SECOND-ORDER CIRCUITS

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SECOND-ORDER CIRCUITS
THE BASIC CIRCUIT EQUATION
Single Loop Use KVL
Single Node-pair Use KCL
Differentiating
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LEARNING BY DOING
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THE RESPONSE EQUATION
IF THE FORCING FUNCTION IS A CONSTANT
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COEFFICIENT OF SECOND DERIVATIVE MUST BE ONE
DAMPING RATIO, NATURAL FREQUENCY
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ANALYSIS OF THE HOMOGENEOUS EQUATION
Iff s is solution of the characteristic equation
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DETERMINE THE GENERAL FORM OF THE SOLUTION
LEARNING EXTENSIONS
Divide by coefficient of second derivative
Roots are real and equal
Roots are complex conjugate
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Form of the solution
LEARNING EXTENSIONS
Classify the responses for the given values of C
HOMOGENEOUS EQUATION
C0.5 underdamped C1.0 critically
damped C2.0 overdamped
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THE NETWORK RESPONSE
DETERMINING THE CONSTANTS
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LEARNING EXAMPLE
STEP 1 MODEL
ANALYZE CIRCUIT AT t0
STEP 2
STEP 3 ROOTS
STEP 4 FORM OF SOLUTION
STEP 5 FIND CONSTANTS
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USING MATLAB TO VISUALIZE THE RESPONSE
script6p7.m plots the response in Example
6.7 v(t)2exp(-2t)2exp(-0.5t)
tgt0 tlinspace(0,20,1000) v2exp(-2t)2exp(-0.
5t) plot(t,v,'mo'), grid, xlabel('time(sec)'),
ylabel('V(Volts)') title('RESPONSE OF OVERDAMPED
PARALLEL RLC CIRCUIT')
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LEARNING EXAMPLE
NO SWITCHING OR DISCONTINUITY AT t0. USE t0
OR t0
model
Form
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USING MATLAB TO VISUALIZE THE RESPONSE
script6p8.m displays the function
i(t)exp(-3t)(4cos(4t)-2sin(4t)) and the
function vc(t)exp(-3t)(-4cos(4t)22sin(4t))
use a simle algorithm to estimate display
time tau1/3 tend10tau tlinspace(0,tend,350)
itexp(-3t).(4cos(4t)-2sin(4t)) vcexp(-3
t).(-4cos(4t)22sin(4t)) plot(t,it,'ro',t,vc
,'bd'),grid,xlabel('Time(s)'),ylabel('Voltage/Curr
ent') title('CURRENT AND CAPACITOR
VOLTAGE') legend('CURRENT(A)','CAPACITOR
VOLTAGE(V)')
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LEARNING EXAMPLE
NO SWITCHING OR DISCONTINUITY AT t0. USE t0 OR
t0
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USING MATLAB TO VISUALIZE RESPONSE
script6p9.m displays the function
v(t)exp(-3t)(16t) tau1/3 tendceil(10tau) t
linspace(0,tend,400) vtexp(-3t).(16t) plot(
t,vt,'rx'),grid, xlabel('Time(s)'),
ylabel('Voltage(V)') title('CAPACITOR VOLTAGE')
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LEARNING EXTENSION
To find initial conditions use steady
state analysis for tlt0
And analyze circuit at t0
Once the switch opens the circuit is RLC series
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To find initial conditions we use steady state
analysis for tlt0
LEARNING EXTENSION
And analyze circuit at t0
For tgt0 the circuit is RLC series
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LEARNING EXTENSION
Analysis at t0