Lecture 3: Superposition and the Thevenin Method

What this lecture covers

This lecture answers how to solve linear circuits faster than grinding through the full KVL/KCL system or even the node method. Building on the node-method example from the previous lecture, it shows that a linear circuit's node voltage is always a linear combination of its independent sources, and uses that fact to motivate two new tools: the method of superposition and the Thevenin method.

By the end, you can find a circuit's response by analyzing the effect of each independent source alone and summing the results, and you can replace an arbitrary linear network, seen from two terminals, with a single equivalent voltage source in series with a single resistor. Both techniques apply only to linear circuits, a restriction the lecture is explicit about, and both are meant to let you solve circuits largely by inspection.

Key ideas

• Linearity: a circuit built only from resistors, independent voltage sources and independent current sources produces outputs that are linear combinations of the source values, with no products or squared terms.
• Homogeneity: scaling every source in a linear circuit by a constant scales the output by the same constant.
• Superposition: the output of a linear circuit equals the sum of the outputs produced by each independent source acting alone, with every other independent source set to zero.
• Zeroing a source: setting a voltage source to zero means replacing it with a short circuit; setting a current source to zero means replacing it with an open circuit.
• Voltage divider pattern: a voltage source across two series resistors produces a node voltage of V * R2 / (R1 + R2), a pattern the lecture says recurs constantly in circuit analysis.
• Thevenin equivalent: as seen from any two terminals, an arbitrary linear network of resistors and independent sources behaves exactly like a single voltage source Vth in series with a single resistor Rth.
• Finding Rth and Vth: Rth is the resistance measured at the two terminals after shorting all voltage sources and opening all current sources; Vth is the open-circuit voltage measured at the same two terminals with nothing else connected.
• Why it matters: once a messy sub-circuit is reduced to its Thevenin equivalent, the rest of the circuit can be solved by simple series-circuit arithmetic instead of re-deriving node equations.

Walkthrough

Review and setting up the example circuit (1:01)

After a review of the lumped matter discipline, KVL/KCL, and the node and composition methods covered so far, the lecture reapplies the node method to a simple circuit with one voltage source V, one current source I, and two resistors. The resulting node equation is shown to be a linear combination of V and I, written in the standard conductance-matrix form, which sets up the rest of the lecture's discussion of linearity.

Linearity, homogeneity and superposition (13:52)

Using an "apples and applesauce" analogy, the lecture explains homogeneity (scaling inputs scales outputs proportionally) and superposition (the response to a mix of inputs equals the sum of the responses to each input applied alone). It states the general rule for zeroing sources, shorting voltage sources and opening current sources, before applying the method formally.

Applying superposition to the example circuit (24:12)

The example circuit is solved by superposition: first with the current source opened, leaving a simple voltage divider that gives eV = V * R2 / (R1 + R2), then with the voltage source shorted, giving eI = I * (R1*R2)/(R1+R2). Summing the two components reproduces exactly the answer obtained earlier with the node method, confirming the method. A live demonstration with a vat of water acting as a distributed resistive network shows a sinusoid and a triangular wave superposing at the output, illustrating superposition physically.

Deriving the Thevenin equivalent from superposition (34:29)

The lecture considers an arbitrary linear network with many resistors, voltage sources and current sources, viewed from two terminals where an external current I is injected. By superposition, the voltage at those terminals is a sum of a term proportional to each independent source plus a term proportional to I multiplied by an equivalent resistance R. This shows that, from the perspective of anything connected at those two terminals, the entire network is indistinguishable from a single voltage source Vth in series with a single resistor R, which is named the Thevenin equivalent.

Finding Rth and Vth, and the Thevenin method (43:51)

The lecture defines Rth as the resistance seen at the terminals with all independent sources shut off (voltage sources shorted, current sources opened), and Vth as the open-circuit voltage measured at the terminals with nothing else connected. It then applies the method to the running example circuit to find the current i1 through R1: boxing off the sub-circuit containing the current source and R2, finding Rth = R2 and Vth = I * R2, and writing i1 = (V - Vth) / (R1 + Rth) directly by inspection, without re-deriving the full node equations.

Before you watch

• Watch Lectures 1 and 2 first; this lecture assumes familiarity with the lumped matter discipline, KVL, KCL, and the node method.
• Comfort with basic algebraic manipulation of linear equations helps, since the lecture leans on recognizing linear combinations of source terms.

Check your understanding

1. What does it mean for a circuit to be linear, and what two properties, homogeneity and superposition, follow from that?
2. How do you "zero" a voltage source versus a current source when applying superposition?
3. In the worked example, what are the two component voltages found by superposition, and how do they combine to match the node-method answer?
4. How do you find Rth and Vth for an arbitrary linear network seen from two terminals?
5. Why are superposition and the Thevenin method restricted to circuits built only from resistors and independent sources?