differntial equations on rlc circuit

7/30/2019 Differntial Equations on RLC Circuit

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1

Differential Equation Solutions

of Transient Circuits


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1st Order Circuits2

Any circuit with a single energy storage element, anarbitrary number of sources, and an arbitrarynumber of resistors is a circuit oforder 1

Any voltage or current in such a circuit is thesolution to a 1st order differential equation


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RLC Characteristics

Element V/I Relation DC Steady-State

Resistor V = I R

Capacitor I = 0; open

Inductor V = 0; short

)()( tiRtv RR

dttvdCti CC )()(

dt

tidLtv LL

)()(

ELI and the ICE man


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A First-Order RC Circuit4

One capacitor and one resistor in series

The source and resistor may be equivalent to acircuit with many resistors and sources

R

Cvs(t)

+

vc(t)

+ vr(t)

+


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The Differential Equation5

KVL around the loop:

vr(t) + vc(t) = vs(t)

vc(t)

R

Cvs(t)

+

+ vr(t)

+


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RC Differential Equation(s)6

)()(1

)( tvdxxiC

tiR s

t

dt

tdvCti

dt

tdiRC s

)()(

)(

dttdvRCtv

dttdvRC sr

r )()()(

Multiply by C;

take derivative

From KVL:

Multiply by R;note vr=Ri


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A First-Order RL Circuit7

One inductor and one resistor in parallel

The current source and resistor may be equivalentto a circuit with many resistors and sources

v(t)is(t) R L

+


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The Differential Equations8

KCL at the top node:

)()(1)( tidxxvLR

tvs

t

v(t)is(t) R L

+


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RL Differential Equation(s)9

)()(1)(

tidxxvLR

tvs

t

dt

tdiLtv

dt

tdv

R

L s )()()(

Multiply by L;take derivative

From KCL:


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1st Order Differential Equation10

Voltages and currents in a 1st order circuit satisfy adifferential equation of the form

wheref(t) is the forcing function (i.e., the independentsources driving the circuit)

)()()( tftxadttdx


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The Time Constant ()11

The complementary solution for any first ordercircuit is

For an RC circuit, =RC

For an RL circuit, =L/R

WhereR is the Thevenin equivalent resistance

/

)(t

c Ketv


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What Does vc(t) Look Like?12

= 10-4


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Interpretation of13

The time constant, , is the amount of time necessaryfor an exponential to decay to 36.7% of its initial

value

-1/ is the initial slope of an exponential with aninitial value of 1


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Applications Modeled bya 1st Order RC Circuit

14

The windings in an electric motor or generator

Computer RAM A dynamic RAM stores ones as charge on a capacitor

The charge leaks out through transistors modeled by largeresistances

The charge must be periodically refreshed


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Important Concepts15

The differential equation for the circuit

Forced(particular) and natural(complementary)solutions

Transientand steady-state responses 1st order circuits: the time constant()

2nd order circuits: natural frequency(0) and thedamping ratio ()


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Every voltage and current is the solution to adifferential equation

In a circuit of order n, these differential equations

have order n The number and configuration of the energy storage

elements determines the order of the circuit

n number of energy storage elements


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Equations are linear, constant coefficient:

The variablex(t) could be voltage or current

The coefficients an through a0 depend on thecomponent values of circuit elements

The functionf(t) depends on the circuit elementsand on the sources in the circuit

)()(...)()(

01

1

1tftxa

dt

txda

dt

txda

n

n

nn

n

n


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Building Intuition18

Even though there are an infinite number ofdifferential equations, they all share commoncharacteristics that allow intuition to be developed:

Particular and complementary solutionsEffects of initial conditions


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Differential Equation Solution19

The total solution to any differential equationconsists of two parts:

x(t) = xp(t) + xc(t)

Particular (forced) solution isxp

(t)

Response particular to a given source Complementary (natural) solution isxc(t)

Response common to all sources, that is,

due to the passive circuit elements


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Forced (or Particular) Solution20

The forced (particular) solution is the solution tothe non-homogeneous equation:

The particular solution usually has the form of asum off(t) and its derivatives That is, the particular solution looks like the forcing

function Iff(t) is constant, thenx(t) is constant

Iff(t) is sinusoidal, thenx(t) is sinusoidal

)()(...)()(

01

1

1tftxa

dt

txda

dt

txda

n

n

nn

n

n


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Natural/Complementary Solution21

The natural (or complementary) solution is thesolution to the homogeneous equation:

Different look for 1st and 2nd order ODEs

0)(...)()(01

1

1

txadt

txdadt

txdan

n

nn

n

n


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First-Order Natural Solution22

The first-order ODE has a form of

The natural solution is

Tau () is the time constant For an RC circuit, = RC

For an RL circuit, = L/R

/)(

t

c Ketx

0)(1)(

txdt

tdxc

c


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Second-Order Natural Solution

The second-order ODE has a form of

To find the natural solution, we solve thecharacteristic equation:

which has two roots: s1 and s2 The complementary solution is (if were lucky)

tsts

c eKeKtx21

21)(

022

00

2 ss

0)()(

2)( 2

002

2

txdt

tdx

dt

txd


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Initial Conditions24

The particular and complementary solutions haveconstants that cannot be determined withoutknowledge of the initial conditions

The initial conditions are the initial value of thesolution and the initial value of one or more of itsderivatives

Initial conditions are determined by initial

capacitor voltages, initial inductor currents, andinitial source values


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2nd Order Circuits25

Any circuit with a single capacitor,a single inductor,an arbitrary number of sources, and an arbitrarynumber of resistors is a circuit oforder 2

Any voltage or current in such a circuit is thesolution to a 2nd order differential equation


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A 2nd Order RLC Circuit26

The source and resistor may be equivalent to acircuit with many resistors and sources

vs(t)

R

C

i(t)

L

+


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KVL around the loop:

vr(t) + vc(t) + vl(t) = vs(t)

vs(t)

R

C

+

vc(t)

+vr(t)

L

+vl(t)

i(t)

+


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RLC Differential Equation(s)28

)()(

)(1

)( tvdt

tdiLdxxi

CtiR s

t

dt

tdv

Ldt

tid

tiLCdt

tdi

L

R s )(1)(

)(

1)(

2

2

Divide by L, and take the derivative

From KVL:


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differntial equations on rlc circuit

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