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電路學(一) Chapter 2 Basic Laws 1 Basic Laws - Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 Ohm’s Law. Nodes, Branches, and Loops. Kirchhoff’s Laws. Series Resistors and Voltage Division. Parallel Resistors and Current Division. Wye-Delta Transformations. 2 2.1 Ohms Law (1) • Ohm’s law states that the voltage across a resistor is directly proportional to the current I flowing through the resistor. • Mathematical expression for Ohm’s Law is as follows: v iR • Two extreme possible values of R: 0 (zero) and (infinite) are related with two basic circuit concepts: short circuit(短路) and open circuit(開路). 3 2.1 Ohms Law (2) • Conductance(電阻) is the ability of an element to conduct electric current; it is the reciprocal of resistance R and is measured in mhos or siemens. 1 i G R v • The power dissipated by a resistor: 2 v p vi i 2 R R 4 2.2 Nodes, Branches and Loops (1) • A branch(分支) represents a single element such as a voltage source or a resistor. • A node(節點) is the point of connection between two or more branches. • A loop(迴路) is any closed path in a circuit. • A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology: b l n 1 5 2.2 Nodes, Branches and Loops (2) Example 1 Original circuit Equivalent circuit How many branches, nodes and loops are there? 6 2.2 Nodes, Branches and Loops (3) Example 2 Should we consider it as one branch or two branches? How many branches, nodes and loops are there? 7 2.3 Kirchhoff’s Laws (1) • Kirchhoff’s current law (KCL) (克希荷夫電流定律) states that the algebraic sum of currents entering a node (or a closed boundary) is zero. N Mathematically, i n 1 n 0 8 2.3 Kirchhoff’s Laws (2) Example 4 • Determine the current I for the circuit shown in the figure below. I + 4-(-3)-2 = 0 I = -5A We can consider the whole enclosed area as one “node”. This indicates that the actual current for I is flowing in the opposite direction. 9 2.3 Kirchhoff’s Laws (3) • Kirchhoff’s voltage law (KVL) (克希荷夫電壓定律) states that the algebraic sum of all voltages around a closed path (or loop) is zero. Mathematically, M v m 1 n 0 10 2.3 Kirchhoff’s Laws (4) Example 5 • Applying the KVL equation for the circuit of the figure below. va − v1 − vb − v2 − v3 = 0 V1 = IR1 v2 = IR2 v3 = IR3 va − vb = I(R1 + R2 + R3) v a vb I R1 R2 R3 11 2.3 Kirchhoff’s Laws (5) Example 6 • Determine vo and i in the circuit. 12 2.3 Kirchhoff’s Laws (6) Example 7 • Find current io and voltage vo in the circuit. 13 2.3 Kirchhoff’s Laws (7) Example 8 • Find current and voltage in the circuit. 14 2.4 Series Resistors and Voltage Division (1) • Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current. • The equivalent resistance (等效電阻) of any number of resistors connected in a series is the sum of the individual resistances. N Req R1 R2 RN Rn n 1 • The voltage divider can be expressed as vn Rn v R1 R2 RN 15 2.4 Series Resistors and Voltage Division (2) Example 9 10V and 5W are in series 16 2.5 Parallel Resistors and Current Division (3) • Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. • The equivalent resistance of a circuit with N resistors in parallel is: 1 1 1 1 Req R1 R2 RN • The total current i is shared by the resistors in inverse proportion to their resistances. The current divider can be expressed as: v iReq in Rn Rn 17 2.5 Parallel Resistors and Current Division (4) Example 10 2W, 3W and 2A are in parallel 18 2.5 Parallel Resistors and Current Division (5) Example 11 • Find Req for the circuit. 19 2.5 Parallel Resistors and Current Division (6) Example 12 • Calculate the equivalent resistance Rab for the circuit. 20 2.5 Parallel Resistors and Current Division (7) Example 13 • Find current io and voltage vo in the circuit. Calculate the power dissipated in the 3-W resistor. 2.5 Parallel Resistors and Current Division (8) Example 14 • Find current io and voltage vo in the circuit. Calculate the power dissipated in the 3-W resistor. 2.6 Wye-Delta Transformations (1) Δ -> Y Rb Rc R1 ( Ra Rb Rc ) Y -> Δ R1 R2 R2 R3 R3 R1 Ra R1 Rc Ra R2 ( Ra Rb Rc ) Rb R1 R2 R2 R3 R3 R1 R2 Ra Rb ( Ra Rb Rc ) Rc R1 R2 R2 R3 R3 R1 R3 R3 23 2.6 Wye-Delta Transformations (2) Example 15 • Obtain the equivalent Rab and the current i.