Voltage Divider Calculator
- What is a voltage divider?
- Voltage divider rule
- Voltage divider formula
- Capacitive and inductive voltage divider equations
- RC and CR filters
- RL and LR filters
- CL and LC as resonant circuits
- Usage of a voltage divider in more complex circuits
- Application of voltage divider
- Pros and cons of voltage dividers
- Current divider
This is a voltage divider calculator - a comprehensive but simple tool, which helps you evaluate output signal (i.e. voltage) that is obtained in a single voltage divider. Read on to learn what is a voltage divider, find out the basic voltage divider formula and how it extends to various equations for different voltage divider types, and learn how it is possible to obtain some fraction of input voltage by applying the voltage divider rule. Additionally, check out the numerous applications of a voltage divider in both basic and incredibly complex systems, and convince yourself that electronic circuits are nothing to worry about!
What is a voltage divider?
A simple voltage divider is a part of a linear circuit that changes input voltage (
V₁) into an output voltage (
V₂) which is a different value. Since the circuit is a passive one, the ratio
V₂ / V₁ is never greater than 1. The general voltage divider is presented in this simple diagram:
Z₂ are some impedances. The impedances can be related to the resistance
R, the capacitance
C, or the inductance
L. We can distinguish a few basic types of voltage dividers for which the voltage divider rule can be applied:
- Resistive divider, RR
- Capacitive divider, CC
- Inductive divider, LL
- RC and CR dividers (also known as RC, CR filters)
- RL and LR dividers (also known as RL, LR filters)
- CL and LC dividers (also known as CL, LC filters)
Voltage divider rule
The principle behind voltage dividers is that the current passing through several elements connected in series is constant, but the voltage is divided somehow between them. To find the exact values, we have to apply Ohm's law to our circuit. Before doing that, one important point must to be stated:
RR composition is the only one applicable to DC circuits. In these cases, any impedance can be treated as a wire with zero resistivity, and capacitances work as a gap in a circuit, so they have infinite resistance. For the rest, they are all used with AC circuits, and the voltage divider rule is applicable for the maximum value of the potential difference. It may also be helpful to find the phase shift for these voltages.
Voltage divider formula
The general voltage divider equation (or formula) for impedances is as follows:
V₂ = Z₂ / (Z₁ + Z₂) * V₁.
A reminder: in general,
V₂ correspond to the amplitudes of signals, e.g. sinusoidal ones.
If we are considering only resistances, the voltage divider formula naturally changes to:
V₂ = R₂ / (R₁ + R₂) * V₁.
Since the resistance doesn't affect the phase of the signal, the formula is the same for both AC and DC cases. Voltage values at a given moment are compared. As mentioned before, the remaining kinds of dividers are considered for to AC circuits, so let's take a look at some examples.
Capacitive and inductive voltage divider equations
For a CC divider, we need to use the impedances of capacitors:
Z = 1 / (j * ω * C), where
j stands for an imaginary number, and
ω is the angular frequency of alternating voltage. Substituting the original voltage divider equation with this expression, we obtain:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = (1 / (jωC₂)) / (1 / (jωC₁) + 1 / (jωC₂)) * V₁,
and multiplying each term by
jωC₁C₂ the result is:
V₂ = C₁ / (C₁ + C₂) * V₁.
A similar procedure can be done for LL dividers, where
Z = j * ω * L. This time the outcome voltage is:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = jωL₂ / (jωL₁ + jωL₂) * V₁.
Dividing numerator and denominator by
jω the final formula is:
V₂ = L₂ / (L₁ + L₂) * V₁.
In both cases, the output voltage is in phase with respect to the input phase.
RC and CR filters
Voltage divider circuits consisting of more than one type of element are not as simple to evaluate as the previous examples. We have to deal with complex number algebra, but trust us, it looks more frightening than it actually is.
For an RC divider, we can extend the voltage divider formula to:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = (1 / (jωC)) / (R + 1 / (jωC)) * V₁ = V₁ / (jωRC + 1).
The result is a complex number, so, in order to estimate the amplitude of the output voltage, we have to find its module:
|V₂| = |V₁ / (jωRC + 1)| = |V₁| / √((ωRC)² + 1).
If the frequency increases, the output amplitude of the voltage decreases, which is the reason why this circuit is also called a low pass filter. The phase shift can be worked out as the inverse tangent of the imaginary part divided by the real part of our complex number:
Δφ = atan(-ωRC).
Analogically, we can find the amplitude and the phase shift for CR circuit. The first step is to evaluate the general formula for output voltage:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = R / (R + 1 / (jωC)) * V₁ = jωRC / (jωRC + 1) * V₁.
Its amplitude can be worked out as:
|V₂| = |jωRC / (jωRC + 1) * V₁| = ωRC / √((ωRC)² + 1) * |V₁|,
and the phase shift is given as:
Δφ = atan(1 / ωRC).
This time we can see that if the frequency tends towards 0, the same happens to the amplitude of
V₂, and, for high values of
ω, it remains the same as the input voltage. The conclusion is that CR can be treated as a high pass filter.
RL and LR filters
Circuits containing resistant and inductive elements aren't much different from RC and CR ones when it comes to calculations, but it's worth repeating each step to realize all the subtle differences.
In case of RL filter we can start, as usual, with the general formula of voltage divider:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = jωL / (R + jωL) * V₁.
To find the amplitude of the output voltage, we need to estimate the module of that value:
|V₂| = |jωL / (R + jωL) * V₁| = ωL / √(R² + (ωL)²) * |V₁|,
and its phase shift is:
Δφ = atan(R / ωL).
For an LR divider, we just replace these elements, so the voltage divider equation results in:
V₂ = Z₂ / (Z₁ + Z₂) * V₁ = R / (R + jωL) * V₁.
Once again, we can determine the amplitude of the output voltage and the phase shift:
|V₂| = |R / (R + jωL) * V₁| = R / √(R² + (ωL)²) * |V₁|,
Δφ = atan(-ωL / R).
Take a look at the outcomes; the
V₂ amplitude for RL filter is very similar to CR, and that of LR resembles the voltage amplitude of RC. So do their phase shifts. It's a very valuable outcome because it turns out that RC and RL filters can be used interchangeably when designed in an appropriate configuration, and the values of conductance and inductance are adequately adjusted. It's especially useful for circuits which have to be resized to the nanometer scale since the application of really small conductors is much easier than having to creating tiny coils.
Anyway, if you have any trouble with calculating the properties of the output signal, you can always come back to these chapters, or you can just try our voltage divider calculator! Choose a suitable option, and the result will be displayed instantly.
Several passive filters have been described above, but voltage divider rule can also be applied with active ones.
CL and LC as resonant circuits
We could perform the same calculations for CL and LC systems, however, some ridiculous outcomes may occur if we simply apply the voltage divider rule. We can cause the amplitude of the output voltage can go into infinity! This is caused by the fact that the connected LC elements are sometimes called resonant circuits. They are used for generating and receiving radio waves, which is most efficient at a resonant frequency, given as:
ω = 1/√(L * C).
In a more realistic picture, we have to consider also some non-zero resistance, so the problem of the resonat circuits is described in the RLC circuit calculator.
Usage of a voltage divider in more complex circuits
What we have already done shows how voltage dividers work for the simplest systems possible. You can obviously imagine that in real life they are used practically nowhere, and, generally, more sophisticated circuits are applied. However, all of the results obtained above can be helpful while simplifying more complex ones. For instance, whenever you can spot resistors in either series or juxtaposed parallel you can treat them as a single resistance. Similarly, capacitors and solenoids work in much the same way. For mixed components, it is evaluated in practically the same way, but we have to take into account impedances
Z instead of
Application of voltage divider
One of the most commonly used devices which works thanks to the concept of the voltage divider is a potentiometer. Another word describing this element is a rheostat. They are usually made out of only resistive components. We can distinguish between both analog and digital ones, but, in any case, the resistance can be set with high precision. Some of the most popular types of potentiometer are slide pots, trimpots, or thumb pots, and these vary by size and structure. The key element is a sliding contact, making the adjustment of the output resistance possible.
High voltage measurement devices - It turns out that it is possible to measure high voltages even if the direct measurement can be destructive for the device. In that case, it's convenient to use the voltage divider in order to lower the voltage to a safe region. For exceptionally high voltages (let's say above 100 kV) it is better to use capacitive instead of resistive ones.
Finding unknown resistance - Is it possible to find some unknown resistance if you only have a voltage source and another resistor with known resistance? You're lucky if you can read its color code, but what if there isn't one? Well, you can just create a simple circuit with both resistors arranged in series, set an input voltage, and measure the voltage over the desired resistor. Afterward, just insert all these values into the voltage divider calculator, and that's it - the mystery is revealed. You can always transform the general voltage divider equation to find
R₂as an unknown parameter:
R₂ = V₂ / (V₁ - V₂) * R₁.
Pros and cons of voltage dividers
Some of you may be wondering why people measure an unknown resistance with a voltage divider when they can simply read the value of amperage that flows through the resistor when an external voltage is applied - just a simple Ohm's law. Well, in general, there should be no significant difference for these methods, but we have to be aware that the resistance of the vast majority of materials is temperature-dependent. Even worse, these dependencies are different for metals, semiconductors, or insulators.
Taking into account metals, their resistance increases with the temperature rise, so, in order to define the resistance at some standardized temperature, e.g.
T = 25°C, we have to find the thermal coefficient (TCR) of the material. This requires precisely measuring the ambient temperature, and carrying out some calculations, all while hoping that there wasn't any mistake made in the meantime. However, we can do in a much simpler way! As you may have guesses, you could use a simple voltage divider!
In the basic version, we have two resistors, and if they are made of the same material, it means their temperature dependencies of resistance is roughly the same. No matter how big is a temperature difference, these resistances change at approximately the same percent, let's say 5% for every 20°C. But, since in general the voltage divider formula has a ratio of impedances, any relative change will cancel and the output voltage should be temperature independent (or at least its impact ought to be significantly reduced). Moreover, if we look at the equation from the previous section, we'll obtain a value of resistance that is the same as the first one at a given temperature - no further calculations are needed!
Secondly, it's convenient to use voltage dividers when designing some complex electrical circuits. Instead of using multiple distinct voltage sources, each of which produces a different potential in the system, we can implement a single source and apply as many voltage dividers as we need.
On the other hand, we have to be aware of the fact that the longer the wires in our circuit are, the more likely a voltage drop is. Well, it's nowhere near the odds of that for long industrial cables, but still, if we need to make some really precise measurements, that factor should be taken into account, and ideally reduced as much as possible.
So far, we've been focused on the processing of a signal - basically, on voltage changes. Still, we can use a similar concept which is looking at the problem from another perspective - called a current divider.
The idea is almost the same, but instead of dividing input voltage into smaller fractions, we want to divide the initial amperage and get some specific value of it as the output. There are only a few differences: firstly, we need a current source instead of a voltage one. Secondly, all of the impedances (in a simple case two, as usual) have to be arranged in parallel, not in series. Actually, these are all crucial differences. With this circuit, we can use Ohm's law once again. The resulting formula is:
Iₓ = Z / (Z + Zₓ) * Iᵢ.
We can spot an interesting and valuable property. For a voltage divider, the higher the output resistance, the bigger the output voltage, while for current divider the outcome behaves the other way round.
Similarly, we can produce different types of current dividers, including coils and capacitors, and all of these are applicable for alternating current, whereas for direct current only the composition of resistors works. In general cases, it is possible to evaluate both the amplitude and the phase shift of a flowing current. We are sure that after reading the step-by-step solutions from this voltage divider calculator it won't be any problem for you to perform similar calculations.