Voltage divider rule
Encyclopedia
In electronics
, a voltage divider (also known as a potential divider) is a simple linear circuit
that produces an output voltage
(Vout) that is a fraction of its input voltage (Vin). Voltage division refers to the partitioning of a voltage among the components of the divider.
The formula governing a voltage divider is similar to that for a current divider, but the ratio describing voltage division places the selected impedance in the numerator, unlike current division where it is the unselected components that enter the numerator.
A simple example of a voltage divider consists of two resistor
s in series or a potentiometer
. It is commonly used to create a reference voltage, or to get a low voltage signal proportional to the voltage to be measured and may also be used as a signal attenuator
at low frequencies.
is created by connecting two electrical impedance
s in series, as shown in Figure 1. The input voltage is applied across the series impedances Z1 and Z2 and the output is the voltage across Z2.
Z1 and Z2 may be composed of any combination of elements such as resistor
s, inductor
s and capacitor
s.
Applying Ohm's Law
, the relationship between the input voltage, Vin, and the output voltage, Vout, can be found:
Proof:
The transfer function
(also known as the divider's voltage ratio) of this circuit is simply:
In general this transfer function is a complex, rational function
of frequency
.
Substituting Z1 = R1 and Z2 = R2 into the previous expression gives:
As in the general case, R1 and R2 may be any combination of series/parallel resistors.
As a more specific and/or practical example, if Vout=6V and Vin=9V (both commonly used voltages), then:
and by solving using algebra
, R2 must be twice the value of R1.
To solve for R1:
To solve for R2:
Any ratio between 0 and 1 is possible. That is, using resistors alone it is not possible to either invert the voltage or increase Vout above Vin.
Consider a divider consisting of a resistor and capacitor
as shown in Figure 3.
Comparing with the general case, we see Z1 = R and Z2 is the impedance of the capacitor, given by
where XC is the reactance of the capacitor, C is the capacitance
of the capacitor, j is the imaginary unit
, and ω (omega) is the radian frequency of the input voltage.
This divider will then have the voltage ratio:
.
The product of τ (tau) = RC is called the time constant
of the circuit.
The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic (first-order) lowpass filter. The ratio contains an imaginary number, and actually contains both the amplitude and phase shift
information of the filter. To extract just the amplitude ratio, calculate the magnitude
of the ratio, that is:
Inductive dividers split AC input according to inductance:
The above equation is for ideal conditions. In the real world the amount of mutual inductance will alter the results.
For an AC input a simple capacitive equation is:
Capacitive dividers are limited in current by the capacitance of the elements used.
This effect is opposite to resistive division and inductive division.
The following example describes the effect when a voltage divider is used to drive an amplifier:
The gain
of an amplifier
generally depends on its source and load terminations, so-called loading effects that reduce the gain. The analysis of the amplifier itself is conveniently treated separately using idealized drivers and loads, and then supplemented by the use of voltage and current division to include the loading effects of real sources and loads. The choice of idealized driver and idealized load depends upon whether current or voltage is the input/output variable for the amplifier at hand, as described next. For more detail on types of amplifier based upon input/output variables, see classification based on input and output variables.
In terms of sources, amplifiers with voltage input (voltage and transconductance
amplifiers) typically are characterized using ideal zero-impedance voltage sources. In terms of terminations, amplifiers with voltage output (voltage and transresistance amplifiers) typically are characterized in terms of an open circuit output condition.
Similarly, amplifiers with current input (current and transresistance amplifiers) are characterized using ideal infinite impedance current sources, while amplifiers with current output (current and transconductance
amplifiers) are characterized by a short-circuit output condition,
As stated above, when any of these amplifiers is driven by a non-ideal source, and/or terminated by a finite, non-zero load, the effective gain is lowered due to the loading effect at the input and/or the output. Figure 3 illustrates loading by voltage division at both input and output for a simple voltage amplifier. (A current amplifier example is found in the article on current division.) For any of the four types of amplifier (current, voltage, transconductance or transresistance), these loading effects can be understood as a result of voltage division and/or current division, as described next.
with Thévenin series impedance RS. For a Thévenin driver, the input voltage vi is reduced from vS by voltage division to a value
where Rin is the amplifier input resistance, and the overall gain is reduced below the idealized gain by the same voltage division factor.
In the same manner, the ideal input current for an ideal driver ii is realized only for an infinite-resistance current driver. For a Norton driver with current iS and source impedance RS, the input current ii is reduced from iS by current division to a value
where Rin is the amplifier input resistance, and the overall gain is reduced below the gain estimated using an ideal driver by the same current division factor.
More generally, complex frequency-dependent impedances can be used instead of the driver and amplifier resistances.
More generally, complex frequency-dependent impedances can be used instead of the load and amplifier resistances.
The resistor ratios in the above expression are called the loading factors.
, the input resistance of the amplifier depends on its load, and the output resistance on the source impedance. The loading factors in these cases must employ the true amplifier impedances including these bilateral effects.
For example, taking the unilateral voltage amplifier of Figure 3, the corresponding bilateral two-port network is shown in Figure 4 based upon g-parameters.The g-parameter two port is the only one of the standard four choices that has a voltage-controlled voltage source on the output side. Carrying out the analysis for this circuit, the voltage gain with feedback Afb is found to be
That is, the ideal current gain Ai is reduced not only by the loading factors, but due to the bilateral nature of the two-port by an additional factorOften called the improvement factor or the desensitivity factor. (1 + β (RS / RL ) Aloaded ), which is typical of negative feedback amplifier circuits. The factor β(RS / RL) is the voltage feedback provided by the current feedback source of current gain β (A/A). For instance, for an ideal voltage source with RS = 0 Ω, the current feedback has no influence, and for RL = ∞ Ω, there is zero load current, again disabling the feedback.
. Another way is to "neutralize" the load impedance by an equivalent negative impedance (INIC).
In terms of the above equations, if current flows into a load resistance RL (attached at the output node where the voltage is Vout), that load resistance RL must be considered in parallel with R2 to determine the voltage at Vout. In this case, the voltage at Vout is calculated as follows:
where RL is a load resistor in parallel with R2. From this result it is clear that Vout is decreased by RL unless R2 // RL ≈ R2 , that is, unless RL >> R2.
In other words, for high impedance
loads it is possible to use a voltage divider as a voltage source, as long as R2 has very small value compared to the load. This technique leads to considerable power
dissipation in the divider.
A voltage divider is commonly used to set the DC Biasing of a common emitter
amplifier, where the current drawn from the divider is the relatively low base current of the transistor.
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, a voltage divider (also known as a potential divider) is a simple linear circuit
Linear circuit
A linear circuit is an electronic circuit in which, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f...
that produces an output voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
(Vout) that is a fraction of its input voltage (Vin). Voltage division refers to the partitioning of a voltage among the components of the divider.
The formula governing a voltage divider is similar to that for a current divider, but the ratio describing voltage division places the selected impedance in the numerator, unlike current division where it is the unselected components that enter the numerator.
A simple example of a voltage divider consists of two resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s in series or a potentiometer
Potentiometer
A potentiometer , informally, a pot, is a three-terminal resistor with a sliding contact that forms an adjustable voltage divider. If only two terminals are used , it acts as a variable resistor or rheostat. Potentiometers are commonly used to control electrical devices such as volume controls on...
. It is commonly used to create a reference voltage, or to get a low voltage signal proportional to the voltage to be measured and may also be used as a signal attenuator
Attenuator (electronics)
An attenuator is an electronic device that reduces the amplitude or power of a signal without appreciably distorting its waveform.An attenuator is effectively the opposite of an amplifier, though the two work by different methods...
at low frequencies.
General case
A voltage divider referenced to groundGround (electricity)
In electrical engineering, ground or earth may be the reference point in an electrical circuit from which other voltages are measured, or a common return path for electric current, or a direct physical connection to the Earth....
is created by connecting two electrical impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
s in series, as shown in Figure 1. The input voltage is applied across the series impedances Z1 and Z2 and the output is the voltage across Z2.
Z1 and Z2 may be composed of any combination of elements such as resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s and capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s.
Applying Ohm's Law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
, the relationship between the input voltage, Vin, and the output voltage, Vout, can be found:
Proof:
The transfer function
Transfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
(also known as the divider's voltage ratio) of this circuit is simply:
In general this transfer function is a complex, rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
of frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
.
Resistive divider
A resistive divider is a special case where both impedances, Z1 and Z2, are purely resistive (Figure 2).Substituting Z1 = R1 and Z2 = R2 into the previous expression gives:
As in the general case, R1 and R2 may be any combination of series/parallel resistors.
Resistive divider
As a simple example, if R1 = R2 thenAs a more specific and/or practical example, if Vout=6V and Vin=9V (both commonly used voltages), then:
and by solving using algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, R2 must be twice the value of R1.
To solve for R1:
To solve for R2:
Any ratio between 0 and 1 is possible. That is, using resistors alone it is not possible to either invert the voltage or increase Vout above Vin.
Low-pass RC filter
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
as shown in Figure 3.
Comparing with the general case, we see Z1 = R and Z2 is the impedance of the capacitor, given by
where XC is the reactance of the capacitor, C is the capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
of the capacitor, j is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, and ω (omega) is the radian frequency of the input voltage.
This divider will then have the voltage ratio:
.The product of τ (tau) = RC is called the time constant
Time constant
In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...
of the circuit.
The ratio then depends on frequency, in this case decreasing as frequency increases. This circuit is, in fact, a basic (first-order) lowpass filter. The ratio contains an imaginary number, and actually contains both the amplitude and phase shift
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
information of the filter. To extract just the amplitude ratio, calculate the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of the ratio, that is:
Inductive divider
Inductive dividers split DC input according to resistive divider rules above.Inductive dividers split AC input according to inductance:
The above equation is for ideal conditions. In the real world the amount of mutual inductance will alter the results.
Capacitive divider
Capacitive dividers do not pass DC input.For an AC input a simple capacitive equation is:
Capacitive dividers are limited in current by the capacitance of the elements used.
This effect is opposite to resistive division and inductive division.
Loading effect
The voltage output of a voltage divider is not fixed but varies according to the load. To obtain a reasonably stable output voltage the output current should be a small fraction of the input current. The drawback of this is that most of the input current is wasted as heat in the resistors.The following example describes the effect when a voltage divider is used to drive an amplifier:
The gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...
of an amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
generally depends on its source and load terminations, so-called loading effects that reduce the gain. The analysis of the amplifier itself is conveniently treated separately using idealized drivers and loads, and then supplemented by the use of voltage and current division to include the loading effects of real sources and loads. The choice of idealized driver and idealized load depends upon whether current or voltage is the input/output variable for the amplifier at hand, as described next. For more detail on types of amplifier based upon input/output variables, see classification based on input and output variables.
In terms of sources, amplifiers with voltage input (voltage and transconductance
Transconductance
Transconductance, also known as mutual conductance, is a property of certain electronic components. Conductance is the reciprocal of resistance; transconductance, meanwhile, is the ratio of the current change at the output port to the voltage change at the input port. It is written as gm...
amplifiers) typically are characterized using ideal zero-impedance voltage sources. In terms of terminations, amplifiers with voltage output (voltage and transresistance amplifiers) typically are characterized in terms of an open circuit output condition.
Similarly, amplifiers with current input (current and transresistance amplifiers) are characterized using ideal infinite impedance current sources, while amplifiers with current output (current and transconductance
Transconductance
Transconductance, also known as mutual conductance, is a property of certain electronic components. Conductance is the reciprocal of resistance; transconductance, meanwhile, is the ratio of the current change at the output port to the voltage change at the input port. It is written as gm...
amplifiers) are characterized by a short-circuit output condition,
As stated above, when any of these amplifiers is driven by a non-ideal source, and/or terminated by a finite, non-zero load, the effective gain is lowered due to the loading effect at the input and/or the output. Figure 3 illustrates loading by voltage division at both input and output for a simple voltage amplifier. (A current amplifier example is found in the article on current division.) For any of the four types of amplifier (current, voltage, transconductance or transresistance), these loading effects can be understood as a result of voltage division and/or current division, as described next.
Input loading
A general voltage source can be represented by a Thévenin equivalent circuitThévenin's theorem
In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...
with Thévenin series impedance RS. For a Thévenin driver, the input voltage vi is reduced from vS by voltage division to a value
where Rin is the amplifier input resistance, and the overall gain is reduced below the idealized gain by the same voltage division factor.
In the same manner, the ideal input current for an ideal driver ii is realized only for an infinite-resistance current driver. For a Norton driver with current iS and source impedance RS, the input current ii is reduced from iS by current division to a value
where Rin is the amplifier input resistance, and the overall gain is reduced below the gain estimated using an ideal driver by the same current division factor.
More generally, complex frequency-dependent impedances can be used instead of the driver and amplifier resistances.
Output loading
For a finite load, RL an output voltage is reduced by voltage division by the factor RL / ( RL + Rout ), where Rout is the amplifier output resistance. Likewise, as the term short-circuit implies, the output current delivered to a load RL is reduced by current division by the factor Rout / ( RL + Rout ). The overall gain is reduced below the gain estimated using an ideal load by the same current division factor.More generally, complex frequency-dependent impedances can be used instead of the load and amplifier resistances.
Loaded gain - voltage amplifier case
Including both the input and output voltage division factors for the voltage amplifier of Figure 4, the ideal voltage gain Av realized with an ideal driver and an open-circuit load is reduced to the loaded gain Aloaded:
The resistor ratios in the above expression are called the loading factors.
Unilateral versus bilateral amplifiers
Figure 3 and the associated discussion refers to a unilateral amplifier. In a more general case where the amplifier is represented by a two portTwo-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
, the input resistance of the amplifier depends on its load, and the output resistance on the source impedance. The loading factors in these cases must employ the true amplifier impedances including these bilateral effects.
For example, taking the unilateral voltage amplifier of Figure 3, the corresponding bilateral two-port network is shown in Figure 4 based upon g-parameters.The g-parameter two port is the only one of the standard four choices that has a voltage-controlled voltage source on the output side. Carrying out the analysis for this circuit, the voltage gain with feedback Afb is found to be
That is, the ideal current gain Ai is reduced not only by the loading factors, but due to the bilateral nature of the two-port by an additional factorOften called the improvement factor or the desensitivity factor. (1 + β (RS / RL ) Aloaded ), which is typical of negative feedback amplifier circuits. The factor β(RS / RL) is the voltage feedback provided by the current feedback source of current gain β (A/A). For instance, for an ideal voltage source with RS = 0 Ω, the current feedback has no influence, and for RL = ∞ Ω, there is zero load current, again disabling the feedback.
Reference voltage
Voltage dividers are often used to produce stable reference voltages. The term reference voltage implies that little or no current is drawn from the divider output node by an attached load. Thus, use of the divider as a reference requires a load device with a high input impedance to avoid loading the divider, that is, to avoid disturbing its output voltage. A simple way of avoiding loading (for low power applications) is to simply input the reference voltage into the non-inverting input of an op-amp bufferBuffer amplifier
A buffer amplifier is one that provides electrical impedance transformation from one circuit to another...
. Another way is to "neutralize" the load impedance by an equivalent negative impedance (INIC).
Voltage source
While voltage dividers may be used to produce precise reference voltages (that is, when no current is drawn from the reference node), they make poor voltage sources (that is, when current is drawn from the reference node). The reason for poor source behavior is that the current drawn by the load passes through resistor R1, but not through R2, causing the voltage drop across R1 to change with the load current, and thereby changing the output voltage.In terms of the above equations, if current flows into a load resistance RL (attached at the output node where the voltage is Vout), that load resistance RL must be considered in parallel with R2 to determine the voltage at Vout. In this case, the voltage at Vout is calculated as follows:
where RL is a load resistor in parallel with R2. From this result it is clear that Vout is decreased by RL unless R2 // RL ≈ R2 , that is, unless RL >> R2.
In other words, for high impedance
High impedance
In electronics, high impedance means that a point in a circuit has a relatively high impedance to other points in the circuit.-Digital electronics:...
loads it is possible to use a voltage divider as a voltage source, as long as R2 has very small value compared to the load. This technique leads to considerable power
Power electronics
Power electronics is the application of solid-state electronics for the control and conversion of electric power.-Introduction:Power electronic converters can be found wherever there is a need to modify a form of electrical energy...
dissipation in the divider.
A voltage divider is commonly used to set the DC Biasing of a common emitter
Common emitter
In electronics, a common-emitter amplifier is one of three basic single-stage bipolar-junction-transistor amplifier topologies, typically used as a voltage amplifier...
amplifier, where the current drawn from the divider is the relatively low base current of the transistor.
External links
- Resistor voltage divider
- Voltage divider or potentiometer calculations
- Voltage divider tutorial video in HD
- Online calculator to choose the values by series E24, E96
- Online voltage divider calculator: chooses the best pair from a given series and also gives the color code
- Voltage divider theory - RC low-pass filterLow-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
example and voltage divider using Thévenin's theoremThévenin's theoremIn circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...