must equal the sum of the currents that are leaving the node. Thus, the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from(into) the node. The two groups must contain the same net current Applying KCL at node b in Fig 3. 1 gives i1(t)+i3()=2(t) A connection of water pipes that has no leaks is a physical analogy of this situation. The net rate at which water is flowing into a joint of two or more pipes must equal the net rate at which water is flowing away from the joint. The joint itself has the property that it only connects the pipes and thereby imposes a structure on ne flow of water, but it cannot store water. This is true regardless of when the flow is measured. Likewise, the nodes of a circuit are modeled as though they cannot store charge.( Physical circuits are sometimes modeled for the purpose of simulation as though they store charge, but these nodes implicitly have a capacitor that for storing the charge. Thus, KCL is ultimately KCL can be stated alternatively as: " the algebraic sum of the branch currents entering(or leaving) any node of a circuit at any instant of time must be zero. In this form, the label of any current whose orientation is away from the node is preceded by a minus sign. The currents entering node b in Fig 3. 1 must satisfy i1(t)-i2(t)+i3(t)=0 In general, the currents entering or leaving each node m of a circuit must satisfy ∑ 1km(D)=0 where ikm(t) is understood to be the current in branch k attached to node m The currents used in this expression are understood to be the currents that would be measured in the branches attached to the node, and thei values include a magnitude and an algebraic sign. If the measurement convention is oriented for the case where currents are entering the node, then the actual current in a branch has a positive or negative sign, depending on whether the current is truly flowing toward the node in question. Once KCL has been written for the nodes of a circuit, the equations can be rewritten by substituting into the equations the voltage-current relationships of the individual components. If a circuit is resistive, the resulting equations will be algebraic. If capacitors or inductors are included in the circuit, the substitution will produce a differential equation. For example, writing KCL at the node for v, in Fig 3. 2 prod +1-13 出+"- FIGURE 3.2 Example of a circuit containing energy storage c 2000 by CRC Press LLC© 2000 by CRC Press LLC must equal the sum of the currents that are leaving the node. Thus, the set of currents in branches attached to a given node can be partitioned into two groups whose orientation is away from (into) the node. The two groups must contain the same net current. Applying KCL at node b in Fig. 3.1 gives i1(t) + i3 (t) = i2 (t) A connection of water pipes that has no leaks is a physical analogy of this situation. The net rate at which water is flowing into a joint of two or more pipes must equal the net rate at which water is flowing away from the joint. The joint itself has the property that it only connects the pipes and thereby imposes a structure on the flow of water, but it cannot store water. This is true regardless of when the flow is measured. Likewise, the nodes of a circuit are modeled as though they cannot store charge. (Physical circuits are sometimes modeled for the purpose of simulation as though they store charge, but these nodes implicitly have a capacitor that provides the physical mechanism for storing the charge. Thus, KCL is ultimately satisfied.) KCL can be stated alternatively as: “the algebraic sum of the branch currents entering (or leaving) any node of a circuit at any instant of time must be zero.” In this form, the label of any current whose orientation is away from the node is preceded by a minus sign. The currents entering node b in Fig. 3.1 must satisfy i1 (t) – i2 (t) + i3 (t) = 0 In general, the currents entering or leaving each node m of a circuit must satisfy where ikm(t) is understood to be the current in branch k attached to node m. The currents used in this expression are understood to be the currents that would be measured in the branches attached to the node, and their values include a magnitude and an algebraic sign. If the measurement convention is oriented for the case where currents are entering the node, then the actual current in a branch has a positive or negative sign, depending on whether the current is truly flowing toward the node in question. Once KCL has been written for the nodes of a circuit, the equations can be rewritten by substituting into the equations the voltage-current relationships of the individual components. If a circuit is resistive, the resulting equations will be algebraic. If capacitors or inductors are included in the circuit, the substitution will produce a differential equation. For example, writing KCL at the node for v3 in Fig. 3.2 produces i2 + i1 – i3 = 0 and FIGURE 3.2 Example of a circuit containing energy storage elements. i km  ( )t = 0 C dv dt v v R C dv dt 1 1 4 3 2 2 2 + 0 - - = R1 C2 R2 i 1 + – v + – v1 + 2 – i 2 C1 v3 vin v4 i 3