4.7 Derivations of Thevenin's and Norton's Theorems

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4.7 Derivations of Thevenin’s and Norton’s Theorems

In this section, we will pro ve Thevenin’s and Norton’s theorems using the superposition principle.

Consider the linear circuit in Fig. 4.46(a). It is assumed that the circuit contains resistors and dependent and independent sources. We have access to the circuit via terminals a and b, through which current from an external source is applied. Our objective is to ensure that the voltagecurrent relation at terminals a and b is identical to that of the Thevenin equivalent in Fig. 4.46(b). F or the sak e of simplicity , suppose the lin ear circuit in Fig. 4.46(a) contains tw o independent voltage sources vs¹ and vs2 and two independent current sources is1 and is2. We may obtain any circuit variable, such as the terminal v oltage v, by applying super position. That is, we consider the contrib ution due to each independent source including the e xternal source i. By superposition, the terminal voltage v is

$$v = A_0 i + A_1 v_{s1} + A_2 v_{s2} + A_3 i_{s1} + A_4 i_{s2}$$

(4.13)

where A0, A1, A2, A3, and A4 are constants. Each term on the right-hand side of Eq. (4.13) is the contrib ution of the related independent source; that is, A0i is the contrib ution to v due to the e xternal current source i, A1vs1 is the contribution due to the voltage source vs1, and so on. We may collect terms for the internal independent sources together as B0, so that Eq. (4.13) becomes

$$v = A_0 i + B_0 \tag{4.14}$$

where B⁰ = A1vs¹ + A2vs² + A3is1 + A4is2. We now want to evaluate the values of constants A0 and B0. When the terminals a and b are opencircuited, i = 0 and v = B0. Thus, B0 is the open-circuit voltage voc, which is the same as VTh, so

$$B_0 = V_{\text{Th}} \tag{4.15}$$

When all the internal sources are turned off, B⁰ = 0. The circuit can then be replaced by an equivalent resistance Req, which is the same as RTh, and Eq. (4.14) becomes

$$v = A_0 i = R_{\text{Th}} i \qquad \Rightarrow \qquad A_0 = R_{\text{Th}} \tag{4.16}$$

Substituting the values of A0 and B0 in Eq. (4.14) gives

$$v = R_{\text{Th}} i + V_{\text{Th}} \tag{4.17}$$

which expresses the voltage-current relation at terminals a and b of the circuit in Fig. 4.46(b). Thus, the two circuits in Fig. 4.46(a) and 4.46(b) are equivalent.

When the same linear circuit is dri ven by a v oltage source v as shown in Fig. 4.47(a), the current flowing into the circuit can be obtained by superposition as

$$i = C_0 v + D_0 \tag{4.18}$$

where C0v is the contrib ution to i due to the e xternal voltage source v and D0 contains the contrib utions to i due to all internal independent sources. When the terminals a-b are short-circuited, v = 0 so that

Derivation of Thevenin equivalent: (a) a current-driven circuit, (b) its Thevenin equivalent.

Figure 4.47 Derivation of Norton equivalent: (a) a voltage-driven circuit, (b) its Norton equivalent.