EGR 2201 Unit 6 Theorems: - PowerPoint Presentation

EGR 2201 Unit 6Theorems: 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 V Th in series with a resistor R Th . There’s a standard procedure for finding the values of V Th and R Th . 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

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.) V TH is the voltage across the two open terminals. Pay attention to polarity! R TH 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. Steps in Finding V TH and R TH

More on Step 3 (Finding RTH) The previous slide said that R TH 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 V TH : Finding RTH :

We found that VTH = 30 V and R TH = 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 I L for six different values of R L . Would you rather analyze the left-hand circuit six times or the right-hand circuit six times? Finding the Thevenin-Equivalent Circuit Example #1: No Dependent Sources (Conclusion) Original Circuit Equivalent Circuit

Finding R TH For a Circuit That Contains Dependent Sources To understand the steps in finding R TH 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 i o flows through the resistor. Then use Ohm’s law to compute the resistance:  

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

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 V TH : Finding RTH :

Finding the Thevenin-Equivalent Circuit Example #2: With Dependent Sources (Conclusion) We found that V TH = 20 V and R TH = 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. 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 V Th in series with a resistor R Th. 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’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 I N 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?) Original Circuit Norton-Equivalent Circuit Thevenin -Equivalent Circuit

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

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 Assuming the source is linear, we can replace the source with its Thevenin -equivalent circuit, giving us: 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:   Question: For fixed values of V Th and R Th , what value of R L will result in maximum load power? The answer is not obvious, since R L 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 R L approaches 0. Also, power approaches 0 as R L 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 ( R L = R Th ) . To see this, find , set it equal to 0, and solve for R L .  

Matching Source and LoadWhen 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.

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. An Important Application of Thevenin’s Theorem: Source Modeling Symbols for ideal voltage sources:

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 . (Ideally, it is 0 Ω .)Source Modeling

Suppose we connect a load resistor across a real voltage source rated at 1 V . If we treat the real source as an ideal source, we’ll conclude that the voltage across the load resistor is 1 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 1 V. Using Source Modeling to Analyze a Circuit

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? A Practical Implication of This 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 .

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. It’s Still Just an Approximation

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 . (Not needed in Lab 6 but will be in future labs.)

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.