Electrical Circuits: Series and Parallel Methods

Study Notes

Three methods of connecting electrical loads: series, parallel, and series-parallel.

Current flows through a single path: ( I_T = I_1 = I_2 ).
Supply voltage equals the sum of individual voltages: ( V_T = V_1 + V_2 ).
Total resistance is the sum of all individual resistances: ( R_T = R_1 + R_2 ); always higher than the highest individual resistance.
Ohm’s Law applies: ( V_T = I_T \times R_T ).

Supply voltage is the same across all branches: ( V_T = V_1 = V_2 ).
Total current is the sum of individual branch currents: ( I_T = I_1 + I_2 ).
Total resistance is calculated by the reciprocal of the sum of the reciprocals of individual resistances:
- ( \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} ).
For two resistors in parallel, use: ( R_T = \frac{R_1 \times R_2}{R_1 + R_2} ); total resistance is always lower than the lowest individual resistance.

Example with ( R_1 = 10 , \Omega ), ( R_2 = 20 , \Omega ), ( V_S = 120 , V ):
- Total resistance: ( R_T = 10 , \Omega + 20 , \Omega = 30 , \Omega ).
- Supply current: ( I_S = \frac{V_S}{R_T} = \frac{120 , V}{30 , \Omega} = 4 , A ).
- Voltage across ( R_1 ): ( V_1 = I_1 \times R_1 = 4 , A \times 10 , \Omega = 40 , V ).

Resistors of ( 50 , \Omega ) and ( 70 , \Omega ) in series with a ( 100 , V ) supply:
- Total resistance: ( R_T = 50 , \Omega + 70 , \Omega = 120 , \Omega ).
- Supply current: ( I_S = \frac{100 , V}{120 , \Omega} \approx 0.83 , A ).

Total resistance for resistors in parallel is calculated with their reciprocal relationships.

Series-parallel circuits incorporate the characteristics of both series and parallel configurations.
They offer varied paths for current, affecting voltage and resistance calculations.