Thévenin’s and Norton’s Theorems Notes Ch2

Simple Example with a Dependent Source – Step 2 We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. On the last slide we found vQ, and we found iQ. We take the ratio of them, and plug in the expressions that we found for each. When we do this, we get i X RX iX RX RX 100[W]      25[W]. iQ iX  3iX 4iX 4 4 vQ Note that ratio is a constant. The ratio has units of resistance, which is what we expect when we take a ratio of a voltage to a current. Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B Simple Example with a Dependent Source – Step 2 (Note) We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. Let’s find the ratio of the voltage across the circuit, labeled vQ, to the ratio of the current through the circuit, labeled iQ. We take the ratio of them, and get i X RX iX RX RX 100[W]      25[W]. iQ iX  3iX 4iX 4 4 vQ The dependent source is in parallel with the resistor RX. Since the parallel combination is 25[W], the dependent source must be behaving as if it were a 33.33[W] resistor. However, this value depends on RX; in fact, it is RX /3. Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B 2nd Simple Example with a Dependent Source – Step 1 We wish to find the equivalent resistance of a second circuit, given below, as seen at terminals A and B. Let’s find the ratio of the voltage across the circuit, labeled vQ, to the ratio of the current through the circuit, labeled iQ. This must be a constant. We note that from Ohm’s Law applied to RX, we can say that vQ  iX RX . Next, we apply KCL at the A node to write that iQ  iX  3iX . Note the change in polarity for the source, from the previous example. Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B 2nd Simple Example with a Dependent Source – Step 2 We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. On the last slide we found vQ, and we found iQ. We take the ratio of them, and plug in the expressions that we found for each. When we do this, we get i X RX iX RX RX 100[W]      50[W]. iQ iX  3iX 2iX 2 2 vQ Note that ratio has changed when we simply changed the polarity of the dependent source. The magnitude is not the only thing that changed; the equivalent resistance is now negative! Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B Simple Example with a Dependent Source – Step 2 (Note) We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. Let’s find the ratio of the voltage across the circuit, labeled vQ, to the ratio of the current through the circuit, labeled iQ. We take the ratio of them, and get i X RX iX RX RX 100[W]      50[W]. iQ iX  3iX 2iX 2 2 vQ The dependent source is in parallel with the resistor RX. Since the parallel combination is -50[W], the dependent source must be behaving as if it were a -33.33[W] resistor. This value depends on RX; in fact, it is -RX /3. Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B Note 1 When we find the equivalent resistance for a Thévenin’s equivalent or a Norton’s equivalent, we set the independent sources equal to zero, and find the equivalent resistance of what remains. We can see that the equivalent resistance can be negative. This is one reason why we have been so careful about polarities all along. We need to get the polarities right to be able to get our signs right. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B Note 2 When we find the equivalent resistance for a Thévenin’s equivalent or a Norton’s equivalent, we set the independent sources equal to zero, and find the equivalent resistance of what remains. In the simple examples that we just did, we were effectively applying a source to the terminals of the circuit. This results in a circuit like others that we have solved before, and we can find the ratio of voltage to current. This is usually easier to think about for most students. It is as if we were applying a source just to test the circuit; we call this method the TestSource Method. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B Test-Source Method – Defined To get the equivalent resistance of a circuit, as seen by two terminals of that circuit, we follow these steps. 1) Set all independent sources equal to zero. 2) Find the equivalent resistance. a) If there are no dependent sources, find this equivalent resistance using the equivalent resistance rules that have been used before. These include series combinations, parallel combinations, and delta-to-wye equivalents. b) If there are dependent sources present, apply a test source to the two terminals. It can be either a voltage source or a current source. 1) If you apply a voltage source, find the current through that voltage source. iT A 2) If you apply a current source, find the voltage across that + Any circuit current source. vT made up of 3) Then, find the ratio of the voltage to the Test resistors and current, which will be the Source equivalent resistance. dependent sources - B Test-Source Method – Note 1 To get the equivalent resistance of a circuit, as seen by two terminals of that circuit, we follow these steps. 1) Set all independent sources equal to zero. 2) Find the equivalent resistance. a) If there are no dependent sources, find this equivalent resistance using Don’t forget this step. It is always applied the equivalent resistance rules that have been used before. These whencombinations, finding equivalent resistance. include series combinations, parallel and delta-to-wye equivalents. b) If there are dependent sources present, apply a test source to the two terminals. It can be either a voltage source or a current source. 1) If you apply a voltage source, find the current through that voltage source. iT A 2) If you apply a current source, find the voltage across that + Any circuit current source. vT made up of 3) Then, find the ratio of the voltage to the Test resistors and current, which will be the Source equivalent resistance. dependent sources - B Test-Source Method – Note 2 To get the equivalent resistance of a circuit, as seen by two terminals of that circuit, we follow these steps. 1) Set all independent sources equal to zero. Note that step 2 has two options (a or b). 2) Find the equivalent resistance. Pick one. You don’t need to do both. a) If there are no dependent sources, find this equivalent resistance using the equivalent resistance rules that have been used before. These include series combinations, parallel combinations, and delta-to-wye equivalents. b) If there are dependent sources present, apply a test source to the two terminals. It can be either a voltage source or a current source. 1) If you apply a voltage source, find the current through that voltage source. iT A 2) If you apply a current source, find the voltage across that + Any circuit current source. vT made up of 3) Then, find the ratio of the voltage to the Test resistors and current, which will be the Source equivalent resistance. dependent sources - B Test-Source Method – Note 3 To get the equivalent resistance of a circuit, as seen by two terminals of that circuit, we follow these steps. You could actually pick option b) every 1) Set all independent sources equal to zero. time, but option a) is easier. Use it if you 2) Find the equivalent resistance. can. a) If there are no dependent sources, find this equivalent resistance using the equivalent resistance rules that have been used before. These include series combinations, parallel combinations, and delta-to-wye equivalents. b) If there are dependent sources present, apply a test source to the two terminals. It can be either a voltage source or a current source. 1) If you apply a voltage source, find the current through that voltage source. iT A 2) If you apply a current source, find the voltage across that + Any circuit current source. vT made up of 3) Then, find the ratio of the voltage to the Test resistors and current, which will be the Source equivalent resistance. dependent sources - B