NETWORKS 2: ECE09'202'01

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CHAPTER 9

• NETWORKS 2 ECE09.202.01
• 28 November 2006 Lecture 12
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2006 Quarter Two

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admin
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Chapter 9 key concepts

• Todays learning objectives
• natural response 2nd order diff eqs
• natural responses
• unforced parallel RLC circuit
• critically damped unforced parallel RLC circuit
• unforced underdamped parallel RLC circuit
• overdamped and undamped RLC circuits
• forced response of RLC circuit
• complete response of RLC circuit

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solution of the 2nd order diff eq the natural
response

• we have now seen that a circuit with two
  irreducible energy storage elements can be
  represented by a 2nd order diff eq of the
  following general form

Where a2, a1 and a0 are known and the forcing
function f(t) is specified
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solution of the 2nd order diff eg the natural
response

• the complete response of a circuit with two
  irreducible energy storage elements x(t) can be
  represented by its two components, namely the
  natural response (xn) and the forced response
  (xf)

Where a2, a1 and a0 are known and the forcing
function f(t) is specified
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solution of the 2nd order diff eg the natural
response

• the natural response (xn) satisfies the unforced
  2nd order diff eq when f(t)0

(1)
Since the exponential function is the only
function that is proportional to all of its
derivatives and integrals we postulate this
general solution
(2)
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solution of the 2nd order diff eq the natural
response

• substituting the value of xn from (2) into (1)

(3)
solving we obtain
(4)
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solution of the 2nd order diff eq the natural
response

• solving for the non-trivial solution (xn ? 0)

(5)
We arrive at the characteristic equation whose
solutions are
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solution of the 2nd order diff eg the natural
response

• there are two distinct roots and two solutions

(6)
The roots of the characteristic equation contain
all the information necessary for determining the
character of the natural response. The roots (s1
s2) are the characteristic roots and are often
called the natural frequencies.
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solution of the 2nd order diff eg the natural
response

• there are two distinct roots and two solutions

(6)
The real roots (s1 s2) are often called the
natural frequencies of the circuit. The
reciprocals of these real characteristic roots
are the circuits time constants.
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example 1

• Find the characteristic equation and the natural
  frequencies for the circuit shown below

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example 1 via the operator method
KCL for the top node is(t) v(t)/4 i(t)
(0.25)sv(t)
KVL right mesh i(t)(6s) v(t) Combine
equations for i and v
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solution of the 2nd order diff eq the natural
response

• solving for the characteristic equation

we set the coefficients of i(t) equal to zero
natural frequencies
Write the circuits time constants as LC1
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Hw solution 9.3-2
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Hw solution 9.3-2 - the natural response

• find the characteristic equation and its roots
  for the circuit in Figure P 9.3-2 (see page 402)

divide through by LC and re-arrange to obtain
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solution of 9.3-2 - the natural response
L100 mH C1/3 mF

• the characteristic equation and its roots are

Write the circuits natural frequencies and time
constants as LC2
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Definitions

• Unforced circuit no forcing function is present
  (interactions are between energy storage elements
  only)
• Undamped circuit a circuits natural response
  will be to oscillate indefinitely
• Damped circuit a circuits natural response will
  be curtailed from oscillating indefinitely and
  depending upon damping may immediately be
  quenched (see over, under, and critically damped)

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natural response unforced parallel RLC circuit
This is the characteristic equation of the
parallel RLC
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more definitions

• For unforced parallel RLC circuits
• Resonant frequency a circuits resonant
  frequency occurs when ?2 1/LC or when

• At resonant frequency a circuits reactance is
  zero, the phase angle is zero and the impedance
  is purely resistive
• NOTE when ? is lower than resonance, the phase
  angle is positive and the reactance is inductive,
  when ? is higher, the phase angle is negative and
  reactance is capacitive

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natural response unforced parallel RLC circuit
The roots we identified in damped circuits will
assume one of 3 possible conditions
Real, distinct overdamped
Real, equal critically damped
Complex underdamped
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the complex s-plane
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Hw example 9.4-2, page 403

• An RLC circuit where v(0)2V. The switch has
  been open for along time before closing at t0.
  Determine and plot v(t).
• Process
• Note initial conditions
• Identify characteristic equation
• Determine natural frequencies and form of v(t)
• Analyze circuit (KVL, KCL), differentiate v(t)
• Solve for A and B

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Hw example 9.4-2, page 403

• An RLC circuit where v(0)2V. The switch has
  been open for along time before closing at t0.
  Determine and plot v(t).
• Process
• Note initial conditions v(0)2V, i(0)0A

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Hw example 9.4-2, page 403

• An RLC circuit where v(0)2V. The switch has
  been open for along time before closing at t0.
  Determine and plot v(t).
• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation

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Hw example 9.4-2, page 403

• An RLC circuit where v(0)2V. The switch has
  been open for along time before closing at t0.
  Determine and plot v(t).
• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation

LC3 Write the equation for this circuit
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Hw example 9.4-2, page 403

• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation
• Determine natural frequencies and v(t)
• Natural frequencies are s1-1, s2-3

LC4 Write the general equation for the natural
response for the circuit of the form
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Hw example 9.4-2, page 403

• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation
• Determine natural frequencies and v(t)
• Analyze circuit (KVL, KCL), differentiate v(t)
• Use natural response and initial conditions to
  determine A and B

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Hw example 9.4-2, page 403

• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation
• Determine natural frequencies and v(t)
• Analyze circuit (KVL, KCL), differentiate v(t)
• Apply KCL to determine A and B

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Hw example 9.4-2, page 403

• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation
• Determine natural frequencies and v(t)
• Analyze circuit (KVL, KCL), differentiate v(t)
• Solve for A and B

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Hw example 9.4-2, page 403

• Process
• Note initial conditions v(0)2V, i(0)0A
• Identify characteristic equation
• Determine natural frequencies and v(t)
• Analyze circuit (KVL, KCL), differentiate v(t)
• Solve for A and B

LC5 Write the final equation for the natural
response for v(t)
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natural response critically damped unforced
parallel RLC circuit
The roots in this case are 2 real, equal roots
Real, equal critically damped
The natural response will be of the form
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critically damped response
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Hw 9.5-2
Find vc(t) for tgt0, assume steady state at t0-
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natural response underdamped unforced parallel
RLC circuit
The roots in this case are 2 complex roots
Complex underdamped
The natural response will be of the form
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underdamped response
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forced response of an RLC circuit
The forced response of a circuit described by a
2nd order differential equation to a forcing
function will often be of the same form as the
forcing function
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forced response of an RLC circuit
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process to find forced response

• KVL, KCL, etc. to get 2nd order diff eq
• divide by LC etc. to get standard form
• substitute component values
• assume a response (of same form)
• solve for unknown

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KVL, KCL, etc. to get 2nd order diff eq
R 6? L 7H C 1/42F is 8e-2t A
KVL v/R i C dv/dt is v L di/dt
dv/dt L di2/dt2
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divide by LC, etc. to get standard form
KVL L di/dt /R i CL di2/dt2 is
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substitute component values
R 6? L 7H C 1/42F is 8e-2t A
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assume a response (of same form)
Response if Be-2t
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solve for unknown
LC6 Write the final equation for the forced
response for i(t)
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• REMINDER
• Lab 1 due at beginning of lab
• HW 6 next tuesday in lab
• Lab 2 starts next week