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Basic Laws Discussion D2.1 Chapter 2 Sections 2-1 – 2-6, 2-10 1 Basic Laws • • • • • Ohm's Law Kirchhoff's Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Source Exchange 2 Georg Simon Ohm (1789 – 1854) German professor who publishes a book in 1827 that includes what is now known as Ohm's law. Ohm's Law: The voltage across a resistor is directly proportional to the currect flowing through it. http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Ohm.html 3 Resistance resistivity in Ohm-meters Resistance R l A l = length Good conductors (low ): Copper, Gold A Good insulators (high ): Glass, Paper 4 v R i + v i R v iR v i1 v i1R (i i1 ) - R + - i v R - + Ohm's Law Units of resistance, R, is Ohms (W) R = 0: short circuit R : open circuit 5 + 1 G R G v - i - + Conductance, G Unit of G is siemens (S), 1 S = 1 A/V i v G i Gv i G v 6 Power A resistor always dissipates energy; it transforms electrical energy, and dissipates it in the form of heat. Rate of energy dissipation is the instantaneous power 2 v (t ) 2 p(t ) v(t )i(t ) Ri (t ) 0 R 2 i (t ) 2 p(t ) v(t )i(t ) Gv (t ) 0 G 7 Basic Laws • • • • • Ohm's Law Kirchhoff's Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Source Exchange 8 Gustav Robert Kirchhoff (1824 – 1887) Born in Prussia (now Russia), Kirchhoff developed his "laws" while a student in 1845. These laws allowed him to calculate the voltages and currents in multiple loop circuits. http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Kirchhoff.html 9 CIRCUIT TOPOLOGY • Topology: How a circuit is laid out. • A branch represents a single circuit (network) element; that is, any two terminal element. • A node is the point of connection between two or more branches. • A loop is any closed path in a circuit (network). • A loop is said to be independent if it contains a branch which is not in any other loop. 10 Fundamental Theorem of Network Topology For a network with b branches, n nodes and l independent loops: b l n 1 Example 7W 1W DC 2W 3W 6W 4W 5W 2A b 9 n 5 l 5 11 Elements in Series Two or more elements are connected in series if they carry the same current and are connected sequentially. I R1 V0 R2 12 Elements in Parallel Two or more elements are connected in parallel if they are connected to the same two nodes & consequently have the same voltage across them. I I1 V R1 I2 R2 13 Kirchoff’s Current Law (KCL) The algebraic sum of the currents entering a node (or a closed boundary) is zero. N i n 1 n 0 where N = the number of branches connected to the node and in = the nth current entering (leaving) the node. 14 Sign convention: Currents entering the node are positive, currents leaving the node are negative. N i n 1 n 0 i2 i1 i5 i3 i4 i1 i2 i3 i4 i5 0 15 Kirchoff’s Current Law (KCL) The algebraic sum of the currents entering (or leaving) a node is zero. Entering: i1 i2 i3 i4 i5 0 Leaving: i1 i2 i3 i4 i5 0 i2 i1 i5 i3 i4 The sum of the currents entering a node is equal to the sum of the currents leaving a node. i1 i2 i4 i3 i5 16 Kirchoff’s Voltage Law (KVL) The algebraic sum of the voltages around any loop is zero. M v m 1 m 0 where M = the number of voltages in the loop and vm = the mth voltage in the loop. 17 Sign convention: The sign of each voltage is the polarity of the terminal first encountered in traveling around the loop. I + R1 V1 + A V0 R2 The direction of travel is arbitrary. Clockwise: V0 V1 V2 0 V2 - Counter-clockwise: V2 V1 V0 0 V0 V1 V2 18 Basic Laws • • • • • Ohm's Law Kirchhoff's Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Source Exchange 19 Series Resistors I + R1 V1 I R1 R2 + A V0 V0 V1 V2 IR1 IR2 R2 IRs V2 Rs R1 R2 - I V Rs 20 Voltage Divider V0 V0 I Rs R1 R2 I R1 V1 R2 V2 A V0 V0 V2 IR2 R2 R1 R2 R2 V2 V0 R1 R2 R1 Also V1 V0 R1 R2 21 Basic Laws • • • • • Ohm's Law Kirchhoff's Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Source Exchange 22 Parallel Resistors I V V I I1 I 2 R1 R2 I1 R1 V R2 1 1 1 Rp R1 R2 I V I2 Rp 1 1 V R1 R2 V Rp R1 R2 Rp R1 R2 23 Current Division i + i1 i(t) R1 i2 R2 v(t) - R2 v(t ) i1 (t ) i (t ) R1 R1 R2 R1 v(t ) i2 (t ) i (t ) R2 R1 R2 R1 R2 v(t ) R p i (t ) i (t ) R1 R2 Current divides in inverse proportion to the resistances 24 Current Division N resistors in parallel 1 1 1 1 Rp R1 R2 Rn Current in jth branch is v(t ) R pi (t ) v(t ) R p i j (t ) i (t ) Rj Rj 25 Basic Laws • • • • • Ohm's Law Kirchhoff's Laws Series Resistors and Voltage Division Parallel Resistors and Current Division Source Exchange 26 Source Exchange ia ' ia + Rs DC vab vs - + vs Rs Rs v ab - We can always replace a voltage source in series with a resistor by a current source in parallel with the same resistor and vice-versa. Doing this, however, makes it impossible to directly find the 27 original source current. Source Exchange Proof ia ' ia + Rs DC RL vL vs + vs Rs Rs RL vL - - RL vL vs Rs RL Rs vs ia ' ia Rs RL Rs vs ia Rs RL RL vL ia ' RL vs Rs RL Voltage across and current through any load are the same28