Series Resistance Calculations

Introduction

In this article we will deduce how to calculate the resulting resistive value by placing resistances in series, i.e. connecting two or more chained resistances. We will also use LT Spice to verify these results.

See also calculations with parallel resistances

two resistors in series in lt spice
Fig. 1: Example of an arrangement with two generic value resistors in series.

If we think about the physical size of the resistors we realize what the result is going to be. Let’s suppose that we have identical resistances of 100 Ohms that measure 1 cm, if we place two of these resistances one after the other we would have two centimeters of resistive material, then intuitively we can say that two resistances of 100 Ohm in series will result in a resistive value of 200 Ohm.

This reasoning leads us to think that if we have resistors connected in series, their Ohmic value will be added.

Deduction

To begin the deduction let’s assume a simple circuit consisting of a V voltage source connected to the array of series resistors, as illustrated in Figure 2.

circuit with source and arrangement of n resistances in series through which a current i circulates.
Fig. 2: Circuit with voltage source and N resistors in series. The same current i circulates through all the resistors.

On this circuit we are going to apply Kirchoff’s circuit Law for voltage, which tells us that a closed circuit the sum of all potential drops gives as a result 0.

In figure 3, in line 1 I have written what this law tells us considering that we have a finite number of resistances.

On line 2 I place all the resistors on the right side of the equation. The next step is to take the current i as a common factor, since this same current i circulates through all the resistances.

Finally we conclude that what is in parentheses on line 3 is the value of a resistance for an equivalent circuit that instead of having n resistors has only one with this value. In figure 4 we see this equivalent circuit.

deduction of the equivalent resistance calculation of a series resistance arrangement
Fig. 3: Deduction of the equivalent resistance considering the circuit of the previous figure.

circuit with voltage source and equivalent resistance of a series array
Fig. 4: Same circuit as above but considering that the voltage source is applied to a single resistor with equivalent value.

LT Spice Simulation

Let’s put together a simple circuit in LT Spice to show that this result is met. If you don’t have LT Spice installed I invite you to read this article in which I show you how to download it, install it and create a simple circuit.

In figure 5 we have simulated a circuit with a voltage source of 1 Volt and two resistances of 100 Ohm connected in series, while in figure 6 we have a circuit with the same voltage source but only a resistance with the equivalent resistance value that would be 200 Ohms.

As can be seen in the posters with the result of the simulation, the current circulating through the voltage source is the same in both cases, so we can conclude that these circuits are equivalent.

circuit with voltage source and resistors, result of the calculations
Fig. 5: Simulation of a circuit with a 1 Volt source and two 100 Ohm resistors in series.

circuit with equivalent voltage source and resistance, result of the calculations
Fig. 6: Simulation of a circuit with a source of 1 Volt and a resistance of 200 Ohms.

Conclusion

We have intuitively raised what the equivalent resistance value of a series resistor array might be.

When we have arrangements of resistances in series the current that circulates is the same in all, applying Kirchoff’s circuit laws we can deduce an expression for the equivalent resistance.

Finally we have made a simulation to show that we can think of an equivalent circuit grouping the resistances.

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