Voltage divider

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Figure 1: Voltage divider

In electronics, a voltage divider (also known as a potential divider) is a simple linear circuit that produces an output voltage (V_out) that is a fraction of its input voltage (V_in). 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 resistors in series or a potentiometer. It is commonly used to create a reference voltage, and may also be used as a signal attenuator at low frequencies.

[edit] General case

A voltage divider referenced to ground is created by connecting two electrical impedances 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 resistors, inductors and capacitors.

Applying Ohm's Law, the relationship between the input voltage, V_in, and the output voltage, V_out, can be found:

$V_\mathrm{out} = \frac{Z_2}{Z_1+Z_2} \cdot V_\mathrm{in}$

Proof:

$V_\mathrm{in} = I\cdot(Z_1+Z_2)$

$V_\mathrm{out} = I\cdot Z_2$

$I = \frac {V_\mathrm{in}}{Z_1+Z_2}$

$V_\mathrm{out} = V_\mathrm{in} \cdot\frac {Z_2}{Z_1+Z_2}$

The transfer function (also known as the divider's voltage ratio) of this circuit is simply:

$H = \frac {V_{out}}{V_{in}} = \frac{Z_2}{Z_1+Z_2}$

In general this transfer function is a complex, rational function of frequency.

[edit] Resistive divider

Figure 2: Simple resistive voltage 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:

$V_\mathrm{out} = \frac{R_2}{R_1+R_2} \cdot V_\mathrm{in}$

As in the general case, R₁ and R₂ may be any combination of series/parallel resistors.

[edit] Examples

[edit] Resistive divider

As a simple example, if R₁ = R₂ then

$V_\mathrm{out} = \frac{1}{2} \cdot V_\mathrm{in}$

As a more specific and/or practical example, if V_out=6V and V_in=9V (both commonly used voltages), then:

$\frac{V_\mathrm{out}}{V_\mathrm{in}} = \frac{R_2}{R_1+R_2} = \frac{6}{9} = \frac{2}{3}$

and by solving using algebra, R₂ must be twice the value of R₁.

To solve for R1:

$R_1 = \frac{R_2 \cdot V_\mathrm{in}}{V_\mathrm{out}} - R_2$

To solve for R2:

$R_2 = \frac{R_1}{\frac{V_\mathrm{in}}{V_\mathrm{out}}-1}$

Any ratio between 0 and 1 is possible. That is, using resistors alone it is not possible to either reverse the voltage or increase V_out above V_in

[edit] Low-pass RC filter

Figure 3: Resistor/capacitor voltage divider

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

$Z_2 = jX_{\mathrm{C}} =\frac {1} {j \omega C}$ $\ ,$

where X_C 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:

${V_\mathrm{out} \over V_\mathrm{in}} = {Z_\mathrm{2} \over Z_\mathrm{1} + Z_\mathrm{2}} = {{1 \over j \omega C} \over {1 \over j \omega C} + R} = {1 \over 1 + j \omega \ R C}$ .

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:

$\left| \frac {V_\mathrm{out}} {V_\mathrm{in}} \right| = \frac {1} {\sqrt { 1 + ( \omega R C )^2 } } \ .$

[edit] Inductive divider

Inductive dividers split DC input according to resistive divider rules above.

Inductive dividers split AC input according to inductance:

$V_{out} = V_{in} \cdot \frac {L2} {L_1 + L_2}$

The above equation is for ideal conditions. In the real world the amount of mutual inductance will alter the results.

[edit] Capacitive divider

Capacitive dividers do not pass DC input.

For an AC input a simple capacitive equation is:

$V_{out} = V_{in} \cdot \frac {C1} {C1 + C2}$

Capacitive dividers are limited in current by the capacitance of the elements used.

This effect is opposite to resistive division and inductive division.

[edit] 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:

Figure 3: A simple voltage amplifier (gray outline) demonstrating input loading (blue outline) and output loading (green outline)

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.^[1] 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.

[edit] Input loading

A general voltage source can be represented by a Thévenin equivalent circuit with Thévenin series impedance R_S. For a Thévenin driver, the input voltage v_i is reduced from v_S by voltage division to a value

$v_i = v_S \frac {R_{in}} {R_S + R_{in}} ,$

where R_in 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 i_i is realized only for an infinite-resistance current driver. For a Norton driver with current i_S and source impedance R_S, the input current i_i is reduced from i_S by current division to a value

$i_i = i_S \frac {R_{S}} {R_S + R_{in}} ,$

where R_in 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.

[edit] Output loading

For a finite load, R_L an output voltage is reduced by voltage division by the factor R_L / ( R_L + R_out ), where R_out is the amplifier output resistance. Likewise, as the term short-circuit implies, the output current delivered to a load R_L is reduced by current division by the factor R_out / ( R_L + R_out ). 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.

[edit] 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 A_v realized with an ideal driver and an open-circuit load is reduced to the loaded gain A_loaded:

$A_{loaded} =\frac {v_L} {v_S} = \frac {R_{in}} {R_S+R_{in}}$ $\frac {R_{L}} {R_{out}+R_{L}} A_v \ .$

The resistor ratios in the above expression are called the loading factors.

Figure 4: G-parameter voltage amplifier two-port; feedback provided by dependent current-controlled current source of gain β A/A

[edit] 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 port, 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.^{[nb 1]} Carrying out the analysis for this circuit, the voltage gain with feedback A_fb is found to be

$A_{fb} = \frac {v_L}{v_S} = \frac {A_{loaded}} {1+ {\beta}(R_S/R_L) A_{loaded}} \ .$

That is, the ideal current gain A_i is reduced not only by the loading factors, but due to the bilateral nature of the two-port by an additional factor^{[nb 2]} (1 + β (R_S / R_L ) A_loaded ), which is typical of negative feedback amplifier circuits. The factor β(R_S / R_L) is the voltage feedback provided by the current feedback source of current gain β (A/A). For instance, for an ideal voltage source with R_S = 0 Ω, the current feedback has no influence, and for R_L = ∞ Ω, there is zero load current, again disabling the feedback.

[edit] Applications

[edit] 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 buffer.

[edit] 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 R_L (attached at the output node where the voltage is V_out), that load resistance R_L must be considered in parallel with R₂ to determine the voltage at V_out. In this case, the voltage at V_out is calculated as follows:

$V_{\text{out}} = \frac {R_2 \| R_{\text{L}}} { R_1+R_2 \| R_{\text{L}} } V_{\text{in}} = \frac {R_2}{R_1 \left( 1 + \frac {R_2}{R_{\text{L}} } \right) + R_2 } V_{\text{in}}$

$\text{where} \quad R_2 \| R_{\text{L}} \triangleq \left( \frac{1}{R_2} + \frac{1}{R_{\text{L}}} \right)^{-1} = \frac{R_2 R_{\text{L}}}{ R_2 + R_{\text{L}} }\, ,$

where R_L is a load resistor in parallel with R₂. From this result it is clear that V_out is decreased by R_L unless R₂ // R_L ≈ R₂ , that is, unless R_L >> 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 bias of a common emitter amplifier, where the current drawn from the divider is the relatively low base current of the transistor.

[edit] Footnotes

^ 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.
^ Often called the improvement factor or the desensitivity factor.^{[citation needed]}

[edit] References

^ A.S. Sedra and K.C. Smith (2004). Microelectronic Circuits (5^th ed.). New York: Oxford University Press. §1.5 pp. 23–31. ISBN 0-19-514251-9.

[edit] Further reading

Paul Horowitz and Winfield Hill, The Art of Electronics, Cambridge University Press, 1989.

[edit] External links

ECE 209: Review of Circuits as LTI Systems — Shows voltage divider theory, gives an RC low-pass filter example, and shows how to turn a complicated circuit into a voltage divider using Thévenin's theorem.
Resistor Voltage Divider Calculator
Online calculator to choose the values by series E24, E96.

[1] 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.

[2] Often called the improvement factor or the desensitivity factor.^{[citation needed]}

[SedraSmith04-0] A.S. Sedra and K.C. Smith (2004). Microelectronic Circuits (5^th ed.). New York: Oxford University Press. §1.5 pp. 23–31. ISBN 0-19-514251-9.

[1]

[nb 1]

[nb 2]

Voltage divider

From Wikipedia, the free encyclopedia

Contents

[edit] General case

[edit] Resistive divider

[edit] Examples

[edit] Resistive divider

[edit] Low-pass RC filter

[edit] Inductive divider

[edit] Capacitive divider

[edit] Loading effect

[edit] Input loading

[edit] Output loading

[edit] Loaded gain - voltage amplifier case

[edit] Unilateral versus bilateral amplifiers

[edit] Applications

[edit] Reference voltage

[edit] Voltage source

[edit] Footnotes

[edit] References

[edit] Further reading

[edit] External links

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