttuuuuuuuuurrrrrrrrraaaaaaaatttttttttttteeeeeeeeeessssssssss bbbbbbbbbbbbbbeeeeeeeeeeeeffffffffffffffoooooooooooooorrrrrrreeee t = 3 s:

o At t = 3 s, vo1 = -3 V.

o Power supply voltage on the 1st integrating amplifier is ±5 V.

o Circuit reaches saturationwhen the 2nd amplifier saturates.


8.5. A Circuit with 2 Integrating Amplifiers2 Integrating Amplifiers with Feedback Resistors

o The Reason the op ampin integrating amplifier saturates

is feedback capacitor's accumulation of charge.

o A resistor is placed in parallel with each feedback capacitorto overcome this problem.


2 Integrating Amplifiers with Feedbacck Resistors

o The Reason thheeee ooooppppp aaammmmppppppin integrattiiiinnngggg aaammmmpppppppppllllliiiffffffffiiiiieeeerrr ssssaaaattttuuuurrrraaaatttteeeessss

is feedback cappaaaaaaaaccciiiitttooooorrr'''ssss aaaaccccccccuuummmmuuuuulllllaaaattttiiiiooooonnnnnn ooooffff cccchhhhhhaaarrrggggee.

o A resistor is placed in parallllell wiithhh eachhh ffeeddbbbbackk capacitorto overcome this problem.



o Final value of vo is input voltage times gain of each stage.

o Capacitors behave as open circuits as :


o Final value of vo iiis iiinnnppuuuuuttttttt vvvvooollltttaaaaaaagggggeeee tttttiiiiiimmmmeeeesssssssss gggggggggaaaaaaiiiinnnn ooooffff eeeeeeeaaaaaaaaaccccchhh ssssssttttttttaaaagggggggeeeeee...

o Capacitors behave as opennn cccccccciiiiiiirrrrrrrcccccccuuuuuuiiiiiiiittttttttssssssss aaaaaaaassssssss :::::


o Solution assumes neither op amp saturates.

o Final value of vo is 5 V, which is less than 6 V.The 2nd op amp does not saturate.

o Final value of vo1 is (250×10-3)(-500/100), or -1.25 V.The 1st op amp does not saturate.



o Solution assumes neithher ooooppppppppppp aaaaaaammmmmmmmppppppppppp ssssssssssaaaaaaaattttttttuuuuuuuurrrrrrrrraaaattttttteeeeeeeeeeesssssss....

o Final value of vo is 5 V, which is less than 6 V.The 2nd op amp does not saturate.

o Final value of vo1 is (250×10-3)(-500/100), or -1.25 V.The 1st op amp does not saturate.








8.6. Summary


8.0. Introduction






8.6. Summary


8.6. Summary


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