Series combination of resistors
A series combination of resistors involves connecting multiple resistors end-to-end to share the same path for electric current, resulting in a single pathway. In a series configuration, the current through each resistor is the same, so the total voltage is the sum of the individual IR drops across each resistor. This is per Ohm’s law. A resistance with lower resistance will have a lower voltage drop across it.
Total Resistance: The formula to calculate the total resistance (RT) of resistance in series is the sum of individual resistances (R1, R2, R3, …):
$$R_{T}=R_1+R_2+R_3+\cdots{}$$
Derivation of resistors in series: As per KVL,
$$V_{AB}=V_{R1}+V_{R2}+V_{R3}$$
Because the same current is flowing through each resistor, we can substitute the V=IR relationship in the above equation,
$$IR_T=IR_1+IR_2+IR_3$$
$$\implies{}R_T=R_1+R_2+R_3$$
Equivalent Resistance: The equivalent resistance in a series combination is always greater than the highest individual resistance. Adding more resistances in series impedes the current more, increasing the overall resistance.
Series combinations of resistors are commonly used to increase the overall resistance in electronic circuits. For a constant current, the voltage drop increases with an increase in series resistance. For a constant voltage, the current reduces with increased series resistance.
Parallel combination of resistors
A parallel combination of resistors involves connecting the terminals of multiple resistors together at the same voltage level, essentially creating multiple pathways for electric current. In a parallel configuration, all resistors have the same voltage across them. Resistors with lesser resistance will allow more current to flow from them.
Total Resistance: The total resistance (Rtotal) of resistors in parallel is reciprocal of the sum of reciprocal of individual resistors (R1, R2, R3, …):
$$\cfrac{1}{R_{total}}=\cfrac{1}{R_1}+\cfrac{1}{R_2}+\cfrac{1}{R_3}+\cdots{}$$
Derivation of resistance in parallel: The equivalent resistance should take sum of all the currents through individual resistors.
$$I_T=I_1+I_2+I_3$$
All the resistances are seeing equal voltage across them. Using the relationship of V=IR in the above equation,
$$\cfrac{V_{AB}}{R_T}=\cfrac{V_{AB}}{R_1}+\cfrac{V_{AB}}{R_2}+\cfrac{V_{AB}}{R_3}$$
$$\implies{}\cfrac{1}{R_T}=\cfrac{1}{R_1}+\cfrac{1}{R_2}+\cfrac{1}{R_3}$$
Equivalent Resistance: The equivalent resistance in a parallel combination is always lesser than the smallest individual resistance. This is because connecting more parallel resistance effectively provides more paths for the current to flow, reducing resistance.
Voltage Rating: When resistors are connected in parallel, the voltage rating of the parallel combination is equal to or lesser than the lowest voltage rating among the individual resistors. This prevents voltage breakdown issues among the resistors.
Parallel combinations of resistors are commonly used to decrease the overall resistance in electronic circuits and increase the circuit’s power rating.