Resistive 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: :V_\mathrm{out} = \frac{R_2}{R_1+R_2} \cdot V_\mathrm{in} If
R1 =
R2 then :V_\mathrm{out} = \frac{1}{2} \cdot V_\mathrm{in} If
Vout = 6 V and
Vin = 9 V (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 = R_2 \cdot \left({\frac{V_\mathrm{in}}{V_\mathrm{out}}-1}\right) To solve for
R2: :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 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 = -\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
ω (omega) is the
radian frequency of the input voltage. This divider will then have the voltage ratio: :\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 R C} \ . The product
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)
low-pass 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}} \ .
Inductive divider Inductive dividers split AC input according to
inductance: V_\mathrm{out} = \frac{L_2}{L_1 + L_2} \cdot V_\mathrm{in} (with components in the same positions as Figure 2.) The above equation is for non-interacting inductors;
mutual inductance (as in an
autotransformer) will alter the results. Inductive dividers split AC input according to the reactance of the elements as for the resistive divider above.
Capacitive divider Capacitive dividers do not pass DC input. For an AC input a simple capacitive equation is: V_\mathrm{out} = \frac{Xc_2}{Xc_1 + Xc_2} \cdot V_\mathrm{in} = \frac{1/C_2}{1/C_1 + 1/C_2} \cdot V_\mathrm{in} = \frac{C_1}{C_1 + C_2} \cdot V_\mathrm{in} (with components in the same positions as Figure 2.) 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==