The RJ Journal - Electronics

Resistors can be connected together, either in series or in parallel. The equivalent resistor value for resistors connected in series is the sum of all resistors.

When resistors are connected in series, the current through all resistors is the same and the voltage over each resistor is the total voltage over all resistors divided between the resistors proportionally with respect to their resistance value. The sum of the voltages over each resistor adds up to the total voltage over all resistors. We can see this if we first calculate the current through the resistors which, according to the U = R * I formula, equals U over all resistors divided by the sum of all resistors. (I = U / R).

If R1 = 100Ω, R2 = 200Ω, R3 = 300Ω and the total voltage over all three resistors is 30V then the current through the resistors is 30V / 600Ω = 50mA. The voltage over each resistor is this current multiplied by the resistance (U = R * I) which equals 5V for R1, 10V for R2 and 15V for R3. We can see that the ratio between 5V, 10V and 15V is the same as the ratio between 100Ω, 200Ω and 300Ω. In fact, we don't have to calculate the current since we can see that over one resistor there will be a voltage which equals the total voltage multiplied by the resistor value divided by the sum of all resistor values. With numbers this equals to 30V * 100Ω / (100Ω + 200Ω + 300Ω) = 5V for R1.

This is commonly used with 2 resistors and is then called a voltage divider. If there is an output voltage from a sensor that swings between 0-10V and our input circuit only can handle 0-2V (an AD converter, for example), we can put a voltage divider between the sensor output and the circuit input. The circuit input should only see 1/5th of the sensor output voltage (2V / 10V) which means that the resistor we will measure over has to be 1/5th of the total resistance in the voltage divider circuit. If we select the input resistor (Ri) to be 10kΩ we can calculate that the measurement resistor (Rm) must be 2.5kΩ since 2.5 / (10 + 2.5) = 0.2.

Note that for this to be true, the input resistance of our measurement circuit (connected to Uout) has to be much higher than Rm. Ri + Rm also has to be much higher than any internal resistance in the sensor in order to not affect the output voltage of the sensor.

Parallel connection

When resistors are connected in parallel, the equivalent resistor value for the circuit is the inverted result of the sum of all inverted resistor values.

In this case all resistors have the same voltage over them and the current through each resistor is the total current through all resistors divided between the resistors proportionally with respect to their inverted resistor value. This makes sense since the higher the resistor value the lower the current. Also, the sum of the currents through all resistors equals the total current through the circuit. So if R1 = 100Ω, R2 = 200Ω, R3 = 300Ω and there is 30V over the resistors, the current through R1 = 30V / 100Ω = 300mA, the current through R2 = 30V / 200Ω = 150mA and the current through R3 = 30V / 300Ω = 100mA. As we can see, the ratio between 300mA, 150mA and 100mA is the same as the ratio between 1 / 100Ω = 0.01, 1 / 200Ω = 0.005 and 1 / 300Ω = 0.00333.

Now that we know the total current through the circuit, which is 300mA + 150mA + 100mA = 550mA, we can calculate the equivalent resistor value for the circuit to 30V / 550mA = 54.5Ω which is the same as we get when using the formula for the equivalent resistor value above.

If there only are 2 resistors connected in parallel the formula for the equivalent resistor value for this circuit can be simplified to R = (R1 * R2) / (R1 + R2). And if all resistors have the same value, the equivalent resistance equals one resistor value divided by the number of resistors in the circuit.

As we can see from the examples above, there are often two ways to calculate the equivalent resistor value, the voltage over one resistor or the current through the resistor in the different circuits above and all calculations always boil down to the original U = R * I formula, taking into account possible constant current or constant voltage which can simplify the calculations.

Other important rules or electric laws, that we have verified with the above calculations, is that the sum of all currents flowing into any junction in an electric circuit is always equal to the sum of all currents flowing out of this junction (resistors connected in parallel) and that the sum of all voltage sources around any closed circuit is equal to the sum of all individual voltage drops over the resistors in this circuit (resistors connected in series). These rules are called Kirchhoff's Current and Voltage laws.

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