Overview
•  Concepts review
•  Practice problems – solutions posted
•  Lab 7 preview
Second Order
Circuits III
o Step response of a series RLC circuit
•  HW 6 circuit behavior graphing review
EGR 220, Chapter 8
March 24, 2016
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RLC Concept Review
RLC Concept Review
•  In a series RLC circuit, setting R = 0 will
produce
•  What are the differences between:
o Series RLC circuit
o Parallel RLC circuit
o General RLC circuit
a)
b)
c)
d)
e)
•  What is meant by
o The behavior of a circuit
o The response of a circuit
an overdamped response
a critically damped response
an underdamped response
an undamped response
none of the above
•  What happens in a parallel RLC circuit
with a very small R?
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RLC Concept Review
RLC Concept Review
•  The response of the circuit will be
a)
b)
c)
d)
•  If vL(0) = 5V, what do we know about dvL/dt?
overdamped
underdamped
critically damped
none of the above
a)
b)
c)
d)
dvL/dt = 5 V/s
nothing
dvL/dt = 0 V/s
none of the above
•  Think of the notation
dvL (t)
dt t=0
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Terminology Summary
RLC Concept Discussion
Discuss with your neighbors:
•  Know how to find:
•  What are the main concepts for dynamic
circuits?
•  What behavior is expected from a dynamic
circuit?
o  Characteristic equation
o  Natural frequencies
o  Resonant frequency
o  1st order vs. 2nd order, vs higher order
o  Sketch the behavior options
o  Damping factor
•  Identify role, or impact, of each parameter in
the sol’n expression on each graph
o  Damping frequency, ωd, for underdamped
systems
o  τ, α, ω0, ωd, initial & final state (or condition)
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2nd Order Circuit Review
Example 1: RLC Circuit
•  Dynamic circuits are either in
•  What are the 3 versions of
this circuit we analyze?
o  Steady-state, or…
o  Transition from one steady-state to the next
o  Draw each circuit
o  Identify how/why each is used
•  Our steady–states
o
o
o
•  Transient behavior (in ‘transition’)
o
o
o
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Example 1: RLC Circuit
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Example 1: RLC Circuit
•  Find v(t) for t > 0
•  Initial conditions
o  vc(0+) = 10V
o  iL(0+) = 2A
o  dvc(0+)/dt = -1/2 A/s
•  Also need:
o  α = 5/2
o  ωo = ½
•  Find v(t) for t > 0
•  Initial Conditions?
•  Type of damping?
o  V∞ = _____?
•  Our natural response is therefore… Overdamped, because
α > ωo (from these we can find the roots, s1 and s2)
•  This means the expression we want is of the form
v(t) = V∞ + A1e s1t + A2 e s2t
•  So we need to find (V∞ and) the coefficients, A1 and A2
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Example 1: RLC Circuit
Solution – Graphing v(t), i(t)
•  Find v(t) for t > 0
1)  Find the initial conditions (the first or initial
steady-state) and mark this on the graph.
2)  Find the final conditions (the 2nd steadystate) and mark this on the graph.
3)  Determine the form of the transition, i.e., the
type of damping.
4)  Sketch the manner in which the voltage
and current make the transition from the
initial to the final conditions.
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Lab 7 Circuit (preview)
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Lab 7 Circuit (preview)
o  Plan your solution approach
R = 2kΩ; L = 47mH; C = 100pF
•  What is happening in each time period of interest?
•  What do you know for certain about this circuit?
•  What might be expressions you want to find?
R = 2kΩ
L = 47mH
C = 100pF
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Series RLC – Lab 7 Concepts
Lab 7: Series RLC Circuit
1)
2)
3)
4)
5)
6)
7)
•  Properties
Step response
Ringing
Resonance
ω 0 and ω d
•  Review the end of §8.3, (excluding the examples)
•  The paragraph beginning with “We conclude this section
by noting the following...”
o  What determines the rate of damping?
o  Adjusting which values shifts the response between
over-, critically- and underdamped?
o  Elements L & C transfer energy back and forth –
leading to ‘ringing’ or ‘oscillations’
o  Settling time – overdamped response has the
longest settling time!
Overshoot
Settling time
Role of “R”
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Example 2: RLC Circuit
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Example 2: RLC Circuit
•  Find i(t) and v(t) for t > 0
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HW6: “Exponential Envelope”
HW 6 Circuit Behavior
&
Graphs
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Homework Problems 24
25
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Homework Problems
-4.68e-37.7t + 64.68e-2.68t V
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Homework Problems
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SAMPLE MATLAB CODE – YOU MAY USE ONLY IF YOU MODIFY TO MAKE IT
YOUR OWN AND COMMENT THAT YOU STARTED WITH THE CLASS SAMPLE
%% HOMEWORK PROBLEM - an underdamped circuit
% Construct the x-axis data vector
t2 = [0:0.001:0.12];
% Define the constants for this circuit
alpha2 = 50;
omega2 = 10^6;
omegaD2 = sqrt(omega2 - alpha2^2);
A21a = 2;
A22a = 0.1;
A21b = 0;
A22b = -400;
% Construct the current and voltage waveforms for this circuit
i2 = exp(-alpha2 *t2) .* (A21a*cos(omegaD2*t2) +
A22a*sin(omegaD2*t2));
v2 = exp(-alpha2 *t2) .* (A21b*cos(omegaD2*t2) +
A22b*sin(omegaD2*t2));
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% Construct the exponential 'envelope' for the shape of the
% exponential decay of the oscillatory behavior
env21a = A21a * exp(-alpha2*t2);
env22a = - A21a * exp(-alpha2*t2);
env21b = A22b * exp(-alpha2*t2);
env22b = - A22b * exp(-alpha2*t2);
% Plot the response of the circuit
figure(2)
plot (t2, i2, 'b', 'LineWidth', 2)
hold on
plot (t2, env21a, 'm-', t2, env22a, 'b-', 'LineWidth', 1);
xlabel ('time (seconds)','FontSize',14)
ylabel ('i(t) (amps)', 'FontSize',14)
title (’i(t) RLC Circuit Underdamped Response','FontSize',14)
legend ('i(t)', 'Exponential decay envelope', 'Exponential
decay envelope')
grid ON
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