rl transient
Post on 20Jan2016
15 views
Embed Size (px)
DESCRIPTION
RL TransientTRANSCRIPT
Transients Analysis
Solution to First Order Differential Equation Consider the general EquationLet the initial condition be x(t = 0) = x( 0 ), then we solve the differential equation:The complete solution consists of two parts: the homogeneous solution (natural solution) the particular solution (forced solution)
The Natural Response Consider the general EquationSetting the excitation f (t) equal to zero, It is called the natural response.
The Forced Response Consider the general EquationSetting the excitation f (t) equal to F, a constant for t 0 It is called the forced response.
The Complete Response Consider the general EquationThe complete response is: the natural response + the forced response Solve for , The Complete solution:called transient responsecalled steady state response
WHAT IS TRANSIENT RESPONSEFigure 5.1
Figure 5.2, 5.3Circuit with switched DC excitationA general model of the transient analysis problem
In general, any circuit containing energy storage elementFigure 5.5, 5.6
Figure 5.9, 5.10(a) Circuit at t = 0(b) Same circuit a long time after the switch is closedThe capacitor acts as open circuit for the steady state condition(a long time after the switch is closed).
(a) Circuit for t = 0(b) Same circuit a long time before the switch is openedThe inductor acts as short circuit for the steady state condition(a long time after the switch is closed).
Why there is a transient response?The voltage across a capacitor cannot be changed instantaneously. The current across an inductor cannot be changed instantaneously.
Figure 5.12, 5.1356Example
Transients Analysis1. Solve firstorder RC or RL circuits.
2. Understand the concepts of transient response and steadystate response.
3. Relate the transient response of firstorder circuits to the time constant.
TransientsThe solution of the differential equation represents are response of the circuit. It is called natural response.
The response must eventually die out, and therefore referred to as transient response. (source free response)
Discharge of a Capacitance through a ResistanceSolving the above equation with the initial conditionVc(0) = Vi
Discharge of a Capacitance through a Resistance
Exponential decay waveformRC is called the time constant.At time constant, the voltage is 36.8%of the initial voltage.Exponential rising waveformRC is called the time constant.At time constant, the voltage is 63.2% of the initial voltage.
RC CIRCUIT for t = 0, i(t) = 0u(t) is voltagestep function
R
C
+
V
C

i(t)
t = 0
+
_
V
R
C
+
V
C

i(t)
t = 0
+
_
Vu(t)
RC CIRCUIT Solving the differential equation
Complete ResponseComplete response = natural response + forced responseNatural response (source free response) is due to the initial conditionForced response is the due to the external excitation.
Figure 5.17, 5.1858a). Complete, transient and steady state responseb). Complete, natural, and forced responses of the circuit
Circuit Analysis for RC CircuitApply KCL vs is the source applied.
DC
Vs
+Vc
+ VR 
R
C
iR
iC
Solution to First Order Differential Equation Consider the general EquationLet the initial condition be x(t = 0) = x( 0 ), then we solve the differential equation:The complete solution consits of two parts: the homogeneous solution (natural solution) the particular solution (forced solution)
The Natural Response Consider the general EquationSetting the excitation f (t) equal to zero, It is called the natural response.
The Forced Response Consider the general EquationSetting the excitation f (t) equal to F, a constant for t 0 It is called the forced response.
The Complete Response Consider the general EquationThe complete response is: the natural response + the forced response Solve for , The Complete solution:called transient responsecalled steady state response
Example
Initial condition Vc(0) = 0V
DC
100V
+Vc
+ VR 
100 k ohms
0.01 microF
iR
iC
Example
Initial condition Vc(0) = 0Vand
DC
100V
+Vc
+ VR 
100 k ohms
0.01 microF
iR
iC
Energy stored in capacitorIf the zeroenergy reference is selected at to, implying that thecapacitor voltage is also zero at that instant, then
Power dissipation in the resistor is:pR = V2/R = (Vo2 /R) e 2 t /RCRC CIRCUITTotal energy turned into heat in the resistor
R
C
RL CIRCUITSInitial condition i(t = 0) = Io
L
R
VR+
+VL
i(t)
RL CIRCUITSInitial condition i(t = 0) = Io
RL CIRCUITPower dissipation in the resistor is:pR = i2R = Io2e2Rt/LRTotal energy turned into heat in the resistorIt is expected as the energy stored in the inductor is
RL CIRCUIT where L/R is the time constant
DC STEADY STATEThe steps in determining the forced response for RL or RC circuits with dc sources are:1. Replace capacitances with open circuits.2. Replace inductances with short circuits.3. Solve the remaining circuit.
Recommended