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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The Differential Equation16
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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The Differential Equation17
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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The Differential Equation27
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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