Oct 13, 2015 Voltage Divider Formula. Voltage divider rule is that rule if a series circuit has more than one resistor; the voltage across of each resistor is the ratio of resistor value multiplied with voltage source to total resistance value. Let us consider above circuit there is three resistances. We have to find out each resistance voltage. A voltage divider is a simple circuit which turns a large voltage into a smaller one. Using just two series resistors and an input voltage, we can create an output voltage that is a fraction of the input. Voltage dividers are one of the most fundamental circuits in electronics.
Figure 1: Schematic of an electrical circuit illustrating current division. Notation RT. refers to the total resistance of the circuit to the right of resistor RX.
In electronics, a current divider is a simple linear circuit that produces an output current (IX) that is a fraction of its input current (IT). Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.
The formula describing a current divider is similar in form to that for the voltage divider. However, the ratio describing current division places the impedance of the considered branches in the denominator, unlike voltage division where the considered impedance is in the numerator. This is because in current dividers, total energy expended is minimized, resulting in currents that go through paths of least impedance, therefore the inverse relationship with impedance. On the other hand, voltage divider is used to satisfy Kirchhoff's Voltage Law. The voltage around a loop must sum up to zero, so the voltage drops must be divided evenly in a direct relationship with the impedance.
To be specific, if two or more impedances are in parallel, the current that enters the combination will be split between them in inverse proportion to their impedances (according to Ohm's law). It also follows that if the impedances have the same value the current is split equally.
- 3Using Admittance
- 4Loading effect
Current divider[edit]
A general formula for the current IX in a resistor RX that is in parallel with a combination of other resistors of total resistance RT is (see Figure 1):
- IX=RTRX+RTIT{displaystyle I_{X}={frac {R_{T}}{R_{X}+R_{T}}}I_{T} }[1]
where IT is the total current entering the combined network of RX in parallel with RT. Notice that when RT is composed of a parallel combination of resistors, say R1, R2, ... etc., then the reciprocal of each resistor must be added to find the total resistance RT:
- 1RT=1R1+1R2+1R3+....{displaystyle {frac {1}{R_{T}}}={frac {1}{R_{1}}}+{frac {1}{R_{2}}}+{frac {1}{R_{3}}}+... .}
General case[2][edit]
Although the resistive divider is most common, the current divider may be made of frequency dependent impedances. In the general case the current IX is given by:
- IX=ZTZXIT,{displaystyle I_{X}={frac {Z_{T}}{Z_{X}}}I_{T} ,}[3]
where ZT refers to the equivalent impedance of the entire circuit.
Using Admittance[edit]
Instead of using impedances, the current divider rule can be applied just like the voltage divider rule if admittance (the inverse of impedance) is used.
- IX=YXYTotalIT{displaystyle I_{X}={frac {Y_{X}}{Y_{Total}}}I_{T}}
Take care to note that YTotal is a straightforward addition, not the sum of the inverses inverted (as you would do for a standard parallel resistive network). For Figure 1, the current IX would be
- IX=YXYTotalIT=1RX1RX+1R1+1R2+1R3IT{displaystyle I_{X}={frac {Y_{X}}{Y_{Total}}}I_{T}={frac {frac {1}{R_{X}}}{{frac {1}{R_{X}}}+{frac {1}{R_{1}}}+{frac {1}{R_{2}}}+{frac {1}{R_{3}}}}}I_{T}}
Example: RC combination[edit]
Figure 2: A low pass RC current divider
Figure 2 shows a simple current divider made up of a capacitor and a resistor. Using the formula below, the current in the resistor is given by:
-
-
IR=1jÏCR+1jÏCIT{displaystyle I_{R}={frac {frac {1}{jomega C}}{R+{frac {1}{jomega C}}}}I_{T}}
- =11+jÏCRIT,{displaystyle ={frac {1}{1+jomega CR}}I_{T} ,}
-
IR=1jÏCR+1jÏCIT{displaystyle I_{R}={frac {frac {1}{jomega C}}{R+{frac {1}{jomega C}}}}I_{T}}
where ZC = 1/(jÏC) is the impedance of the capacitor and j is the imaginary unit.
The product Ï = CR is known as the time constant of the circuit, and the frequency for which ÏCR = 1 is called the corner frequency of the circuit. Because the capacitor has zero impedance at high frequencies and infinite impedance at low frequencies, the current in the resistor remains at its DC value IT for frequencies up to the corner frequency, whereupon it drops toward zero for higher frequencies as the capacitor effectively short-circuits the resistor. In other words, the current divider is a low pass filter for current in the resistor.
Loading effect[edit]
Figure 3: A current amplifier (gray box) driven by a Norton source (iS, RS) and with a resistor load RL. Current divider in blue box at input (RS,Rin) reduces the current gain, as does the current divider in green box at the output (Rout,RL)
The gain of an amplifier generally depends on its source and load terminations. Current amplifiers and transconductance amplifiers are characterized by a short-circuit output condition, and current amplifiers and transresistance amplifiers are characterized using ideal infinite impedance current sources. When an amplifier is terminated by a finite, non-zero termination, and/or driven by a non-ideal source, the effective gain is reduced due to the loading effect at the output and/or the input, which can be understood in terms of current division.
Figure 3 shows a current amplifier example. The amplifier (gray box) has input resistance Rin and output resistance Rout and an ideal current gain Ai. With an ideal current driver (infinite Norton resistance) all the source current iS becomes input current to the amplifier. However, for a Norton driver a current divider is formed at the input that reduces the input current to
-
- ii=RSRS+RiniS,{displaystyle i_{i}={frac {R_{S}}{R_{S}+R_{in}}}i_{S} ,}
which clearly is less than iS. Likewise, for a short circuit at the output, the amplifier delivers an output current io = Ai ii to the short-circuit. However, when the load is a non-zero resistor RL, the current delivered to the load is reduced by current division to the value:
-
- iL=RoutRout+RLAiii.{displaystyle i_{L}={frac {R_{out}}{R_{out}+R_{L}}}A_{i}i_{i} .}
Combining these results, the ideal current gain Ai realized with an ideal driver and a short-circuit load is reduced to the loaded gainAloaded:
-
- Aloaded=iLiS=RSRS+Rin{displaystyle A_{loaded}={frac {i_{L}}{i_{S}}}={frac {R_{S}}{R_{S}+R_{in}}}}RoutRout+RLAi.{displaystyle {frac {R_{out}}{R_{out}+R_{L}}}A_{i} .}
The resistor ratios in the above expression are called the loading factors. For more discussion of loading in other amplifier types, see loading effect.
Unilateral versus bilateral amplifiers[edit]
Figure 4: Current amplifier as a bilateral two-port network; feedback through dependent voltage source of gain β V/V
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 current amplifier of Figure 3, the corresponding bilateral two-port network is shown in Figure 4 based upon h-parameters.[4] Carrying out the analysis for this circuit, the current gain with feedback Afb is found to be
-
- Afb=iLiS=Aloaded1+β(RL/RS)Aloaded.{displaystyle A_{fb}={frac {i_{L}}{i_{S}}}={frac {A_{loaded}}{1+{beta }(R_{L}/R_{S})A_{loaded}}} .}
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 factor[5] ( 1 + β (RL / RS ) Aloaded ), which is typical of negative feedback amplifier circuits. The factor β (RL / RS ) is the current feedback provided by the voltage feedback source of voltage gain β V/V. For instance, for an ideal current source with RS = â Ω, the voltage feedback has no influence, and for RL = 0 Ω, there is zero load voltage, again disabling the feedback.
References and notes[edit]
- ^Nilsson, James; Riedel, Susan (2015). Electric Circuits. Edinburgh Gate, England: Pearson Education Limited. p. 85. ISBN978-1-292-06054-5.
- ^'Current Divider Circuits | Divider Circuits And Kirchhoff's Laws | Electronics Textbook'. Retrieved 2018-01-10.
- ^Alexander, Charles; Sadiku, Matthew (2007). Fundamentals of Electric Circuits. New York, NY: McGraw-Hill. p. 392. ISBN978-0-07-128441-7.
- ^The h-parameter two port is the only two-port among the four standard choices that has a current-controlled current source on the output side.
- ^Often called the improvement factor or the desensitivity factor.
See also[edit]
External links[edit]
- Divider Circuits and Kirchhoff's Laws chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.
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Figure 1: A simple voltage divider
In electronics, a voltage divider (also known as a potential divider) is a passivelinear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.
Resistor voltage dividers are commonly used to create reference voltages, or to reduce the magnitude of a voltage so it can be measured, and may also be used as signal attenuators at low frequencies. For direct current and relatively low frequencies, a voltage divider may be sufficiently accurate if made only of resistors; where frequency response over a wide range is required (such as in an oscilloscope probe), a voltage divider may have capacitive elements added to compensate load capacitance. In electric power transmission, a capacitive voltage divider is used for measurement of high voltage.
- 2Examples
- 4Applications
General case[edit]
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.
If the current in the output wire is zero then the relationship between the input voltage, Vin, and the output voltage, Vout, is:
- Vout=Z2Z1+Z2â Vin{displaystyle V_{mathrm {out} }={frac {Z_{2}}{Z_{1}+Z_{2}}}cdot V_{mathrm {in} }}
Proof (using Ohm's Law):
- Vin=Iâ (Z1+Z2){displaystyle V_{mathrm {in} }=Icdot (Z_{1}+Z_{2})}
- Vout=Iâ Z2{displaystyle V_{mathrm {out} }=Icdot Z_{2}}
- I=VinZ1+Z2{displaystyle I={frac {V_{mathrm {in} }}{Z_{1}+Z_{2}}}}
- Vout=Vinâ Z2Z1+Z2{displaystyle 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:
- H=VoutVin=Z2Z1+Z2{displaystyle H={frac {V_{mathrm {out} }}{V_{mathrm {in} }}}={frac {Z_{2}}{Z_{1}+Z_{2}}}}
In general this transfer function is a complex, rational function of frequency.
Examples[edit]
Resistive divider[edit]
Figure 2: Simple resistive voltage divider
A resistive divider is the case where both impedances, Z1 and Z2, are purely resistive (Figure 2).
Substituting Z1 = R1 and Z2 = R2 into the previous expression gives:
- Vout=R2R1+R2â Vin{displaystyle V_{mathrm {out} }={frac {R_{2}}{R_{1}+R_{2}}}cdot V_{mathrm {in} }}
If R1 = R2 then
- Vout=12â Vin{displaystyle V_{mathrm {out} }={frac {1}{2}}cdot V_{mathrm {in} }}
If Vout=6V and Vin=9V (both commonly used voltages), then:
- VoutVin=R2R1+R2=69=23{displaystyle {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:
- R1=R2â VinVoutâR2=R2â (VinVoutâ1){displaystyle R_{1}={frac {R_{2}cdot V_{mathrm {in} }}{V_{mathrm {out} }}}-R_{2}=R_{2}cdot left({{frac {V_{mathrm {in} }}{V_{mathrm {out} }}}-1}right)}
To solve for R2:
- R2=R1â 1(VinVoutâ1){displaystyle R_{2}=R_{1}cdot {frac {1}{left({{frac {V_{mathrm {in} }}{V_{mathrm {out} }}}-1}right)}}}
Any ratio Vout/Vin greater than 1 is not possible. That is, using resistors alone it is not possible to either invert the voltage or increase Vout above Vin.
Low-pass RC filter[edit]
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
- Z2=âjXC=1jÏC,{displaystyle Z_{2}=-mathrm {j} X_{mathrm {C} }={frac {1}{mathrm {j} omega C}} ,}
where XC is the reactance of the capacitor, C is the capacitance of the capacitor, j is the imaginary unit, and Ï (omega) is the radian frequency of the input voltage.
This divider will then have the voltage ratio:
- VoutVin=Z2Z1+Z2=1jÏC1jÏC+R=11+jÏRC.{displaystyle {frac {V_{mathrm {out} }}{V_{mathrm {in} }}}={frac {Z_{mathrm {2} }}{Z_{mathrm {1} }+Z_{mathrm {2} }}}={frac {frac {1}{mathrm {j} omega C}}{{frac {1}{mathrm {j} omega C}}+R}}={frac {1}{1+mathrm {j} omega RC}} .}
The product Ï (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:
- |VoutVin|=11+(ÏRC)2.{displaystyle left|{frac {V_{mathrm {out} }}{V_{mathrm {in} }}}right|={frac {1}{sqrt {1+(omega RC)^{2}}}} .}
Inductive divider[edit]
Inductive dividers split AC input according to inductance:
Vout=L2L1+L2â
Vin{displaystyle V_{mathrm {out} }={frac {L_{2}}{L_{1}+L_{2}}}cdot V_{mathrm {in} }}
The above equation is for non-interacting inductors; mutual inductance (as in an autotransformer) will alter the results.
Inductive dividers split DC input according to the resistance of the elements as for the resistive divider above.
Capacitive divider[edit]
Capacitive dividers do not pass DC input.
For an AC input a simple capacitive equation is:
Vout=C1C1+C2â
Vin{displaystyle V_{mathrm {out} }={frac {C_{1}}{C_{1}+C_{2}}}cdot V_{mathrm {in} }}
Any leakage current in the capactive elements requires use of the generalized expression with two impedances. By selection of parallel R and C elements in the proper proportions, the same division ratio can be maintained over a useful range of frequencies. This is the principle applied in compensated oscilloscope probes to increase measurement bandwidth.
Loading effect[edit]
The output voltage of a voltage divider will vary according to the electric current it is supplying to its external electrical load. The effective source impedance coming from a divider of Z1 and Z2, as above, will be Z1 in parallel with Z2 (sometimes written Z1 // Z2), that is: (Z1Z2) / (Z1 + Z2)=HZ1.
To obtain a sufficiently stable output voltage, the output current must either be stable (and so be made part of the calculation of the potential divider values) or limited to an appropriately small percentage of the divider's input current. Load sensitivity can be decreased by reducing the impedance of both halves of the divider, though this increases the divider's quiescent input current and results in higher power consumption (and wasted heat) in the divider. Voltage regulators are often used in lieu of passive voltage dividers when it is necessary to accommodate high or fluctuating load currents.
Applications[edit]
Voltage dividers are used for adjusting the level of a signal, for bias of active devices in amplifiers, and for measurement of voltages. A Wheatstone bridge and a multimeter both include voltage dividers. A potentiometer is used as a variable voltage divider in the volume control of many radios.
Sensor measurement[edit]
Voltage dividers can be used to allow a microcontroller to measure the resistance of a sensor.[1] The sensor is wired in series with a known resistance to form a voltage divider and a known voltage is applied across the divider. The microcontroller's analog-to-digital converter is connected to the center tap of the divider so that it can measure the tap voltage and, by using the measured voltage and the known resistance and voltage, compute the sensor resistance.An example that is commonly used involves a potentiometer (variable resistor) as one of the resistive elements. When the shaft of the potentiometer is rotated the resistance it produces either increases or decreases, the change in resistance corresponds to the angular change of the shaft. If coupled with a stable voltage reference, the output voltage can be fed into an analog-to-digital converter and a display can show the angle. Such circuits are commonly used in reading control knobs. Note that it is highly beneficial for the potentiometer to have a linear taper, as the microcontroller or other circuit reading the signal must otherwise correct for the non-linearity in its calculations.
High voltage measurement[edit]
High voltage resistor divider probe.
A voltage divider can be used to scale down a very high voltage so that it can be measured by a volt meter. The high voltage is applied across the divider, and the divider outputâwhich outputs a lower voltage that is within the meter's input rangeâis measured by the meter. High voltage resistor divider probes designed specifically for this purpose can be used to measure voltages up to 100 kV. Special high-voltage resistors are used in such probes as they must be able to tolerate high input voltages and, to produce accurate results, must have matched temperature coefficients and very low voltage coefficients. Capacitive divider probes are typically used for voltages above 100 kV, as the heat caused by power losses in resistor divider probes at such high voltages could be excessive.
Logic level shifting[edit]
A voltage divider can be used as a crude logic level shifter to interface two circuits that use different operating voltages. For example, some logic circuits operate at 5V whereas others operate at 3.3V. Directly interfacing a 5V logic output to a 3.3V input may cause permanent damage to the 3.3V circuit. In this case, a voltage divider with an output ratio of 3.3/5 might be used to reduce the 5V signal to 3.3V, to allow the circuits to interoperate without damaging the 3.3V circuit. For this to be feasible, the 5V source impedance and 3.3V input impedance must be negligible, or they must be constant and the divider resistor values must account for their impedances. If the input impedance is capacitive, a purely resistive divider will limit the data rate. This can be roughly overcome by adding a capacitor in series with the top resistor, to make both legs of the divider capacitive as well as resistive.
References[edit]
- ^'A very quick and dirty introduction to Sensors, Microcontrollers, and Electronics'(PDF). Retrieved 2 November 2015.
See also[edit]
External links[edit]
- Divider Circuits and Kirchhoff's Laws chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.
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