The resistor combination is the way of grouping several resistors to study their collective effect within a circuit.
When we look at a complex schematic, our initial goal is usually to simplify it. We want to reduce that entire tangle of components into a single Equivalent Resistance (
If we achieve this, calculating the total current is as easy as applying Ohm’s Law just once.
To do this, there are two basic ways of connecting components: in series or in parallel.
Series Combination
Two or more resistors are in series when they are connected one after another, forming a single branch without intermediate junctions.
- Topological Condition: They share an exclusive node that does not connect to anything else.
- Physical Consequence: The same Current (
) flows through all of them.
Calculating the Equivalent Resistance
The total resistance is simply the sum of the individual resistances. By placing obstacles one after another, the total difficulty increases.
Check Rule: In series, the equivalent resistance will always be greater than the largest resistance in the group.
The Voltage Divider
This is possibly the most widely used subcircuit in electronics. Since the current is the same for all, but the resistances are different, the total voltage is distributed proportionally among them.
If we have two resistors (R₁ and R₂) in series connected to an input voltage (
- Typical Use: Reading resistive sensors (like an LDR or a thermistor) with a microcontroller. We convert the change in resistance into a change in voltage that the circuit can read.
The formula assumes the output is unloaded. If you connect an impedance to
Parallel Combination
Two or more resistors are in parallel when their terminals are connected to the same two nodes.
- Topological Condition: They “embrace” the same voltage.
- Physical Consequence: All of them have the same Voltage (
) across their terminals.
Calculating the Equivalent Resistance
Here we are opening multiple paths for the current. The more paths we open, the easier it is for current to flow. Therefore, the total resistance decreases.
The general formula is the sum of the reciprocals (conductances):
The simplified case (only 2 resistors): If we only have two resistors (the most common case), we can use this quick formula:
Check Rule: In parallel, the equivalent resistance will always be less than the smallest resistance in the group.
If you put a 1MΩ resistor in parallel with a 1Ω resistor, the result will be less than 1Ω.
The current prefers to take the easy path (1Ω) and almost completely ignores the difficult path.
The Current Divider
Upon reaching a node, the total current is divided. The smaller the resistance of a branch, the more current it will draw.
The current flowing through a branch
Or, for the case of two resistors (R₁ and R₂), the current flowing through
Notice that
Mixed Combinations
In real life, circuits are a mix of both. To solve them, we use the “onion” strategy: we solve from the inside out.
Resolution Algorithm:
Look for resistors that are clearly in Series or Parallel (with nothing strange in between).
Calculate their equivalent and redraw the circuit, replacing them with a single resistor.
Repeat step 1 in the new simplified circuit.
Continue until only one resistor connected to the source remains.
This procedure does not directly work for bridge networks or other topologies that do not contain pure series/parallel groups. In those cases, we will use Kirchhoff’s laws.
Example: Light bulbs in series or parallel
To solidify concepts, think about house lights compared to old Christmas lights.
- Christmas Lights in Series: If one bulb burns out (the circuit opens), all of them go out. The current cannot flow.
- House Lights in Parallel: If the living room bulb burns out, the kitchen bulb still works. They are independent branches.
