Electrical Circuit Symbols Electrical circuits often contain one
Electrical Circuit Symbols 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
Resistances in Series Resistors are said to be connected in series when there is a single path for the current. I R 1 VT R 2 R 3 Only one current For series connections: 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: I = I 1 = I 2 = I 3 VT = V 1 + V 2 + V 3
Equivalent Resistance: Series The equivalent resistance Re of a number of resistors connected in series is equal to the sum of the individual resistances. VT = V 1 + V 2 + V 3 ; (V = IR) I R 1 VT R 2 R 3 Equivalent Resistance IT R e = I 1 R 1 + I 2 R 2 + I 3 R 3 But. . . IT = I 1 = I 2 = I 3 Re = R 1 + R 2 + R 3
Example 1: Find the equivalent resistance Re. What is the current I in the circuit? 2 W 3 W 1 W 12 V Re = R 1 + R 2 + R 3 Re = 3 W + 2 W + 1 W = 6 W Equivalent Re = 6 W The current is found from Ohm’s law: V = IRe I=2 A
Sources of EMF in Series The output direction from a source of emf is from + side: - a + b E Thus, from a to b the potential increases by E; From b to a, the potential decreases by E. A R AB: DV = +9 V – 3 V = +6 V 3 V BA: DV = +3 V - 9 V = -6 V B - 9 V + + Example: Find DV for path AB and then for path BA.
Summary: Single Loop Circuits: R 2 Resistance Rule: Re = SR R 1 Voltage Rule: SE = SIR E 2 E 1
Parallel Connections Resistors are said to be connected in parallel when there is more than one path for current. Parallel Connection: 2 W 4 W 6 W Series Connection: 2 W 4 W 6 W For Parallel Resistors: V 2 = V 4 = V 6 = V T I 2 + I 4 + I 6 = I T For Series Resistors: I 2 = I 4 = I 6 = I T V 2 + V 4 + V 6 = V T
Equivalent Resistance: Parallel VT = V 1 = V 2 = V 3 IT = I 1 + I 2 + I 3 VT Parallel Connection: Ohm’s law: The equivalent resistance for Parallel resistors: R 1 R 2 R 3
Example 3. Find the equivalent resistance Re for the three resistors below. VT R 1 2 W R 2 4 W R 3 6 W Re = 1. 09 W For parallel resistors, Re is less than the least Ri.
Example 3 (Cont. ): Assume a 12 -V emf is connected to the circuit as shown. What is the total current leaving the source of emf? VT R 1 2 W R 2 4 W R 3 6 W VT = 12 V; Re = 1. 09 W V 1 = V 2 = V 3 = 12 V IT = I 1 + I 2 + I 3 12 V Ohm’s Law: Total current: IT = 11. 0 A
Example 3 (Cont. ): Show that the current leaving the source IT is the sum of the currents through the resistors R 1, R 2, and R 3. VT R 1 2 W R 2 4 W R 3 6 W 12 V 6 A + 3 A + 2 A = 11 A IT = 11 A; Re = 1. 09 W V 1 = V 2 = V 3 = 12 V IT = I 1 + I 2 + I 3 Check !
Short Cut: Two Parallel Resistors The equivalent resistance Re for two parallel resistors is the product divided by the sum. Example: VT R 1 6 W R 2 3 W Re = 2 W
Series and Parallel Combinations In complex circuits resistors are often connected in both series and parallel. R 1 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. V T R 2 VT R 3 Re
Example 3. Find the equivalent resistance for the circuit drawn below (assume VT = 12 V). 4 W VT 3 W 6 W Re = 4 W + 2 W Re = 6 W 4 W 12 V 2 W 12 V 6 W
Example 3 (Cont. ) Find the total current IT. Re = 6 W 4 W VT 3 W 6 W IT = 2. 00 A 4 W 12 V 2 W 12 V IT 6 W
Example 3 (Cont. ) Find the currents and the voltages across each resistor. I 4 = I T = 2 A 4 W VT 3 W 6 W V 4 = (2 A)(4 W) = 8 V The remainder of the voltage: (12 V – 8 V = 4 V) drops across EACH of the parallel resistors. V 3 = V 6 = 4 V This can also be found from V 3, 6 = I 3, 6 R 3, 6 = (2 A)(2 W) (Continued. . . )
Example 3 (Cont. ) Find the currents and voltages across each resistor. V 4 = 8 V V 6 = V 3 = 4 V I 3 = 1. 33 A I 6 = 0. 667 A 4 W VT 3 W I 4 = 2 A Note that the junction rule is satisfied: SI (enter) = SI (leaving) IT = I 4 = I 3 + I 6 6 W
Kirchoff’s Laws for DC Circuits Kirchoff’s first law: The sum of the currents entering a junction is equal to the sum of the currents leaving that junction. Junction Rule: SI (enter) = SI (leaving) 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. Voltage Rule: SE = SIR
Sign Conventions for Emf’s § 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. § If tracing from B to A, this emf is considered negative. A E + B B
Summary Kirchoff’s Laws Kirchoff’s first law: The sum of the currents entering a junction is equal to the sum of the currents leaving that junction. Junction Rule: SI (enter) = SI (leaving) 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. Voltage Rule: SE = SIR
Summary of Formulas: Rules for a simple, single loop circuit containing a source of emf and resistors. Resistance Rule: Re = SR D Single Loop - 2 W SE = SIR C - 18 V + + Voltage Rule: 3 W 3 V A B
Summary (Cont. ) For resistors connected in series: For series connections: I = I 1 = I 2 = I 3 VT = V 1 + V 2 + V 3 Re = R 1 + R 2 + R 3 Re = SR 2 W 3 W 1 W 12 V
Summary (Cont. ) Resistors connected in parallel: V = V 1 = V 2 = V 3 IT = I 1 + I 2 + I 3 For parallel connections: Parallel Connection R 1 R 2 R 3 VT 2 W 12 V 4 W 6 W
CONCLUSION: Chapter 28 A Direct Current Circuits
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