EEE ( ) - ACTIVE LEARNING ASSIGNMENT Presented by: Divyang Vadhvana( ) Branch: Information Technology

Electrical circuits often contain one or more resistors grouped together and attached to an energy source, such as a battery. The following symbols are often used: Ground Battery - + Resistor

Resistors are said to be connected in series when there is a single path for the current. The current I is the same for each resistor R 1, R 2 and R 3. The energy gained through E is lost through R 1, R 2 and R 3. The same is true for voltages: For series connections: I = I 1 = I 2 = I 3 V T = V 1 + V 2 + V 3 R1R1 I VTVT R2R2 R3R3 Only one current

The equivalent resistance R e of a number of resistors connected in series is equal to the sum of the individual resistances. V T = V 1 + V 2 + V 3 ; (V = IR) I T R e = I 1 R 1 + I 2 R 2 + I 3 R 3 But... I T = I 1 = I 2 = I 3 R e = R 1 + R 2 + R 3 R1R1 I VTVT R2R2 R3R3 Equivalent Resistance

The output direction from a source of emf is from + side: E +- a b Thus, from a to b the potential increases by ; From b to a, the potential decreases by. Thus, from a to b the potential increases by E ; From b to a, the potential decreases by E. Example: Find  V for path AB and then for path BA. R 3 V3 V V9 V A B AB:  V = +9 V – 3 V = +6 V BA:  V = +3 V - 9 V = -6 V

Consider the simple series circuit drawn below: 2  3 V V A C B D 4  Path ABCD: Energy and V increase through the 15-V source and decrease through the 3-V source. The net gain in potential is lost through the two resistors: these voltage drops are IR 2 and IR 4, so that the sum is zero for the entire loop.

Resistance Rule: R e =  R Voltage Rule:  E =  IR R2R2 E1E1 E2E2 R1R1

A complex circuit is one containing more than a single loop and different current paths. R2R2 E1E1 R3R3 E2E2 R1R1 I1I1 I 3 I2I2 mn At junctions m and n: I 1 = I 2 + I 3 or I 2 + I 3 = I 1 Junction Rule:  I (enter) =  I (leaving) Junction Rule:  I (enter) =  I (leaving)

Resistors are said to be connected in parallel when there is more than one path for current. 2  4 4  6  Series Connection: For Series Resistors: I 2 = I 4 = I 6 = I T V 2 + V 4 + V 6 = V T Parallel Connection: 6  2 2  4  For Parallel Resistors: V 2 = V 4 = V 6 = V T I 2 + I 4 + I 6 = I T

V T = V 1 = V 2 = V 3 I T = I 1 + I 2 + I 3 Ohm’s law: The equivalent resistance for Parallel resistors: Parallel Connection: R3R3 R2R2 VTVT R1R1

The equivalent resistance R e for two parallel resistors is the product divided by the sum. R e = 2  Example: R2R2 VTVT R1R1 6  3 

In complex circuits resistors are often connected in both series and parallel. VTVT R2R2 R3R3 R1R1 In such cases, it’s best to use rules for series and parallel resistances to reduce the circuit to a simple circuit containing one source of emf and one equivalent resistance. VTVT ReRe

Kirchoff’s first law: The sum of the currents entering a junction is equal to the sum of the currents leaving that junction. Kirchoff’s second law: The sum of the emf’s around any closed loop must equal the sum of the IR drops around that same loop. Junction Rule:  I (enter) =  I (leaving) Voltage Rule:  E =  IR

 When applying Kirchoff’s laws you must assume a consistent, positive tracing direction.  When applying the voltage rule, emf’s are positive if normal output direction of the emf is with the assumed tracing direction.  If tracing from A to B, this emf is considered positive. E A B+  If tracing from B to A, this emf is considered negative. E A B+

 When applying the voltage rule, IR drops are positive if the assumed current direction is with the assumed tracing direction.  If tracing from A to B, this IR drop is positive.  If tracing from B to A, this IR drop is negative. I A B+ I A B+

R3R3 R1R1 R2R2 E2E2 E1E1 E3E3 1. Assume possible consistent flow of currents. 2. Indicate positive output directions for emf’s. 3. Indicate consistent tracing direction. (clockwise) + Loop I I1I1 I2I2 I3I3 Junction Rule: I 2 = I 1 + I 3 Voltage Rule:  E =  IR E 1 + E 2 = I 1 R 1 + I 2 R 2 Voltage Rule:  E =  IR E 1 + E 2 = I 1 R 1 + I 2 R 2

4. Voltage rule for Loop II: Assume counterclockwise positive tracing direction. Voltage Rule:  E =  IR E 2 + E 3 = I 2 R 2 + I 3 R 3 Voltage Rule:  E =  IR E 2 + E 3 = I 2 R 2 + I 3 R 3 R3R3 R1R1 R2R2 E2E2 E1E1 E3E3 Loop I I1I1 I2I2 I3I3 Loop II Bottom Loop (II) + Would the same equation apply if traced clockwise? - E 2 - E 3 = -I 2 R 2 - I 3 R 3 Yes!

5. Voltage rule for Loop III: Assume counterclockwise positive tracing direction. Voltage Rule:  E =  IR E 3 – E 1 = -I 1 R 1 + I 3 R 3 Voltage Rule:  E =  IR E 3 – E 1 = -I 1 R 1 + I 3 R 3 Would the same equation apply if traced clockwise? E 3 - E 1 = I 1 R 1 - I 3 R 3 Yes! R3R3 R1R1 R2R2 E2E2 E1E1 E3E3 Loop I I1I1 I2I2 I3I3 Loop II Outer Loop (III) + +

6. Thus, we now have four independent equations from Kirchoff’s laws: R3R3 R1R1 R2R2 E2E2 E1E1 E3E3 Loop I I1I1 I2I2 I3I3 Loop II Outer Loop (III) + + I 2 = I 1 + I 3 E 1 + E 2 = I 1 R 1 + I 2 R 2 E 2 + E 3 = I 2 R 2 + I 3 R 3 E 3 - E 1 = -I 1 R 1 + I 3 R 3

Resistance Rule: R e =  R Voltage Rule:  E =  IR Rules for a simple, single loop circuit containing a source of emf and resistors. 2  3 V V A C B D 3  Single Loop

For resistors connected in series: R e = R 1 + R 2 + R 3 For series connections: I = I 1 = I 2 = I 3 V T = V 1 + V 2 + V 3 R e =  R 2  12 V 1  3 

Resistors connected in parallel: For parallel connections: V = V 1 = V 2 = V 3 I T = I 1 + I 2 + I 3 R3R3 R2R2 12 V R1R1 2  4  6  VTVT Parallel Connection

Kirchoff’s first law: The sum of the currents entering a junction is equal to the sum of the currents leaving that junction. Kirchoff’s second law: The sum of the emf’s around any closed loop must equal the sum of the IR drops around that same loop. Junction Rule:  I (enter) =  I (leaving) Voltage Rule:  E =  IR