EGR 2201 Unit 5 Linearity, Superposition, & Source Transformation Read Alexander & Sadiku, Sections 4.1 to 4.4. Homework #5 and Lab #5 due next week. Quiz next week.

Techniques That Can Simplify Circuit Analysis Nodal analysis and mesh analysis are powerful tools for analyzing circuits. But you should also know other techniques that may simplify the equations that you get when you perform nodal or mesh analysis, or may save you from having to perform nodal or mesh analysis at all.

Example of a Simplifying Technique One such technique is our method of finding the equivalent resistance of several series-parallel resistors. Example: In this circuit, if you just want to find i, using this technique will get you the answer much faster than doing a full nodal or mesh analysis of the circuit. How many meshes in this circuit? How many non-reference nodes?

Other Useful Techniques The current-divider rule and voltage-divider rule are two other useful tools that can save you work when analyzing circuits. Chapter 4 presents several new techniques: Linearity Superposition Source transformation Thevenin’s theorem Norton’s theorem Maximum-power-transfer theorem

Linear Circuits A linear circuit is a circuit whose output is linearly related (or directly proportional) to its input. Typically the input is an independent voltage source or current source, and the output is a resistor’s voltage or current. So, in a linear circuit, if the input voltage or current increases by a factor of k, then the output voltage or current must increase by that same factor.

Why Linearity is Useful Linearity is a useful property because it lets us easily answer certain “what-if” questions about linear circuits. Example: Suppose we know that in the circuit shown, i = 2 A when vs = 10 V. We might have had to do a lot of work to learn this. Then we know that doubling vs to 20 V will also double i to 4 A.

How to Tell if a Circuit is Linear Any circuit that contains only the following elements is a linear circuit: Resistors Capacitors Inductors Linear dependent sources All of the circuits that we will study in this course are linear. In other courses you will study non-linear circuits. Linear expression Do practice question 1.

Caution: Power Is Not Related Linearly to Voltage and Current Recall that a resistor’s power is non-linearly related to its current and voltage by the equations p = i 2  R p = v 2  R Therefore, for example, doubling vs in the circuit shown will not double R’s power.

Techniques That Can Simplify Circuit Analysis Chapter 4 presents several new techniques: Linearity Superposition Source transformation Thevenin’s theorem Norton’s theorem Maximum-power-transfer theorem

Superposition Principle (1 of 2) The superposition principle says that to find the total effect of two or more independent sources in a linear circuit, you can find the effect of each source acting alone, and then combine those effects. Note: Superposition has nothing to do with supernodes or supermeshes.

Superposition Principle (2 of 2) To find the effect of an independent source acting alone, you must turn off all the other independent sources. To turn off a voltage source, replace it by a short circuit. To turn off a current source, replace it by an open circuit.

Steps in Using the Superposition Principle Select one independent source and turn off all the others, as explained above. Find the desired voltage or current due to this source using techniques from earlier chapters. Repeat step 1 for each of the other independent sources. Add the values obtained from the steps above, paying careful attention to the signs (+ or ) of each value.

Example: Using Superposition (1 of 2) Suppose we wish to find v in the circuit shown. Redrawn circuit, with the voltage source acting alone: Redrawn circuit, with the current source acting alone:

Example: Using Superposition (2 of 2) In this circuit, use the voltage-divider rule to find v1 = 2 V. In this circuit, use the current-divider rule to find i3 = 2 A, then Ohm’s law to find v2 = 8 V. Therefore, in the original circuit, v = 2 V + 8 V = 10 V.

Wouldn’t Nodal Analysis Be Easier? For this circuit, you might argue that nodal (or mesh) analysis is easier than superposition. And you might be right. But looking ahead to the second half of this course, if one source were a DC source and the other were an AC source, we would have no choice but to use superposition. Do practice question 2.

Techniques That Can Simplify Circuit Analysis Chapter 4 presents several new techniques: Linearity Superposition Source transformation Thevenin’s theorem Norton’s theorem Maximum-power-transfer theorem

Combining Series or Parallel Sources Before we look at source transformation, note the following: Just as we can combine series or parallel resistors into a single equivalent resistor, we can sometimes combine sources into a single equivalent source. In particular: We can combine series voltage sources into an equivalent voltage source. We can combine parallel current sources into an equivalent current source. The book never explicitly discusses this, but at times assumes it.

Series-Aiding Voltage Sources If two series-connected voltage sources produce current in the same direction, they are said to be series-aiding. The net effect on the circuit is the same as that of a single source whose voltage equals the sum of the two voltage sources. Example: Most flashlights have two 1.5-V batteries in series to give 3 V. Do practice question 3a.

Series-Opposing Voltage Sources If two series-connected sources produce current in opposite directions, they’re said to be series-opposing. The effect is the same as that of a single source equal in magnitude to the difference between the source voltages and having the same polarity as the larger of the two. Do practice question 3b.

Parallel Voltage Sources? Although there are exceptions, in general we don’t connect voltage sources in parallel with each other. To see why not, think about this: what is the voltage between points a and b in the circuit below?

Parallel Current Sources Current sources can be connected in parallel. The parallel-connected sources can be replaced by a single equivalent current source that produces a current equal to the algebraic sum of the sources. Do practice questions 3c, 3d. Case 1: Both sources push current in same direction through rest of circuit. Case 2: Sources push current in opposite directions through rest of circuit.

Series Current Sources? In general we don’t connect current sources in series with each other. To see why not, think about this: how much current flows around the loop in the circuit below?

Source Transformation Sometimes when you're analyzing a circuit, it can be useful to substitute a voltage source (and a resistor in series with it) by a current source (and a resistor in parallel with it), or vice versa. Either substitution is called a source transformation. Two questions: Why would this be useful? Are we allowed to do it? Keep in mind throughout this discussion that we're simply talking about a mathematical conversion on paper (or in your head). We're not physically converting one actual device into another actual device.

Example of Why Source Transformation Can Be Useful Suppose we wish to find vo in the circuit to the right. This is not difficult, but it requires a bit of work. We may get the answer more easily if we can first replace the current-source-and- parallel-resistor with a voltage-source-and-series-resistor. Do practice question 4a.

Are We Allowed to Do It? Yes, we’re allowed to make this substitution, as long as we use the right values for the elements. The resistor R in series with the voltage source must be the same size as the one in parallel with the current source. The current source’s and voltage source’s values must satisfy the following: 𝑣 𝑠 = 𝑖 𝑠 𝑅 or 𝑖 𝑠 = 𝑣 𝑠 𝑅 -Note the similarity of these equations to Ohm’s law. -Do practice questions 4b, 4c.

Source Transformation: A More Complicated Example (1 of 2) In some cases we can use source transformation repeatedly to convert a complicated problem into a much simpler one. The book’s Example 4.6 shows such a case: Original circuit (Our goal is to find vo): After transforming both sources: Continued on next slide…

Source Transformation: A More Complicated Example (2 of 2) …From previous slide: After combining 4- with 2- and transforming 12-V source: After combining 6- with 3- and combining 2-A with 4-A: