Introduction
Current divider equations and voltage divider formulas help you better understand resistor functions in electronic circuits. The current divider circuit is a parallel circuit in which the source current or power supply current divided into a multiple parallel paths. In a parallel circuit, the terminals of all components are connected together, sharing the same two end nodes. This results in the current to flow or pass through different paths and branches. However, the current through each component can have a different value. While a voltage divider circuit is a very common circuit that takes a higher voltage and converts it to a lower one by using a pair of resistors.
The main feature of a parallel circuit is that, although the branch circuit currents are different, the voltages of all connection paths are the same. Therefore, there is no need to find the voltage of each resistor, so that the branch current can be easily found by Kirchhoff's current law (KCL) and Ohm's law.
Figure 1. Voltage and Current Divider Current Circuits
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In this section, through the discussion of the commonly used resistor series voltage divider circuit and resistors in parallel divider circuit to find their rules. This article contains plenty of equations based on Kirchhoff's current law and Ohm's law for you to master voltage divider and current divider circuits.
A Discussion of the Useful Voltage Divider and Current Divider Circuits.
Ⅰ Resistive Voltage Divider Circuit
In electronics, a voltage divider (also known as a potential divider) is a passive linear circuit that produces an output voltage that is a fraction of its input voltage. It is a simple circuit which turns a large voltage into a smaller one. The basic voltage divider circuit with two resistors in series as shown in the Figure 2. is analyzed, and some useful formulas are obtained:
Figure 2. Basic Voltage Divider Circuit
The following equation is given by Kirchoffs Current Law (KCL):
The following equation is given by Kirchoffs Voltage Law (KVL):
Equation of voltage current relation of circuit elements:
Substituting the Ohm's law of the resistance element into the KVL equation, the calculation formula for the current i is obtained:
Then substitute it into the Ohm's law of the resistance element, the voltage division formula for calculating the resistance voltage is obtained:
Generally speaking, when N resistors are connected in series, the voltage on the Kth resistor can be calculated according to the following voltage division formula:
The resistor series voltage divider formula shows the relationship between the voltage of a certain resistor and the total voltage. The voltage division formula expresses that the voltage of a resistor is proportional to its resistance value, that is, when the resistance increases, the voltage also increases.
According to the voltage reference direction obtained by the above voltage divider circuit formula, it can be seen that it has nothing to do with the selection of the current reference direction. When the reference direction of the voltage variable uk or us involved in the formula changes, a negative sign will appear in the formula.
As shown in the Figure 3, find the voltage Uab when R=0Ω, 4Ω, 12Ω, ...∞.
Figure 3. Voltage Reference Direction
The voltage Uac and Ubc can be obtained by using the resistor series voltage divider formula:
Substituting the resistance R into the above formula, after obtaining the voltage Ubc, then using KVL to obtain the voltage Uab, the calculation result is as follows:
It can be seen from the calculation results that as the resistance R increases, the voltage Ubc gradually decreases, and the voltage Uab changes from negative to positive, indicating that its actual direction will varies with the change of the resistance R.
The Figure 4. below shows the dual-supply DC voltage divider circuit. Try to find the range of potential change at point a when the sliding end of the potentiometer moves.
Figure 4. Dual-supply DC Voltage Divider Circuit
Solution: Replace the two potentials of +12V and -12V with two voltage sources to obtain the circuit shown in Figure 4. (b).
When the sliding end of the potentiometer moves to the bottom end, the potential at point a is the same as that at point c:
When the sliding end of the potentiometer moves to the top, the potential at point a is the same as that at point b:
When the sliding end of the potentiometer gradually moves from bottom to top, the potential at point a will continuously change between -10V to 10V.
Here discusses the change law of load current i and voltage u when an actual supply powers to a variable resistor load. As shown in the Figure 5, RL is a variable resistance load, and R0 represents the internal resistance of the power supply:
Figure 5. Variable Resistor Load
Load current i:
Among them, k=RL/R0 represents the ratio of the load resistance to the internal resistance of the power supply, and isc=us/R0 represents the current when the load is short-circuited.
Load voltage u:
Among them, uoc=us represents the voltage when the load is open.
Power absorbed by load resistor:
When the coefficient k=RL/R0 takes different values, a series of relative values of current, voltage and power are calculated, as shown in the following table:
K=R_{L}/R_{0} |
0 |
0.2 |
0.4 |
0.6 |
0.8 |
1.0 |
2.0 |
3.0 |
4.0 |
5.0 |
∞ |
i/i_{sc} |
1 |
0.833 |
0.714 |
0.625 |
0.555 |
0.5 |
0.333 |
0.25 |
0.2 |
0.167 |
0 |
u/u_{oc} |
0 |
0.167 |
0.286 |
0.375 |
0.444 |
0.5 |
0.667 |
0.75 |
0.8 |
0.833 |
1 |
p/p_{imax} |
0 |
0.556 |
0.816 |
0.938 |
0.988 |
1 |
0.889 |
0.75 |
0.64 |
0.556 |
0 |
According to the above data, the curve of voltage, current and power changing with load resistance can be drawn, as shown in the Figure 6:
They show:
1. When the load resistance gradually increases from zero, the load current gradually changes from the maximum value isc=us/R0 to zero. When the load resistance is equal to the internal resistance of the power supply, the current is equal to half of the maximum value.
2. When the load resistance gradually increases from zero, the load voltage gradually increases from zero to the maximum value uoc=us. When the load resistance is equal to the internal resistance of the power supply, the voltage is equal to half of the maximum value.
3. When the load resistance is equal to the internal resistance of the power supply, the current is equal to half of the maximum value, the voltage is equal to half of the maximum value, and the power absorbed by the load resistance reaches the maximum value, and pmax=0.25uocisc.
The non-linear change law of the current when the load resistance changes can be seen from the resistance scale of an ordinary multimeter. The circuit model of a multimeter electric blocking is a series connection of a voltage source and a resistor. When we use a multimeter to measure unknown resistance, we should first short-circuit the multimeter and adjust the zero potentiometer pointer to 0Ω. At this time, the current is the largest and the meter pointer is fully deflected. When the short-circuit wire is removed, the pointer of the multimeter returns to ∞, and the measured current is zero at this time.
When the multimeter is connected to the measured resistor, as the resistance value changes, the current of the meter head will change accordingly, the pointer will be deflected to the corresponding position, and the measured resistance value can be directly read according to the scale on the surface. There is a special case, when the measured resistance value is just equal to the internal resistance of the multimeter, the current is half of the full deflection current, and the pointer stays in the middle position. Conversely, the internal resistance can be known from the reading in the middle of the multimeter’s electrical barrier scale. For example, the reading of a 500-type multimeter when the pointer stays in the middle position is 10, the internal resistance when using a ×1k electrical barrier is 10kΩ, and the internal resistance is 1kΩ when using a ×100 electrical barrier.
If necessary, use Voltage Divider Calculator to calculate the output voltage of a resistor divider circuit for a given set of resistor values and source voltage.
Ⅱ Resistive Current Divider Circuit
A current divider is defined as a linear circuit that produces an output current that is a fraction of its input current. The following formula describing a current divider is similar in form to that for the voltage divider. The Figure 7. shows a circuit in which a current source supplies power to two parallel resistors, and some useful formulas are drawn from its analysis.
Figure 7. Resistive Current Divider Circuit
The following equation is given by Kirchoffs Voltage Law (KVL):
The following equation is given by Kirchoffs Current Law (KCL):
Equation of current voltage relation of circuit elements:
Substituting the Ohm's law of the resistance element into the KCL equation, the calculation formula for the voltage u is obtained:
Then substitute Ohm's law into the resistive current divider equation for calculating the resistor current:
The resistive current divider formula of two parallel resistors expressed by resistance parameters is:
Generally speaking, when n resistors are connected in parallel, the current on the Kth resistor can be calculated according to the following formula:
The resistive current divider in parallel formula indicates the relationship between the current of a resistor and the total current. It shows that the resistance current is proportional to its conductance value. For example, the current will increases when the conductance increases.
According to the current reference direction obtained by the above formula, it can be seen that it has nothing to do with the selection of the voltage reference direction. When the reference direction of the current is or ik changes, a negative sign will appear in the formula.
Figure 8. Resistive Divider Circuit
According to the characteristics of two resistors in parallel, the current in the 3Ω and 6Ω resistors is obtained:
Then, the current in the 12Ω and 6Ω resistors is obtained:
Calculate the current i5 in the short-circuit line according to the KCL equation of node a:
i5 can also be calculated according to the KCL equation of node b:
It should be noted that the current i5=1A in the short circuit is different from the total current.
If necessary, the Current Divider Calculator can be used to determine the current going through any branch in a parallel circuit. Enter a current source and resistance values to calculate the current through each resistor. The calculator will display the current through each resistor entered.
Ⅲ Duality (Electrical Circuits)
According to the above-mentioned analysis of the resistive voltage divider circuit and current divider circuit, there is a certain similarity between them.
Figure 9. Duality Circuit Examples
The equations of the resistor divider circuit are listed as follows:
It can be seen that the equations of these two circuits have a dual relationship. If the current i in the KCL equation of a certain circuit is replaced with the voltage u, the KVL equation of another circuit is obtained; the voltage u in the KVL equation of a certain circuit is replaced with the current i, and the KCL equation of another circuit is obtained. This similar relationship in circuit structure is called topological duality. Similarly, replace u in the VCR equation of a certain circuit with i, i with u, R with G, G with R, etc., you can get the VCR equation of another circuit. This similar relationship of the element VCR equation is called element duality. If two circuits are both topological duality and component duality, they are called dual circuits.
The circuit equations of the dual circuit are dual, and the various formulas and results derived therefrom are also dual. For example, the dual formula derived for the dual circuit of Figure 9 (a) and (b) is as follows:
This section is a simple analysis of dual circuits, dual formulas, dual theorems and dual analysis methods in order to better grasp the basic concepts of circuit theory and various analysis methods. Here are a few test questions that can be used to test how well you learn about voltage divider and current divider circuits:
1) Find the voltages u_{1} and u_{2} in the circuit shown in the following figure:
2) Find the current i_{1} and i_{2} in the circuit shown in the following figure:
3) Find the current i_{2}, is and voltage u in the circuit shown in the following figure:
Frequently Asked Questions about Resistor Voltage Divider and Current Divider Rules and Formulas
1. What is VDR and CDR?
The Voltage Divider Rule formula (VDR) shows how the voltage distributes among different resistors in a series circuit. Similarly, the Current Divider Rule formula (CDR) shows how current distributes in a parallel circuit.
2. What is the current divider rule with examples?
When two resistors are connected in a parallel circuit, the current in any branches will be a fraction of the total current (IT)). If both the resistors are of equal value, then the current will divide equally through both the branches.
3. Why does a voltage divider need two resistors?
One resistor can be used to drop voltage (if the load draws current) but to divide voltage you need something to create a division ratio. To be a voltage divider the output voltage needs to be a constant proportion of the input voltage. ... Note that this need for two resistors only applies to DC.
4. Where are current divider used?
By using a current divider, the current flowing through a component can be minimized and thus smaller component size can be used. For example, in a case where larger resistor wattage is required; adding multiple resistors in parallel decreases the heat dissipation, and smaller wattage resistors can do the same job.
5. What is voltage divider formula?
A voltage divider is applying a voltage across a series of two resistors. We may draw in a few different ways, but they should always essentially be the same circuit. Thus formula is given as follows: V_{out} = \frac{R_b}{R_a+R_b} \times V_{in}
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