1. ENGR 111 Lecture 4 Reading: Chapters 19, Class notes

2. Lecture 4: DC Fundamentals Review of Last Class: More/less electrons => Charge Potential charge difference results in charge flow or current Potential charge difference = voltage Different materials offer different resistance to current Voltage V(volts), Current I (Amperes), Resistance R (ohms)

3. Water Analogy Charge flow through a wire similar to water flow in a pipe Harder to push water through a thinner pipe (smaller current, higher resistance) For water to flow, there has to be pressure difference at ends of pipe  Voltage has to exist across a wire for current

4. Some basic laws (Kirchoff) Kirchoff’s Current Law (KCL):  Current flowing into and out of a node should be equal  Conservation principle

5. Kirchoff’s voltage Law Voltages around a closed circuit should sum to zero  When you come to the same point, voltage difference should be zero Start End V1 V2 V3 V4 V5 V1 + V2 + V3 +V4 + V5 = 0

6. Ohm’s law relates resistance, voltage and current V = I * R Higher resistance, need higher voltage for the same amount of current to flow Water Analogy, higher pressure at ends of pipe, higher flow of water Ohm’s Law

7. Resistors Connected in series I I R1 R2 KCL => current entering R1 must leave R1 Current entering R2 = current leaving R1 V1 = I * R1, V2 = I *R2 V = V1 + V2 = I * R1 + I * R2 = I (R1+R2) = IR Resistors in series R = R1 + R2

8. Resistors in Series 100 ohms in series with 100 ohms = 200 ohms equivalent resistance 100 ohms in series with 1 ohm = ?  101 ohms from the calculator  100 ohms taking significant digits into account  Resistors are calibrated to 5 or 10% accuracy 100 ohms in series with 100 ohms = ? 100 ohms in series with 1M ohms = ?

9. Resistors in Parallel The current gets divided among the two paths. KVL tells us V = I1 * R1 = I2 * R2 KCL => I = I1 + I2 = V/R1 + V/R2 = V (1/R1 + 1/R2) I = V (R2 + R1)/R1R2 V = I (R1 * R2)/(R1 + R2) Equivalent Resistance R = R1 * R2/(R1 + R2) Easier to Remember 1/R = 1/R1 + 1/R2 Voltage across the two resistors must be equal. R1 R2 I I1 I2 I

10. Resistors in Parallel 100 ohms in parallel with 100 ohms 1/R = 1/100 + 1/100 = 2/100 = 1/50 R = 50 ohms, Resistance is smaller!! Water Analogy, two pipes in parallel, more opportunity for water to flow, less resistance 100 ohms in parallel with 1000 ohms 1/R = 1/100 + 1/1000, R = 90.90 = 91Ω

11. Voltage Dividers Resistors in series provide a mechanism The resistors determine the output Voltage KCL says same current in R1 and R2 Vout = V1 * R2/(R1+R2)

12. Current Dividers Resistors in parallel provide a mechanism The resistors determine the current in each path I1 * R1 = I2 * R2, I2 = I1 * R1/R2 I = I1 + I2 => I1 = I * R2/(R1+R2) I I1 I2 R1 R2

13. Example Dividers Given 10V, Need to provide 3V, how? Resistors in Series R2/(R1+R2) = 3/10, choose R2 = 300 KΩ R1 = 700 KΩ Why should R1, R2 be high? What happens when we connect a resistor R3 across R2?

14. Example Dividers Want to divide current into two paths, one with 30% --how? Resistors in parallel R2/(R1+R2) = 0.3, Choose R2 = 300 KΩ R1 = 700 KΩ Why should R1, R2 be high? What happens when we connect a resistor R3 in series with R2?

15. Summary Ohm’s Law V = I * R KCL/KVL and Ohm’s law allow us to compute equivalent resistances Resistances in series R = R1 + R2 Resistances in parallel 1/R = 1/R1 + 1/R2 Resistances in series => Voltage Dividers Resistances in parallel => Current dividers

16. Example 1: KVL & Ohm’s Law

17. Example 2: Resistors

18. Example 3: Resistors

19. Example 4: Voltage Divider

20. Example 5: Current Divider