Ohm’s Law

• Materials in general have a characteristic behavior of resisting the flow of electric charge. This is known as resistance and is represented by the symbol $R$.
• The circuit element used to model the current-resisting behavior of a material is called resistor.
• Ohm’s Law: the voltage $V$ across a resistor is directly proportional to the current $I$ flowing through the resistor ($V \propto I$). $$\boxed{V=IR}$$ Where $R$ is the resistance of an element denotes its ability to resist the flow of electric current. It is measured in ohms ($\ohm$)
• To apply Ohm’s law, the direction of $I$ and the polarity of $V$ must conform with the passive sign convention.
• There are two possible extreme values of $R$.
  
  1. A short circuit is a circuit element with resistance approaching zero ($R = 0$).
     • For a short circuit, $V = IR = 0$, shows that the voltage is zero but the current could be anything.
  2. An open circuit is a circuit element with resistance approaching infinity ($R = \infty$).
     • For an open circuit, $I = \lim_{R\to\infty}\frac{V}{R}=0$, indicates that the current is zero although the voltage could be anything.
• The reciprocal of resistance $R$ is known as conductance and denoted by $G$. $$G=\frac{1}{R}=\frac{I}{V}$$ Where the unit of $G$ is the mho ($\mho$) or siemens ($S$).
• The power dissipated by a resistor can be represented in terms of $R$ as follows.
  • กำลังไฟฟ้าและพลังงานไฟฟ้าที่สูญเสียในตัวต้านทาน
  $$\boxed{\ce{P}=\ce{IV}=\ce{I^2R}=\frac{\ce{V^2}}{\ce{R}}}$$

Node, Branches and Loops

• In network topology, we study the properties relating to the placement of elements in the network and the geometric configuration of the network. Such elements include branches, nodes, and loops.
• A branch is a single element such as voltage source or a resistor. A branch represents any two terminals element.
  • จำนวนอุปกรณ์แค่นั้นแหละ (จะ Dependent, Independent นับหมด)
• A node is the point of connection between two or more branches. It is usually indicated by a dot in a circuit.
  • จุดต่อของอุปกรณ์มากกว่า 2 ชิ้นขึ้นไป
  • Reference node, เป็น Node ที่เอาไว้วิเคราะห์ใน Node Analysis → Voltage = 0
    • ปกติแล้วจะเลือกด้านล่าง + จุดที่มี Branch connect กันมากที่สุด
• Loop is any closed path in a circuit. A closed path formed by starting at a node, passing through a set of nodes and returning to starting node without passing through any node more than once.
• Mesh is a closed path in a circuit with no other paths inside it. In other words, a loop with no other loops inside it.

• Series: two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current.
• 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.

Kirchhoff’s Laws

• ใช้แก้โจทย์วงจรไฟฟ้าที่มีหลาย Loop ซึ่งไม่สามารถยุบให้เหลือ Loop เดียวได้
• Ohm’s law by itself is not sufficient to analyze circuits. → ก็เลยต้องมี Kirchhoff’s Laws มาช่วย

Kirchhoff’s Current Law (KCL)

• ==A Kirchhoff’s Current Law (KCL): based on the law of conservation of charge, the algebraic sum of current entering a node (or a closed boundary) is zero.== It is mathematically represented as follows.
  • คิดเป็น Closed boundary ก็ได้ ไม่จำเป็นต้องเป็นจุด ๆ เดียว
  $$\sum_{n=1}^{N}I_{n}=0$$ $$\sum I_{in}=\sum I_{out}$$ Where $N$ is the number of branches connected to the node and in is the $n$th current entering (or leaving) the node.
• The sum of currents entering a node is equal to the sum of the currents leaving the node.
• Point Rule: ที่จุดใด ๆ ในวงจร ผลรวมของกระแสไฟฟ้าที่ไหลเข้าเท่ากับที่ไหลออก

Kirchhoff’s Voltage Law (KVL)

• ==A Kirchhoff’s Voltage Law (KVL): based on the principle of conservation of energy, the algebraic sum of all voltage around a closed path (or loop/mesh) is zero.== It is mathematically represented as follows. $$\sum_{m=1}^{M}V_{m}=0$$ Where $M$ is the number of voltages in the loop (or the number of branches in the loop) and $V_m$ is the $m$th voltage.
• It can be interpreted as; in a loop, the sum of voltage drops is equal to the sum of voltage rises.

อย่าลืมถ้าเค้าไม่กำหนดขั้วของ $R$ มาให้ ไหลเข้าให้เป็น + เสมอ จำไว้!

• อย่างข้อนี้ ให้หา $V_1,V_2,V_3$ จะหายังไง ก็บอกแล้ว $\Sigma V = 0$ ใช้ได้กับ Closed Path, ก็เลือก Loop ที่มันเป็นมีค่าบอกอยู่แล้ว ไม่ต้องไปคิด Loop เล็ก ติด 3 ตัวแปร มีแค่ 1 สมการ!

Series Resistors and Voltage Division

• The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances. For $N$ resistors in series,
  • ก็แค่รวม $R$ ธรรมดาแบบอนุกรม
  $$R_{eq}=R_1+R_2+\dotso+R_N=\sum_{n=1}^{N}R_n$$
• The principle of voltage division states that the source voltage $V$ is divided among the resistors in direct proportion to their resistances.
• In general, if a voltage divider has $N$ resistors in series with the source voltage $V$, the $n$th resistor $R_n$ will have a voltage drop as follows. $$V_n=\frac{R_n}{R_1+R_2+\dotso+R_n}V$$

Parallel Resistors and Current Division

• The equivalent resistance for the general case of a circuit with $N$ resistors in parallel is as follows.
  • ก็แค่รวม $R$ ธรรมดาแบบขนาน
  $$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\dotso+\frac{1}{R_N}$$ $$G_{eq}=G_1+G_2+\dotso+G_N$$
• The principle of current divider states that the total current $i$ is shared by the resistors in inverse proportion to their resistances.
• Current Divider ($N=2$)
  • ก็เหมือนเอา $V=IR$ มา Substitute เข้า $V$ อีกรอบนึง แล้วก็กระจายคูณส่วนบวก

I_2=\frac{V}{R_2}=\frac{R_{eq}}{R_2}I=\frac{R_1}{R_1+R_2}I

R_1=\frac{R_b\times R_c}{\Delta}

R_2=\frac{R_a\times R_c}{\Delta}

R_3=\frac{R_a\times R_b}{\Delta}

R_a=\frac{R_1R_2+R_2R_3+R_1R_3}{R_1}

R_b=\frac{R_1R_2+R_2R_3+R_1R_3}{R_2}

R_c=\frac{R_1R_2+R_2R_3+R_1R_3}{R_3}