Resistors in Parallel Circuits

Resistors in Parallel

• Resistors in parallel are connected to the same two points in a circuit.

Parallel Circuits

• Parallel circuits have more than one current path, or branch.
• These branches are connected to a common voltage source.

Parallel Circuit Rule for Voltage

• All components in a parallel circuit are connected across the same voltage source.
• Therefore, the voltage across each component is the same.
• For example, if a source voltage is 5.0 V, a voltmeter reads 5.0 V across each resistor.

Parallel Circuit Rule for Resistance

• The total resistance of resistors in parallel is the reciprocal of the sum of the reciprocals of the individual resistors
• For example, a parallel circuit with resistors of 680 Ω, 1.5 kΩ, and 2.2 kΩ has a total resistance of 386 Ω.

Special Case for Resistance of Two Parallel Resistors

• The total resistance of two parallel resistors can be calculated using: R_T = 1 / (1/R₁ + 1/R₂) or R_T = (R₁R₂) / (R₁ + R₂).
• For example, if R₁ = 27 kΩ and R₂ = 56 kΩ, the total resistance is 18.2 kΩ.

Parallel Circuit Parameters

• Use current, resistance, voltage, and power tabulations.
• Used to summarize the parameters in a parallel circuit, given the other components

Kirchhoff's Current Law (KCL)

• KCL says the sum of currents entering a node is equal to the sum of currents leaving the node.
• Σ currents in - Σ currents out = 0 or Σ currents in = Σ currents out

Application of KCL

• In a parallel circuit, the current from the source equals the sum of the branch currents.

Current Divider

• Current flowing into two parallel resistors divides according to their resistance values.
• The formula for calculating the current (i₁) through resistor R₁ is: i₁ = i * (R₂ / (R₁ + R₂)), where 'i' is total current.

Current Divider Formula

• The current divider formula is applicable only to two branch resistances.
• The branch with the smallest resistance has the largest current flowing through it.
• R₂ = 0 implies that i₁ = i and i₂ = 0. It is a short
• R₂ = ∞ implies R₂ is an open with i₁ = 0 and i₂ = i, then R_eq = R.
• To find i₁ and i₂ in terms of is: i₁ = i_s * (R₂ / (R₁ + R₂)) and i₂ = i_s * (R₁ / (R₁ + R₂)).

Application of Current Divider

• For a 2.2 kΩ resistor (R₁) in parallel with a 4.7 kΩ resistor (R₂), with a total current of 8.0 mA.
• The current through R₁ is 5.45 mA.
• The current through R₂ is 2.55 mA.
• The larger resistor has the smaller current.

Power in Parallel Circuits

• Power in each resistor is calculated using P = V² / R.
• Total power is the sum of the powers dissipated in each resistor.
• A parallel combo of of R₁ = 270 Ω, R₂ = 150 Ω with applied 10V has a total power of 1.04 W

Kirchhoff's Voltage Law (KVL)

• KVL, or Kirchhoff's Current Law, is generally stated as the algebraic sum of voltages around each loop is zero
• Σ voltage drops - Σ voltage rises = 0 or Σ voltage drops = Σ voltage rises

Application of KVL

• Consider a 24 V circuit:
• The equivalent resistance is 10 Ω.
• The current is 2.4 A.
• The voltage across a 4 Ω resistor is 9.6V.
• The voltage across a 6 Ω resistor is 14.4V.
• V₁+V₂ = V_s = 24 V

Voltage Divider

• The Voltage Divider Rule (VDR) determines voltage drop across a resistance within a series circuit.
• The general form of the VDR formula is: V_Rx = (R_x / R_T) * V_T
• In this equation, R_T is the total resistance of the series circuit.
• R_x is the resistance across which you are calculating the voltage drop.

Application of VDR

• The voltage across a 3 kΩ resistor in series with a 4 kΩ and a 5 kΩ resistor is: V_R1 = (3 kΩ / (3kΩ+4kΩ+5kΩ)) * 12 V = 3 V
• The voltage across a 2 kΩ resistor in series with a 4 kΩ and a 6 kΩ resistor is: V_R1 = (2 kΩ / (2kΩ+4kΩ+6kΩ)) * 24 V = 4 V

Combining Voltage and Current Division

• Voltage and current are combined and applied with their divisions to find V_o and i_o:
• i = 12/(4+2) = 2 A
• V_o = 2/(2+4)(12 V) = 4 V
• i_o = 6/(6+3)i = 2/3(2 A) = 4/3 A

Complex Circuit Parameters

• The total current in a circuit with a 12 V source and resistors is calculated as follows:
• R_T = R₁ + R₂ + R₃
• R_T = 560 + 680 + 1000
• R_T = 2240 ohms
• E_T=I_T/R_T
• I_T=12/2240
• I_T = 0.0054 amp or 5.4 milliamp

Parallel Resistance

• For a two branch parallel circuits: Since the resistances are connected in parallel, the voltage across each resistor is the same.
• The current I is the total of the currents in the two branches. Then, I = I₁ + I₂

Superposition Principle

• Based on linearity property
• The voltage across (current through) an element is the algebraic sum of the voltage across (current through) that element due to each independent source acting alone.
• Only consider 1 independent source at a time
• Replace voltage sources with a wire (0 V).
• Replace current sources with an open circuit (no current can flow).

Steps in Applying the Superposition Principle

• Turn off all independent sources except one and find the output.
• Repeat this for each independent source.
• Find the total output by algebraically adding all results.

Thevening's Theorem

• Thevening's Theorem states any resistive circuit or network, no matter how complex, can be represented as a voltage source in series with a source resistance
• Steps to Determine Thevenin Equivalent:
  • Identify the Portion of the Circuit: Select the portion of the circuit where the Thevenin equivalent is to be found.
  • Remove the Load Resistor: Temporarily remove the load resistor.
  • Calculate Thevenin Voltage (Vth): Determine the open-circuit voltage across the terminals.
  • Calculate Thevenin Resistance (Rth): Deactivate all independent sources and calculate the equivalent resistance seen from the open terminals.
  • Construct Thevenin Equivalent Circuit: Use Vth as the voltage source and Rth as the series resistance, then reconnect the load resistor.

Norton's Theorem

• Turns that any circuit can be made up of currents in parallel
• To Determine Currents: Steps to Determine IN and RN
  • Steps to Determine IN and RN Identify the load, which may be a resistor or a part of the circuit. Replace the load with a short circuit. Calculate Isc. This is IN. Turn off all independent voltage and currents sources in the linear 2-term circuit. Calculate the equivalent resistance of the circuit. This is RN.
• The current through and voltage across the load in parallel with IN and RN is the load's actual current and voltage in the original circuit.