As the number of energy-storage elements in a circuit increases, one can therefore expect that higher-order differential equations will result.. Phasors and Impedance In this section, we
Trang 1or, equivalently, by observing that the current flowing in the series circuit is related to the capacitor
voltage by i(t) = CdvC /dt, and that Eq (11.55) can be rewritten as
(11.57)
Note that although different variables appear in the preceding differential equations, both Eqs (11.55) and (11.57) can be rearranged to appear in the same general form as follows:
(11.58)
where the general variable y(t) represents either the series current of the circuit of Fig 11.49 or the
capacitor voltage By analogy with Eq (11.54), we call Eq (11.58) a second-order ordinary differential equation with constant coefficients As the number of energy-storage elements in a circuit increases, one
can therefore expect that higher-order differential equations will result
Phasors and Impedance
In this section, we introduce an efficient notation to make it possible to represent sinusoidal signals as
complex numbers, and to eliminate the need for solving differential equations.
Phasors
Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector
whose argument, or angle, is given by (ωt + φ) and whose length, or magnitude, is equal to the peak
amplitude of the sinusoid The complex phasor corresponding to the sinusoidal signal Acos(ωt + φ)
is therefore defined to be the complex number Ae jφ:
(11.59)
1 Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form
and a frequency-domain (or phasor) form
2 A phasor is a complex number, expressed in polar form, consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal referenced to a cosine signal.
3 When using phasor notation, it is important to make a note of the specific frequency, ω, of the sinusoidal signal, since this is not explicitly apparent in the phasor expression
Impedance
We now analyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation.
The result will be a new formulation in which resistors, capacitors, and inductors will be described in the same notation A direct consequence of this result will be that the circuit theorems of section 11.3 will be extended to AC circuits In the context of AC circuits, any one of the three ideal circuit elements
RC dv C dt
- LC d
2
v C( )t
dt2
- v C( )t
a2d
2
y t( )
dt2
- a1dy t( )
dt
- a0y t( )
Ae jf = complex phasor notation for Acos(wt+f)
v t( ) = Acos(wt+f)
V jw( ) = Ae jf
Trang 2or, equivalently, by observing that the current flowing in the series circuit is related to the capacitor
voltage by i(t) = CdvC /dt, and that Eq (11.55) can be rewritten as
(11.57)
Note that although different variables appear in the preceding differential equations, both Eqs (11.55) and (11.57) can be rearranged to appear in the same general form as follows:
(11.58)
where the general variable y(t) represents either the series current of the circuit of Fig 11.49 or the
capacitor voltage By analogy with Eq (11.54), we call Eq (11.58) a second-order ordinary differential equation with constant coefficients As the number of energy-storage elements in a circuit increases, one
can therefore expect that higher-order differential equations will result
Phasors and Impedance
In this section, we introduce an efficient notation to make it possible to represent sinusoidal signals as
complex numbers, and to eliminate the need for solving differential equations.
Phasors
Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector
whose argument, or angle, is given by (ωt + φ) and whose length, or magnitude, is equal to the peak
amplitude of the sinusoid The complex phasor corresponding to the sinusoidal signal Acos(ωt + φ)
is therefore defined to be the complex number Ae jφ:
(11.59)
1 Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form
and a frequency-domain (or phasor) form
2 A phasor is a complex number, expressed in polar form, consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal referenced to a cosine signal.
3 When using phasor notation, it is important to make a note of the specific frequency, ω, of the sinusoidal signal, since this is not explicitly apparent in the phasor expression
Impedance
We now analyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation.
The result will be a new formulation in which resistors, capacitors, and inductors will be described in the same notation A direct consequence of this result will be that the circuit theorems of section 11.3 will be extended to AC circuits In the context of AC circuits, any one of the three ideal circuit elements
RC dv C dt
- LC d
2
v C( )t
dt2
- v C( )t
a2d
2
y t( )
dt2
- a1dy t( )
dt
- a0y t( )
Ae jf = complex phasor notation for Acos(wt+f)
v t( ) = Acos(wt+f)
V jw( ) = Ae jf
Trang 3Engineering Thermodynamics
12.1 Fundamentals
Basic Concepts and Definitions • Laws of Thermodynamics
12.2 Extensive Property Balances
Mass Balance • Energy Balance • Entropy Balance • Control Volumes at Steady State • Exergy Balance
12.3 Property Relations and Data
12.4 Vapor and Gas Power Cycles
Although various aspects of what is now known as thermodynamics have been of interest since antiquity, formal study began only in the early nineteenth century through consideration of the motive power of heat: the capacity of hot bodies to produce work Today the scope is larger, dealing generally with energy and entropy, and with relationships among the properties of matter Moreover, in the past 25 years engineering thermodynamics has undergone a revolution, both in terms of the presentation of funda-mentals and in the manner that it is applied In particular, the second law of thermodynamics has emerged
as an effective tool for engineering analysis and design
12.1 Fundamentals
Classical thermodynamics is concerned primarily with the macrostructure of matter It addresses the gross characteristics of large aggregations of molecules and not the behavior of individual molecules The microstructure of matter is studied in kinetic theory and statistical mechanics (including quantum thermodynamics) In this chapter, the classical approach to thermodynamics is featured
Basic Concepts and Definitions
Thermodynamics is both a branch of physics and an engineering science The scientist is normally interested in gaining a fundamental understanding of the physical and chemical behavior of fixed, quiescent quantities of matter and uses the principles of thermodynamics to relate the properties of matter Engineers are generally interested in studying systems and how they interact with their surround-ings To facilitate this, engineers have extended the subject of thermodynamics to the study of systems through which matter flows
System
In a thermodynamic analysis, the system is the subject of the investigation Normally the system is a specified quantity of matter and/or a region that can be separated from everything else by a well-defined surface The defining surface is known as the control surface or system boundary The control surface may
be movable or fixed Everything external to the system is the surroundings A system of fixed mass is Michael J Moran
The Ohio State University