1. lecture 121 Norton's Theorem (4.3) Prof. Phillips February 26, 2003

2. lecture 122 Introduction Any Thevenin equivalent circuit is in turn equivalent to a current source in parallel with a resistor [source transformation]. A current source in parallel with a resistor is called a Norton equivalent circuit. Finding a Norton equivalent circuit requires essentially the same process as finding a Thevenin equivalent circuit.

3. lecture 123 Independent Sources Circuit with one or more independent sources R Th Norton equivalent circuit I sc

4. lecture 124 No Independent Sources Circuit without independent sources R Th Norton equivalent circuit

5. lecture 125 Finding the Norton Equivalent Circuits with independent sources: –Find V oc and I sc –Compute R Th Circuits without independent sources: –Apply a test voltage (current) source –Find resulting current (voltage) –Compute R Th

6. lecture 126 Example: Strain Gauge Strain is the amount of deformation of a body due to an applied force-it is defined as the fractional change in length. Strain can be positive (tensile) or negative (compressive). One type of strain gauge is made of a foil grid on a thin backing.

7. lecture 127 A Strain Gauge The strain gauge’s resistance varies as a function of the strain:  R = G F  R  is the strain, R is the nominal resistance, G F is the Gauge Factor Backing Foil

8. lecture 128 Typical values Measured strain values are typically fairly small-usually less than 10 -3. G F is usually close to 2. Typical values for R are 120 , 350 , and 1000 . A typical change in resistance is  R = 210 -3 120  = 0.24 

9. lecture 129 Measuring Small Changes in R To measure such small changes in resistance, the strain gauge is placed in a Wheatstone bridge circuit. The bridge circuit uses an excitation voltage source and produces a voltage that depends on  R.

10. lecture 1210 The Bridge Circuit R+  R V ex R R R +– V out +–+–

11. lecture 1211 Norton Equivalent for Any 

12. lecture 1212 Thevenin/Norton Analysis 1. Pick a good breaking point in the circuit (cannot split a dependent source and its control variable). 2. Thevenin: Compute the open circuit voltage, V OC. Norton: Compute the short circuit current, I SC. For case 3(b) both V OC =0 and I SC =0 [so skip step 2]

13. lecture 1213 Thevenin/Norton Analysis 3. Compute the Thevenin equivalent resistance, R Th. (a) If there are only independent sources, then short circuit all the voltage sources and open circuit the current sources (just like superposition). (b) If there are only dependent sources, then must use a test voltage or current source in order to calculate R Th = V Test /I test (c) If there are both independent and dependent sources, then compute R Th from V OC /I SC.

14. lecture 1214 Thevenin/Norton Analysis 4. Thevenin: Replace circuit with V OC in series with R Th. Norton: Replace circuit with I SC in parallel with R Th. Note: for 3(b) the equivalent network is merely R Th, that is, no voltage (or current) source. Only steps 2 & 4 differ from Thevenin & Norton!

15. lecture 1215 Class Examples