NETWORKS 2: ECE 09'202'01

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CHAPTER 9

• NETWORKS 2 ECE 09.202.01
• 4 December 2007 Lecture 9
• ROWAN UNIVERSITY
• College of Engineering
• Dr Peter Mark Jansson, PP PE
• DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
• Autumn Semester 2007 Quarter Two

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admin

• Final Exam Tuesday 18 Dec 2007
• 10.15 1215PM (Rowan Lab 1st flr)
• HW 6 due beginning of lecture tomorrow

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Chapter 9 key concepts

• Todays learning objectives
• ecomms and power systems
• diff eqs for two energy storage elements
• Cramers Rule review
• natural response 2nd order diff eqs
• natural responses
• unforced parallel RLC circuit
• critically damped unforced parallel RLC circuit

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electrical communications and power systems
May include wires, earths atmosphere and/or free
space
Signal or power IN
Signal or power OUT
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key ecomms/power innovations (1864-1924)

• First prediction of EM waves (light speed) 1864
  Maxwell
• First DC power grid Pearl Street, NYC, US 1879
  Edison
• First coherer to detect EM waves 1885 Branley
• First gt100kW electric machine 1885 Siemens
• First observation that E-M waves dont diminish
  w/ R2 1886 Hertz
• First 2 3 phase AC motors/alternators patented
  1887 Tesla
• First AC power grid design 1888 Tesla
  Ferrari
• First oscillator designed to vary EM wave
  frequency 1888 Hertz
• First electrical engineering course developed
  1889 Columbia
• First publication of theory on operators for diff
  eqns 1892 Heaviside
• First RF transmitter circuit designed 1894
  Marconi
• First radio system invented 1901 Tesla
  Marconi
• First commercial radio 1920 WKDA Pittsburgh
• First television system invented 1924 RCA
  Zworykin

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key ecomms/power innovations (1936-1995)

• First FM radio developed and announced 1936
  Armstrong
• First commercial television 1939 New York
  City
• First radar and microwave systems developed
  1938-1945 WW II
• First telecommunications satellite 1962
  Telstar
• First fiber optic telephone communications system
  1983
• First cellular mobile telephone 1984
• First global internet 1995

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ecomms/power circuits

• an electrical system is an interconnection of
  electrical elements and circuits to achieve a
  desired objective
• both communications and power systems are
  electrical systems that contain capacitance and
  inductance throughout their circuits
• this chapter will enable us to determine the
  natural and forced responses of 2nd order
  circuits - circuits with two irreducible energy
  storage elements (NOTE irreducible means that
  all parallel or series connections of C L
  elements have been made, reducing C L
  components to their irreducible form)

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First, well learn two methods for determining
2nd order differential equations for RLC circuits

• Method 1 direct method
• write node (or mesh) equation
• substitute equation for L or C into it
• Method 2 operator method
• we will develop this after an example of method 1

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example 1
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example 1 direct method
KCL at top node -is v/R iL Cdv/dt 0
v1 -
Equation for inductor v Ldi/dt Substituting
value of v from inductor into KCL
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example 2
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example 2 lets try the direct method
KVL for the loop -vs Ldi/dt vC Ri 0
Equation for capacitor i Cdv/dt Substituting
value of i from capacitor into KVL
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Hw problem 9.2-1
Find the differential equation for the circuit
show using the direct method
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Hw problem 9.2-1
substituting KCL values in KVL equation
combine common terms and substitute values to get
Write your answer for X as LC1, for Y as LC2,
and Z as LC3
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the operator method

• Method 2 operator method
• write node (or mesh) equation(s)
• 2) use operators sd/dt or 1/s?dt to make
  algebraic equations
• 3) use Cramers rule to solve for desired value
• 4) convert operators back to find differential
  eqns

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example 3

• Find the differential equation for the node
  voltage using the operator method

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example 3 lets try the operator method
KCL for the top node (v-vs)/R1 i Cdv/dt 0
2nd Equation Ri Ldi/dt v Substitute
operators for i and v
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Ex. 3 operator method w/Cramers Rule
Set up matrix and solution
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Ex. 3 operator method w/Cramers Rule
A
b
1s 1000 vs -1000 1000s 0
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Lets recall Cramers Rule (App A-4)

• See pages 830-831

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Ex. 3 operator method w/Cramers Rule
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example 3 operator method continued
Multiplying through by 1000 yields
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example 3 operator method finishing up
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example 3 operator method - the end
Write your answer for coefficients of vs terms as
LC4
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example 4

• Find the 2nd order differential equation for
  circuit shown in terms of v using the operator
  method

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example 4 via the operator method
KCL for the top node is(t) v(t)/1 i(t)
(0.5)dv(t)/dt
2nd Equation 1di/dt v(t) Substitute operators
for i and v
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example 4 completing the operator method
Multiplying through by 2s we get.
Write the final 2nd order diff eq as LC5
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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 5

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

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example 5 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 LC6
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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 9.4-2 (see page 387)

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 LC7