CHPTR 26 Notes: Kirchoff's Rules & RC Circuits

Kirchhoff’s Rules

• Application of Kirchhoff’s Rules:

  • Analyzing complex circuits is simplified by the use of Kirchhoff’s rules.

• Current (or Junction) Rule:

  • Definition: The sum of the currents entering/leaving any junction point is zero.

  • Mathematical Representation: In the example at junction ‘a’:
    I1 + I2 - I_3 = 0

  • Implication: Indicates conservation of charge at any junction.

• Voltage (or Loop) Rule:

  • Definition: The sum of potential differences across all elements around any closed loop is zero.

  • Mathematical Representation:
    ext{Sum of } igtriangleup V = 0

  • Implication: Indicates energy conservation in closed circuits.

Voltage Across Elements

• Voltage Change Representation:

  • The voltage changes across each element as it is traversed from left to right is shown.

  • Important for applying Kirchhoff’s Voltage Rule in practical scenarios.

Solving Circuit Problems

• Equation Requirement:

  • To solve a circuit problem, the number of independent equations needed equals the number of unknown quantities.

RC Circuits

• Definition of RC Circuit:

  • A circuit consisting of a resistor (R) and a capacitor (C) connected in series.

  • Purpose: To control how quickly electric current increases or decreases within the circuit.

  • Key Characteristic: Do not deal with a constant value of electric current.

• Applications of RC Circuits:

  • Commonly utilized as “high-pass” and “low-pass” filters for specific frequencies.

  • Other applications include:

  • Blinking lights

  • Turn signals

  • Windshield wipers

  • Pacemakers

  • Strobe lights

  • Flashbulbs

  • “ON/OFF” processes

• Kirchhoff’s Application in RC Circuits:

  • When Kirchhoff’s rule is applied to a resistor and capacitor in series with a voltage source, the following is derived:

  • The product of R and C is referred to as the “time constant” (denoted as $ au$) of the circuit.

  • Mathematical Formulation:
    RC = au
    rac{dq}{dt} + rac{q}{RC} = - rac{ ext{ε}}{RC}
    Where $ ext{ε}$ is the electromotive force (emf) applied to the circuit.

• Distinct Stages in RC Circuits:

  • There are two key phases:

  • Charging the Capacitor:

    • Involves the rate of charge into the capacitor.

  • Discharging the Capacitor:

    • Involves the rate of charge leaving the capacitor.

Charging Capacitors through a Resistor

• Charge Function Over Time:

  • Formula applying to the charge on the capacitor as a function of time:
    Q(t) = C ext{ε}(1 - e^{-t/RC})

  • Current During Charging:

  • Current can be expressed as:
    i(t) = rac{ε}{R} e^{-t/RC}

  • Where $Q$ is the charge at any given time $t$ and $C$ is the capacitance.

Discharging a Capacitor

• Discharge Dynamics:

  • As the capacitor discharges through the resistor, the equation governing the circuit is:
    R rac{di}{dt} + rac{q}{C} = 0

• Current and Voltage for Discharging Capacitor:

  • The equations representing the current and voltage during a discharging phase are:

  • Current:
    i(t) = - rac{Q}{RC} e^{-t/RC}

  • Voltage:
    V(t) = Q imes rac{1}{C} e^{-t/RC}