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Thevenin and Norton equivalent circuits [1-3] are a valuable analytical tool for circuit designers and researchers. How to obtain the Thevenin resistance and Norton conductance of a network with available network theorems?

In this Design Idea, the substitution theorem of network analysis has been used to obtain the equivalent resistance/conductance using the SPICE circuit simulation software [4-5]. The Thevenin equivalent representation of a network with an arbitrary load resistance of value (k·R_{T}) with k >0, is shown in the Figure 1a. The Thevenin resistance R_{T} is to be determined between terminals A,B. The load resistance k·R_{T} can be, by the substitution theorem, replaced by an equivalent voltage source *E _{c}·k*/(

**Figure 1 **The Thevenin equivalent representation of a network with an arbitrary load resistance of value (k · R_{T}) with k >0, is shown (1a); the product of E_{C} with k/(k+1) is realized by (in SPICE) a voltage dependent voltage source connected as shown in 1b.

The value of R_{T} for the network (**Figure 3**) can be obtained [6-7] by dividing this voltage by the product of current I_{k} through the voltage dependent voltage source and k, with a value defined as: *E _{c}/(I_{k}*(

**Figure 3** A Network whose Thevenin resistance across terminals A,B is to be determined.

The Thevenin resistance calculated for various values of k is given in **Table I** for Figure 3:

The Norton equivalent circuit for a linear network with an arbitrary load conductance of value (G_{NOR}/k) with k>0, is shown in the Figure 2a for the network, whose Norton conductance (G_{NOR}) is to be determined between terminals C, D. The load conductance G_{NOR}/k in the Norton equivalent representation, by the substitution theorem, can be replaced (Figure 2b) by an equivalent current source *I _{SC}/(k*+1) , where I

**Figure 2** The Norton equivalent circuit for a linear network with an arbitrary load conductance of value ( G_{NOR}/k) with k>0, is shown in 2a for the network; the load conductance G_{NOR}/k in the Norton equivalent representation, by the substitution theorem, can be replaced (2b) by an equivalent current source.

The final Norton conductance value can be obtained by the expression: *k*·*I _{SC}/((k*+1)·

The voltage V_{k} (the voltage across the current controlled current source) can be derived using the SPICE program. **Figure 4** shows a network whose Norton conductance across terminals C, D is to be determined.

**Figure 4** A network whose Norton conductance across terminals C, D is to be determined.

The Norton conductance G_{NOR }(in millimhos) for various values of k for Figure 4 is given in **Table 2**:

**Description of SPICE file to obtain Thevenin resistance (R _{T})**

The sub-circuit THEV is used to describe the network (Figure 3) in **Table 3**:

The Thevenin open circuit voltage is available at node 1 at the subcircuit X1. The voltage source ESUBS as required by the substitution theorem is connected at node 3 of sub-circuit X2. The current through the zero voltage source V23 gives the current I_{k } through the voltage dependent voltage source ESUBS. The current I_{k} is converted to a DC voltage of the same value (I_{k}= v(4)) at node 4 using current controlled current source FSUBS and a 1-ohm resistor, RR40. The expression *E _{c}/((k*+1)·

**Description of SPICE file to obtain Norton conductance (G _{NOR})**

The network (Figure 4) whose Norton conductance (G_{NOR}) is to be determined, is described under sub-circuit name NORT in **Table 4**:

The short circuit Norton current I_{SC} is obtained by the current through the zero voltage source V10 connected across nodes 1 and ground 0. This short circuit current is converted to a DC voltage (v(4)=I_{SC}) by a current controlled current source FNORT using a 1-ohm resistor. The equivalent current source *I _{SC}/(k*+1

**References**

- E. Van Valkenburg, Network Analysis, 3
^{rd}Edition, Chapter IX, pp. 259-261, Prentice-Hall 1976 - P. Malvino, Electronic Principles,Chapter I, pp. 4-7, Tata-McGraw Hill 1998
- Karpaty,”Simple Method uses PSpice for Thevenin-equivalent Circuits”, EDN, Design Ideas, pp. 47-48, April 23, 2009
- Vladimirescu, K. Zhang, A. R. Newton, D. O. Pederson, A. Sangiovanni-Vincentelli, SPICE Version 2G User’s Guide, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, CA, USA
- MicroSim PSpice & Basics, Circuit Analysis Software, User’s Guide, Version 8.0, June 1997
- Epler, SPICE2 application notes for dependent sources, IEEE Circuits & Devices Magazine, pp. 36-44,3(5), 1984
- Bharath kumar,”Inverse ABCD parameter determination using SPICE”, International Journal of Analog Integrated Circuits, IJAIC, Vol-3, issue 1, pp. 1-6, 2017

*This article was originally published on **EDN**.*

**Bharath Kumar** obtained B. Tech degree in E & CE with highest honours from JNT University, Anantapur, India, in 1981 and a Master’s Tech degree from Indian Institute of Technology, Kharagpur,India in the area of Microwave and Optical Communication in the year 1983. He later joined Indian Telephone Industries, Bangalore, India and worked in the area of Fiber optics, and in the year 1990 obtained M. S. degree in electrical engineering from the Illinois Institute of Technology, Chicago, USA and joined the Oki Electric, Japan as a researcher in the semiconductor laboratory. He has over twenty publications in the area of simulation and modeling of electronics circuits in various national and international journals. Now he is retired from Oki electric,Japan and his current address is 10-365, Sarojini Road, Anantapur, AP 515 001,India and can be reached at email:bbkrishnapuram3@yahoo.com.