electric signals
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electric signalsTRANSCRIPT
UEEA1253 SIGNALS, CIRCUITS & SYSTEMS
SEE 2015
Faculty of Engineering And Science
Y.C.See
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SIGNAL I
Analysis of signals is important
When we are dealing with dynamics of system and networks, the signals are function of time.
DC signal
AC signal – sinusoidal signal
Only periodic signal which retain its shape under a linear operation -> +, - ,
A sinusoidal current pass through an inductor will produce a sinusoidal voltage at a same freq. Capacitor?
Exponential signal
General term
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, ddt∫
( )
A & S = in general complexSt have to be dimensionless
Stf t Ae=
2
SIGNAL II
E.g.
In the complex plane you have different kind of behavior as the time dependant signal is concern
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=2Ae cos( )t tσ ω φ+
( )Ae j j teφ σ ω+ +
ω
σ
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SINGULARITY FUNCTION
Discontinuous functions
Unit step function
Unit ramp function
Unit impulse function/ delta/dirac function
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UNIT STEP FUNCTION I
t=0 ?
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UNIT STEP FUNCTION II SEE 2015
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UNIT RAMP FUNCTION – CAUSAL SIGNAL SEE 2015
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UNIT RAMP FUNCTION – CAUSAL SIGNAL
IS r(t) = t ????
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CHALLENGING (SADIKU 7.26(D) PG 304
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UNIT IMPULSE FUNCTION Delta dirac function
Properties
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Sifting Property of the Impulse----If b>a, then
UNIT IMPULSE FUNCTION II
What are the relation among unit, ramp and impulse?
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ENERGY STORAGE ELEMENTS I (FAST)
Capacitors store energy in an electric field which consists of two conducting surfaces separated by a dielectric material.
Inductors store energy in a magnetic field which cossists of a conducting wire in a form of a coil
Capacitors and inductors are passive elements: Can store energy supplied by circuit Can return stored energy to circuit Cannot supply more energy to circuit than is stored.
Voltages and currents in a circuit without energy storage elements are solutions to algebraic equations.
Voltages and currents in a circuit with energy storage elements are solutions to linear, constant coefficient differential equations.
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ENERGY STORAGE ELEMENTS II (FAST)
Electrical engineers (and their software tools) usually do not solve the differential equations directly.
Instead, they use: Laplace transforms – you will learn the beauty of it
AC steady-state analysis
These techniques covert the solution of differential equations into algebraic problems.
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ENERGY STORAGE ELEMENTS III (FAST)
Energy storage elements model electrical loads: Capacitors model computers and other electronics
(power supplies). Inductors model motors.
Capacitors and inductors are used to build filters and amplifiers with desired frequency responses.
Capacitors are used in A/D converters to hold a sampled signal until it can be converted into bits.
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CAPACITORS (FAST) SEE 2015
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CAPACITORS (FAST)
Capacitance occurs when two conductors are separated by a dielectric (insulator).
Charge on the two conductors creates an electric field that stores energy.
The voltage difference between the two conductors is proportional to the charge.
The proportionality constant C is called capacitance.
Capacitance is measured in Farads (F).
Given
( ) ( )tvCtq =
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CAPACITORS (FAST)
i(t) +
-
v(t)
The rest of the
circuit
dttdvCti )()( =
∫∞−
=t
dxxiC
tv )(1)(
∫+=t
t
dxxiC
tvtv0
)(1)()( 0
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ENERGY STORED IN A CAPACITOR (FAST) The energy stored on a capacitor can be expressed in terms of the work done by the battery The voltage V is proportional to the amount of charge which is already on the capacitor.
If Q is the amount of charge stored when the whole battery voltage appears across the capacitor, then the stored energy is obtained from the integral:
It can be rewrite so that,
Element of energy stored:
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DC IN CAPCITOR(FAST)
A dc voltage does not changes with time, therefore the current is zero – block DC – OPEN CIRCUIT
From
What if we have instantaneous change in capacitor voltage?
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CAPACITORS – WHAT YOU HAVE LEARNED! (FAST)
In steady state, the reactance
In terms of Laplace transform,
CjX C ω
1=
sCX C
1=
Parallel & Serial capacitors :
∑=k
kCC
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INDUCTORS (FAST) SEE 2015
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INDUCTORS (FAST)
Inductance occurs when current flows through a (real) conductor.
The current flowing through the conductor sets up a magnetic field that is proportional to the current.
The voltage difference across the conductor is proportional to the rate of change of the magnetic flux.
The proportionality constant is called the inductance, denoted L.
Inductance is measured in Henrys (H).
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INDUCTORS (FAST)
dttdiLtv )()( =
i(t)
+
-
v(t)
The rest of the
circuit
L
∫∞−
=t
dxxvL
ti )(1)(
∫+=t
t
dxxvL
titi0
)(1)()( 0
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Energy stored in an inductor
21( ) ( )2Lw t Li t=
INDUCTORS – WHAT YOU HAVE LEARNED! (FAST)
What happen if dc current flowing through an inductor? The current in an inductor cannot change
instantaneously.
In steady state, the reactance of an inductor
In terms of Laplace transform, the reactance of an inductor
LjX L ω=
sLX L =
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INDUCTORS – WHAT YOU HAVE LEARNED! (FAST)
Series inductors ∑=
=N
kks LL
1
Parallel inductors
∑=
=N
k kp LL 1
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P 7.3-6 (PG 298 DORF)
The initial capacitor voltage of the circuit shown in Figure is vc(0-) = 3 V. Determine (a) the voltage v(t) and (b) the energy stored in the capacitor at t= 0.2 s and t= 0.8 s when
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P 7.8-12 (DORF PAGE 308)
The circuit shown in Figure has reached steady state before the switch closes at time t = 0.
(a) Determine the values of iL(t ), vc(t ), and vR(t) immediately before the switch closes. (b) Determine the value of vR(t ) immediately after the switch closes.
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E.G. 3
Find Vs, assuming the waveform for Vc is as shown
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REFERENCES
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Dorf, Richard C., & Svoboda, James A. (2011). Introduction to electric circuits. (8th ed.). Hoboken, NJ: John Wiley & Sons.
Alexander, Charles K., & Sadiku, Matthew N.O. (2007). Fundamentals of electric circuits. Boston: McGraw-Hill Higher Education.
Hayt, William; Kemmerly, Jack E. (1971), Engineering Circuit Analysis (2nd ed.), McGraw-Hill, ISBN 0-07-027382-0
John Bird. (2007) Electrical circuit theory and technology.Newness
J.David Irwin, R.Mark Nelms (2010) Basic Engineering Circuit Analysis. Wiley