Title: NETWORKS 2: 090920201
1 CHAPTER 9
- NETWORKS 2 0909202-01
- 6 December 2005 Lecture 11
- ROWAN UNIVERSITY
- College of Engineering
- Dr Peter Mark Jansson, PP PE
- DEPARTMENT OF ELECTRICAL COMPUTER ENGINEERING
- Autumn Semester 2005 Quarter Two
2admin
- Reworking the Mid-Term
- Due this Friday by noon
- Honor Code in Force
- No collaborations
- Final Exam Thursday 15 Dec 2005
- 8AM 10AM (Rowan Auditorium)
- HW 5 due today, 6 tomorrow , 7 next week
3Chapter 9 key concepts
- Todays learning objectives
- ecomms and power systems
- diff eqs for two energy storage elements
- natural response 2nd order diff eqs
- natural responses
- unforced parallel RLC circuit
- critically damped unforced parallel RLC circuit
4electrical communications and power systems
May include wires, earths atmosphere and/or free
space
Signal or power IN
Signal or power OUT
5key 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
eqs 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
6key 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
7ecomms/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)
8First, 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
9example 1
10example 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
11example 2
12example 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
13Hw problem 9.3-1
14Hw problem 9.3-1
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
15the operator method
Method 2 operator method 1) 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
16example 3
- Find the differential equation for the node
voltage -
17example 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
18example 3 operator method continued
Multiplying through by 1000 yields
19example 3 operator method finishing up
20example 3 operator method - the end
Write your answer for coefficients of vs terms as
LC4
21example 4
- Find the 2nd order differential equation for
circuit shown in terms of v using the operator
method -
22example 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
23example 4 completing the operator method
Multiplying through by 2s we get.
Write the final 2nd order diff eq as LC5
24solution 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
25solution 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
26solution 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)
27solution of the 2nd order diff eq the natural
response
- substituting the value of xn from (2) into (1)
(3)
solving we obtain
(4)
28solution 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
29solution 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.
30solution 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.
31example 5
- Find the characteristic equation and the natural
frequencies for the circuit shown below -
32example 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
33solution 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
34Hw solution 9.4-2
35Hw solution 9.4-2 - 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
36solution of 9.4-2 - 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
37- REMINDER
- HW 5 now
- HW 6 tomorrow
- HW 7 next Tuesday