(real) axis defines the time-domain signal corresponding to the real part of v(n),i.e, A cos(ot projection onto the vertical (imaginary) axis defines the time-domain signal corresponding to the part of v(t), i.e., A sin(at +o]. The composite signal v(n) is a mathematical entity, it cannot be seen with an oscilloscope. Its value lies in the fact that when a circuit is in the steady state, its voltages and currents are uniquely determined by their corresponding phasors, and these in turn satisfy Kirchhoff's voltage and current laws! Thus, we are able to write ∑hm=0 where Ikm is the phasor of ikm(O), the sinusoidal current in branch k attached to node m. An equation of this form can be written at each node of the circuit. For example, at node b in Fig. 3.1 KCL would have the form Consequently, a set of linear, algebraic equations describe the phasors of the currents and In a circu in the sinusoidal steady state, i.e., the notion of time is suppressed(see Section 3. 2).The of the set of quations yields the phasor of each voltage and current in the circuit, from which the expressions can be extracted On It can also be shown that KCL can be extended to apply to the Fourier transforms and the Laplace transforms the currents in a circuit. Thus, a single relationship between the currents at the nodes of a circuit applies to all of the known mathematical representations of the currents [ Ciletti, 1988] Kirchhoff's Voltage Law Kirchhoff's voltage law(KVL) describes a relationship among the voltages measured across the branches in any closed, connected path in a circuit. Each branch in a circuit is connected to two nodes. For the purpose of applying KVL, a path has an orientation in the sense that in"walking"along the path one would enter one of the nodes and exit the other. This establishes a direction for determining the voltage across a branch in the path: the voltage is the difference between the potential of the node entered and the potential of the node at which the path exits. Alternatively, the voltage drop along a branch is the difference of the node voltage at the entered node and the node voltage at the exit node. For example, if a path includes a branch between node"a and node " b, the voltage drop measured along the path in the direction from node"a"to node"b"is denoted by vab and is given by vab=v-w Given vab, branch voltage along the path in the direction from node"b"to node "a"is vi=v Kirchhoff's voltage law, like Kirchhoff's current law, is true at any time. KVL can also be stated in terms of voltage rises instead of voltage drops. KVL can be expressed mathematically as"the algebraic sum of the voltages drops around any closed path of a circuit at any instant of time is zero. This statement can also be cast as an equation: ∑ vhm,()=0 where vm(t) is the instantaneous voltage drop measured across branch k of path m. By convention, the voltage drop is taken in the direction of the edge sequence that forms the path. The edge sequence le, e2, e,, e4, e, e,) forms a closed path in Fig 3. 1. The sum of the voltage drops taken around the path must satisfy KVL vab(t)+vc(t)+ va(t)+ vae(t)+ ve (t)+ ya(t)=0 Since va(n)=-(t), we can also write c 2000 by CRC Press LLC© 2000 by CRC Press LLC (real) axis defines the time-domain signal corresponding to the real part of vc(t), i.e., A cos[wt + f], and its projection onto the vertical (imaginary) axis defines the time-domain signal corresponding to the imaginary part of vc(t), i.e., A sin[wt + f]. The composite signal vc(t) is a mathematical entity; it cannot be seen with an oscilloscope. Its value lies in the fact that when a circuit is in the steady state, its voltages and currents are uniquely determined by their corresponding phasors, and these in turn satisfy Kirchhoff’s voltage and current laws! Thus, we are able to write where Ikm is the phasor of ikm(t), the sinusoidal current in branch k attached to node m. An equation of this form can be written at each node of the circuit. For example, at node b in Fig. 3.1 KCL would have the form I 1 – I 2 + I 3 = 0 Consequently, a set of linear, algebraic equations describe the phasors of the currents and voltages in a circuit in the sinusoidal steady state, i.e., the notion of time is suppressed (see Section 3.2). The solution of the set of equations yields the phasor of each voltage and current in the circuit, from which the actual time-domain expressions can be extracted. It can also be shown that KCL can be extended to apply to the Fourier transforms and the Laplace transforms of the currents in a circuit. Thus, a single relationship between the currents at the nodes of a circuit applies to all of the known mathematical representations of the currents [Ciletti, 1988]. Kirchhoff’s Voltage Law Kirchhoff’s voltage law (KVL) describes a relationship among the voltages measured across the branches in any closed, connected path in a circuit. Each branch in a circuit is connected to two nodes. For the purpose of applying KVL, a path has an orientation in the sense that in “walking” along the path one would enter one of the nodes and exit the other. This establishes a direction for determining the voltage across a branch in the path: the voltage is the difference between the potential of the node entered and the potential of the node at which the path exits. Alternatively, the voltage drop along a branch is the difference of the node voltage at the entered node and the node voltage at the exit node. For example, if a path includes a branch between node “a” and node “b”, the voltage drop measured along the path in the direction from node “a” to node “b” is denoted by vab and is given by vab = va – vb. Given vab, branch voltage along the path in the direction from node “b” to node “a” is vba = vb – va = –vab . Kirchhoff’s voltage law, like Kirchhoff’s current law, is true at any time. KVL can also be stated in terms of voltage rises instead of voltage drops. KVL can be expressed mathematically as “the algebraic sum of the voltages drops around any closed path of a circuit at any instant of time is zero.” This statement can also be cast as an equation: where vkm(t) is the instantaneous voltage drop measured across branch k of path m. By convention, the voltage drop is taken in the direction of the edge sequence that forms the path. The edge sequence {e1, e2 , e3 , e4 , e6 , e7} forms a closed path in Fig. 3.1. The sum of the voltage drops taken around the path must satisfy KVL: vab (t) + vbc(t) + vcd (t) + vde(t) + vef(t) + vfa(t) = 0 Since vaf(t) = –vfa (t), we can also write I  km = 0 vkm  ( )t = 0