1. Lecture 221 Series and Parallel Resistors/Impedances

2. Lecture 222 Introduction For analysis, series resistors/impedances can be replaced by an equivalent resistor/ impedance. Parallel resistors/impedances can be replaced by an equivalent resistor/ impedance.

3. Lecture 223 Introduction Complicated networks of resistors/ impedances can be replaced by a single equivalent resistor/impedance.

4. Lecture 224 Equivalent Resistance i(t) + - v(t) i(t) + - v(t) R eq R eq is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics.

5. Lecture 225 Equivalent Resistance The rest of the circuit cannot tell whether the resistor network or the equivalent resistor is connected to it. The equivalent resistance cannot be used to find voltages or currents internal to the resistor network.

6. Lecture 226 Equivalent Impedance I + - V I + - V Z eq Z eq is equivalent to the network on the left in the sense that they have the same phasor I-V characteristics at the frequency .

7. Lecture 227 Series Resistance R1R1 R3R3 R2R2 R eq R eq = R 1 + R 2 + R 3

8. Lecture 228 Series Two elements are in series if the current that flows through one must also flow through the other. R1R1 R2R2 Series R1R1 R2R2 Not Series

9. Lecture 229 Z eq Series Impedance Z1Z1 Z eq = Z 1 + Z 2 + Z 3 Z3Z3 Z2Z2

10. Lecture 2210 Example: Series Inductors What is the equivalent impedance of two series inductors? L2L2 L1L1

11. Lecture 2211 Series Inductors The equivalent impedance is j  (L 1 +L 2 ) Two inductors in series are equivalent to a single inductor whose inductance is the sum of the two inductances.

12. Lecture 2212 Parallel Resistance R eq 1/R eq = 1/R 1 + 1/R 2 + 1/R 3 R3R3 R2R2 R1R1

13. Lecture 2213 Parallel Two elements are in parallel if they are connected between the same two nodes. Parallel Not Parallel R1R1 R2R2 R1R1 R2R2

14. Lecture 2214 Parallel Impedance 1/Z eq = 1/Z 1 + 1/Z 2 + 1/Z 3 Z3Z3 Z1Z1 Z2Z2 Z eq

15. Lecture 2215 Example: Parallel Capacitors What is the equivalent impedance of two parallel capacitors? C1C1 C2C2

16. Lecture 2216 Parallel Capacitors The equivalent impedance is Two capacitors in parallel are equivalent to a single capacitor whose capacitance is the sum of the two capacitances.

17. Lecture 2217 Example: Ladder Network Ladder networks are used in analog-to- digital converters to provide reference voltages that are 1/2, 1/4, 1/8, etc. of a source voltage.

18. Lecture 2218 Ladder Network 1k  2k  1k  2k  Find the equivalent resistance.

19. Lecture 2219 Ladder Network Find the equivalent resistance by making combinations of series and parallel resistors until you have only one resistor left.

20. Lecture 2220 Ladder Network 2k  1k  2k 

21. Lecture 2221 Ladder Network 1k  2k 

22. Lecture 2222 Ladder Network 2k 

23. Lecture 2223 Ladder Network 1k  The equivalent resistance of the ladder network is 1k 

24. Lecture 2224 10  769pF 159  H Bandpass Filter For  = 2.86  10 6, find the equivalent impedance.

25. Lecture 2225 10  -j455  j455  Compute Impedances Now combine series impedances

26. Lecture 2226 -j455  10  j455  Bandpass Filter Now combine parallel impedances 455.1   88.7 

27. Lecture 2227 Bandpass Filter

28. Lecture 2228 Loaded Bandpass Filter For  = 2.86  10 6, find the equivalent impedance. 10  769pF 159  H 50k 