Combining Resistors & Voltage/Current Division

Series Resistors: If resistors are connected in series with each other, that is if they share the same node back-to-back exclusively, than the sum of their resistance values is the equivalent resistance and both resistors can be replaced by the equivalent resistor.

• Req = R1 + R2 + ....... + Rn
  

Parallel Resistors: If resistors are connected in parallel with each other, that is if both ends of the resistors share the same node exclusively, than their equivalent resistance is the product of both resistances divided by the sum of the resistances.

• Req = (R1)(R2) / (R1 + R2)

Voltage Division: If the voltage across resistor n is to be found, the value of that resistor is divided by the sum of the resistors in series and then multiplied by the voltage source.

• Vn = [Rn / (R1 + R2 + ... + Rn)] * V

Current Division: for resistors in parallel the total current is split between the resistors. The value of current through each resistor (in parallel) is proportional to their resistances. Current travels through the path of least resistance, so the resistor that has a lower resistance will have a larger current flow through it.



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Laws

Ohm's Law: voltage across a resistor is proportional to the current through it.

• Voltage = current multiplied by resistance. V = iR

Kirchhoff's Current Law(KCL): the sum of currents entering a node is zero.

Kirchhoff's Voltage Law(KVL): the sum of voltages around a loop is zero.

• Branch: any single element (example: resistor, voltage source, etc...).
• Node: point of connection between two or more branches.
• Loop: closed path in a circuit.

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Basics

Circuit analysis is the study of electric circuits, which is the interconnection of circuit elements. An element is any component of the circuit(resistor, voltage source, etc..)

Fundamental concepts:

• Voltage(volts): the energy/work to move charge through an element. 1 volt = 1 joule per coulomb
  
• Current(amperes): the rate of charge flow. 1 amp = 1 coulomb per second
  
• Charge(coulombs): the charge on an proton/electron is +/- 1.602*10^-19
  
• Power(watts): the rate of energy being supplied or absorbed. Power = voltage multiplied by current (P=vi).
  
• Energy(joule): the capacity to do work.

In a circuit there can be two types of elements:

• Active: these elements generate energy (such as a battery). There are two sources of active elements:

1. Independent source: voltage/current source delivers a specific amount of voltage/current to the circuit.
2. Dependent source: the amount of voltage/current is controlled by other elements in the circuit.
   

• Passive: these elements do not generate energy (such as a resistor).

Ideal voltage source: provides any amount of current to maintain the stated terminal voltage.
Ideal current source: provides any amount of voltage to maintain the stated current flow.




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