EGR 2201 Unit 6 Theorems: Thevenin’s, Norton’s, Maximum Power Transfer Read Alexander & Sadiku, Sections 4.5 to 4.11. Homework #6 and Lab #6 due next week. Quiz next week.

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

Thevenin’s Theorem Thevenin’s theorem says that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of an independent voltage source VTh in series with a resistor RTh. There’s a standard procedure for finding the values of VTh and RTh. But first, what exactly does this mean and why is it useful?

Load Often an electrical circuit or system contains an element (called the “load”) that can be varied while the rest of the circuit or system stays fixed. Example: When you unplug a lamp at home and plug in a hair-dryer, you’ve varied one small part of a huge electrical circuit that extends all the way back to the power plant. The lamp or hair-dryer is the “load” part of the circuit.

What Thevenin’s Theorem Means Thevenin’s theorem lets us replace everything on one side of a pair of terminals by a very simple equivalent circuit consisting of just a voltage source and a resistor. Original Circuit Equivalent Circuit

Why It’s Useful This greatly simplifies computation when you wish to find values of voltage or current for several different possible values of a load resistance. Original Circuit Equivalent Circuit

Steps in Finding VTH and RTH Open the circuit at the two terminals where you wish to find the Thevenin-equivalent circuit. (In the circuit shown, this means removing the load.) VTH is the voltage across the two open terminals. Pay attention to polarity! RTH is the resistance looking into the open terminals with all independent sources turned off. Recall that to turn off a voltage source we replace it by a short. To turn off a current source we replace it by an open.

More on Step 3 (Finding RTH) The previous slide said that RTH is the resistance looking into the open terminals with all independent sources turned off. What about dependent sources? We don’t turn those off. As shown in the following two examples, how you find RTH depends on whether the circuit contains any dependent sources. If no dependent sources, just combine resistors in series and parallel to find equivalent resistance. If you have dependent sources, it’s trickier.

Finding the Thevenin-Equivalent Circuit Example #1: No Dependent Sources Goal: Find the Thevenin- equivalent circuit for the circuit in the book’s Figure 4.27: Finding VTH: Finding RTH:

We found that VTH = 30 V and RTH = 4 . Finding the Thevenin-Equivalent Circuit Example #1: No Dependent Sources (Conclusion) We found that VTH = 30 V and RTH = 4 . So as far as the load is concerned, the original circuit below is equivalent to the simpler circuit on the right. Question: Suppose you need to find the load current IL for six different values of RL. Would you rather analyze the left-hand circuit six times or the right-hand circuit six times? Do practice question 1. Original Circuit Equivalent Circuit

Finding RTH For a Circuit That Contains Dependent Sources To understand the steps in finding RTH for a circuit that contains dependent sources, think about this question: Suppose you have lab equipment including a voltage source (of known value vo ) and an ammeter. If I give you a resistor of unknown value, how can you find its resistance? Answer: Connect the voltage source across the resistor and measure how much current io flows through the resistor. Then use Ohm’s law to compute the resistance: 𝑅= 𝑣 𝑜 𝑖 𝑜

Finding RTH For a Circuit Containing Dependent Sources If a circuit contains dependent sources, follow these steps to find RTH. Turn off all independent sources as previously described. Connect an independent voltage source of any value vo (say, 1 V) across the open terminals. Find the current io flowing into the circuit from the voltage source that you connected in Step 2. To find RTh, divide the voltage vo by the current io. In symbols, RTh = vo ÷ io.

Finding the Thevenin-Equivalent Circuit Example #2: With Dependent Sources Goal: Find the Thevenin- equivalent circuit for the circuit in the book’s Figure 4.31: Finding VTH: Finding RTH:

Finding the Thevenin-Equivalent Circuit Example #2: With Dependent Sources (Conclusion) We found that VTH = 20 V and RTH = 6 . So we conclude that, as far as any load connected to terminals a-b is concerned, the original circuit on the left below is equivalent to the much simpler circuit on the right below. Do practice question 2. Original Circuit Equivalent Circuit

Summary: Thevenin’s Theorem As we’ve seen, Thevenin’s theorem says that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of an independent voltage source VTh in series with a resistor RTh. Original Circuit Thevenin-Equivalent Circuit

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

Norton-Equivalent Circuit Norton’s Theorem Norton’s theorem says that any linear two-terminal circuit can be replaced by an equivalent circuit consisting of an independent current source IN in parallel with a resistor RN. Original Circuit Norton-Equivalent Circuit

Not Surprising, Is It? Norton’s theorem follows from Thevenin’s theorem, since we can substitute a voltage-source-plus-series-resistor by a current- source-plus-parallel-resistor, or vice versa. (Remember source transformation?) Thevenin-Equivalent Circuit Original Circuit Norton-Equivalent Circuit

Finding IN and RN The book describes how to find IN and RN. The procedure is similar to how we find VTh and RTh. But you don’t need to learn this new procedure. Instead, just find VTh and RTh, and then apply a source transformation: 𝑅 N = 𝑅 Th and 𝐼 N = 𝑉 Th 𝑅 Th Do practice question 3. Thevenin-Equivalent Circuit Norton-Equivalent Circuit

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

Source and Load In many cases, we can think of an electrical system as being composed of a source of power and a load connected to that source. Examples of sources: amplifier, generator, power supply. Examples of loads: loudspeaker, electric motor, antenna.

Maximizing the Load Power Replacing the source with its Thevenin-equivalent circuit, we have the following situation: In many applications, we wish to maximize the power transferred from a fixed source to a variable load. Thevenin-equivalent of source Variable load resistance

The Load’s Power Depends on the Load Resistance For this circuit, the load resistor’s power is given by: 𝑝= 𝑖 2 𝑅 𝐿 = 𝑉 𝑇ℎ 𝑅 𝑇ℎ + 𝑅 𝐿 2 𝑅 𝐿 Question: For fixed values of VTh and RTh, what value of RL will result in maximum load power? The answer is not obvious, since RL appears in both the numerator and the denominator.

Variation in the Load’s Power as RL Varies A graph of the equation looks like this: Note that power approaches 0 as RL approaches 0. Also, power approaches 0 as RL approaches . What is the value of RL where the graph peaks?....

Maximum Power Transfer Theorem The maximum power transfer theorem says that maximum power is transferred to a load when the load resistance equals the source’s Thevenin resistance (RL = RTh). To see this, find 𝑑𝑝 𝑑 𝑅 𝐿 , set it equal to 0, and solve for RL.

Matching Source and Load When source and load have the same resistance, they are said to be matched. In many practical applications, a major part of a circuit designer’s effort is to ensure that components are matched. Do practice question 4.

An Important Application of Thevenin’s Theorem: Source Modeling We’ve been using ideal voltage sources. These are theoretical models that maintain the same output voltage no matter what you attach to their terminals. Such sources do not exist in real life. What about real voltage sources, such as batteries or power supplies? To a first approximation, we treat them as ideal voltage sources. Symbols for ideal voltage sources: Have them go to lab bench, set power supply to produce 1.00 V; then connect a 10 k-ohm resistor across it and measure again; then connect a 10 ohm resistor across it and measure again. Typically much more of an issue with AC sources; repeat exercise above with trainer’s func generator.

Source Modeling For a better approximation, we treat a real voltage source (also called a “practical” voltage source) as an ideal voltage source in series with a resistor. This resistor is called the source’s internal resistance: the lower it is, the better.

Using Source Modeling to Analyze a Circuit Suppose we connect a load resistor across a real voltage source rated at 10 V. If we treat the real source as an ideal source, we’ll conclude that the voltage across the load resistor is 10 V. But if we take into account the source’s internal resistance, we’ll find that the voltage across the load resistor is less than 10 V. Do practice question 5.

A Practical Implication of This We’ve seen that a real voltage source’s voltage will decrease when you connect it to resistors. Therefore, which is the better way to build a circuit on the breadboard? Method A Adjust the voltage source to the correct value. Build the circuit. Connect the voltage source to the circuit. Method B Build the circuit. Connect the voltage source to the circuit. Adjust the voltage source to the correct value.

It’s Still Just an Approximation This model does not perfectly predict all aspects of a real voltage source’s behavior, but it gives more accurate results than treating the real source as an ideal source.

Making Graphs in Word 2013 (1 of 4) Select Insert > Chart on Word’s menu bar. Select X Y (Scatter). Select Scatter with Smooth Lines and Markers. Click OK.

Making Graphs in Word 2013 (2 of 4) Type your data values in this window. You can create a new plot on the same chart by typing a new column of data.

Making Graphs in Word 2013 (3 of 4) Close the data-editing window by clicking X. If you need to re-open that window to edit your data, select Edit Data on Word’s menu bar.

Making Graphs in Word 2013 (4 of 4) Add axis titles and a chart title by clicking the + and checking the boxes. Edit your axis titles and chart title by clicking them and typing. Also show how to insert symbols and mention significant digits.