KCL for the node between C2 and RI can be written to eliminate variables and lead to a solution describing the capacitor voltages. The capacitor voltages, together with the applied voltage source, determine the remaining voltages and currents in the circuit Nodal analysis(see Section 3. 2)treats the systematic modeling and analysis of a circuit under the influence of its sources and energy storage elements Kirchhoff,s Current Law in the Complex Domain Kirchhoff's current law is ordinarily stated in terms of the real (time-domain) currents flowing in a circuit, because it actually describes physical quantities, at least in a macroscopic, statistical sense. It also applied, however, to a variety of purely mathematical models that are commonly used to analyze circuits in the so-called comp domain For example, if a linear circuit is in the sinusoidal steady state, all of the currents and voltages in the circuit are sinusoidal. Thus, each voltage has the form v(t)=Asin(ot+φ) and each current has the form i(t)=Bsin(ot+θ) where the positive coefficients A and B are called the magnitudes of the signals, and o and e are the phas angles of the signals. These mathematical models describe the physical behavior of electrical quantities, and instrumentation,such as an oscilloscope, can display the actual waveforms represented by the mathematical model. Although methods exist for manipulating the models of circuits to obtain the magnitude and phase coefficients that uniquely determine the waveform of each voltage and current, the manipulations are cumbe some and not easily extended to address other issues in circuit analysis Steinmetz [Smith and Dorf, 1992] found a way to exploit complex algebra to create an elegant framework for representing signals and analyzing circuits when they are in the steady state. In this approach, a model is developed in which each physical sign is replaced by a"complex" mathematical signal. This complex signal in polar, or exponential, form is represented as v(t)= Ae(jot +o) The algebra of complex exponential signals allows us to write this as v(t)=Aepejor and Euler's identity gives the equivalent rectangular form: v(t)=A[cos(ot +o)+j sin(ot +o) So we see that a physical signal is either the real(cosine)or the imaginary (sine) component of an abstrac complex mathematical signal. The additional mathematics required for treatment of complex numbers allows us to associate a phasor, or complex amplitude, with a sinusoidal signal. The time-invariant phasor associate V=A Notice that the phasor ve is an algebraic constant and that in incorporates the parameters A and o of the ding time-domain sinusoidal signal. Phasors can be thought of as being vectors in a two-dimensional plane. If the vector is allowed to rotate about the origin in the counterclockwise direction with frequency a, the projection of its tip onto the horizontal c 2000 by CRC Press LLC© 2000 by CRC Press LLC KCL for the node between C2 and R1 can be written to eliminate variables and lead to a solution describing the capacitor voltages. The capacitor voltages, together with the applied voltage source, determine the remaining voltages and currents in the circuit. Nodal analysis (see Section 3.2) treats the systematic modeling and analysis of a circuit under the influence of its sources and energy storage elements. Kirchhoff’s Current Law in the Complex Domain Kirchhoff’s current law is ordinarily stated in terms of the real (time-domain) currents flowing in a circuit, because it actually describes physical quantities, at least in a macroscopic, statistical sense. It also applied, however, to a variety of purely mathematical models that are commonly used to analyze circuits in the so-called complex domain. For example, if a linear circuit is in the sinusoidal steady state, all of the currents and voltages in the circuit are sinusoidal. Thus, each voltage has the form v(t) = A sin(wt + f) and each current has the form i(t) = B sin(wt + q) where the positive coefficients A and B are called the magnitudes of the signals, and f and q are the phase angles of the signals. These mathematical models describe the physical behavior of electrical quantities, and instrumentation, such as an oscilloscope, can display the actual waveforms represented by the mathematical model. Although methods exist for manipulating the models of circuits to obtain the magnitude and phase coefficients that uniquely determine the waveform of each voltage and current, the manipulations are cumber￾some and not easily extended to address other issues in circuit analysis. Steinmetz [Smith and Dorf, 1992] found a way to exploit complex algebra to create an elegant framework for representing signals and analyzing circuits when they are in the steady state. In this approach, a model is developed in which each physical sign is replaced by a “complex” mathematical signal. This complex signal in polar, or exponential, form is represented as vc(t) = Ae(jwt + f) The algebra of complex exponential signals allows us to write this as vc(t) = Aejfejwt and Euler’s identity gives the equivalent rectangular form: vc(t) = A[cos(wt + f) + j sin(wt + f)] So we see that a physical signal is either the real (cosine) or the imaginary (sine) component of an abstract, complex mathematical signal. The additional mathematics required for treatment of complex numbers allows us to associate a phasor, or complex amplitude, with a sinusoidal signal. The time-invariant phasor associated with v(t) is the quantity Vc = Ae jf Notice that the phasor vc is an algebraic constant and that in incorporates the parameters A and f of the corresponding time-domain sinusoidal signal. Phasors can be thought of as being vectors in a two-dimensional plane. If the vector is allowed to rotate about the origin in the counterclockwise direction with frequency w, the projection of its tip onto the horizontal