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Passive Electronic Circuits

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    Remember that signal-to-noise ratios can be enhanced if the signal frequency is different than the noise frequency. You will be introduced to these frequency-dependent analog filters at the end of this section. For now, let’s start very simply...

    Resistor Fundamentals

    The simplest circuit involving a resistor and a voltage source is shown below. The dotted lines are just there to represent where a high-quality voltmeter would be connected if we wished to measure the voltage across the resistor. Calculating the current flowing through this resistor requires the use of Ohm’s Law.

    circuit#1.png

    Circuit #1

    According to Ohm's Law \(\mathrm{I = \dfrac{V}{R} = \dfrac{1.0\: Volt}{20\: Ohms} = 0.050\: Amperes}\)

    • V = 1.0 Volt
    • R = 20 Ohms

    Resistors in Series

    Practically speaking, we are not limited to a single resistor. Circuit #1 could also be represented by Circuit #2 below:

    circuit#2.png

    • Resistors placed in a "head-to-tail" configuration are in series.
    • The total resistance is the sum of all the individual resistances
    • Putting resistors together in series gives a larger total resistance

    \[\mathrm{R_T = (R_1 + R_2) = (10\,Ω + 10\,Ω) = 20\,Ω}\]

    Resistors in Parallel

    Resistors placed in a “side-to-side” configuration are in parallel.

    circuit#3.png

    The total resistance is the reciprocal of the sum of each reciprocal resistance. So for a pair of resistors as shown in Circuit #3 above:

    \[\mathrm{R_T = \dfrac{1}{\dfrac{1}{R_1} + \dfrac{1}{R_2}} = \dfrac{R_1R_2}{R_1 + R_2}}\]

    Applying this to Circuit #3:

    \[\mathrm{R_T = \dfrac{1}{\dfrac{1}{40} + \dfrac{1}{40}} = \dfrac{40 * 40} {40 + 40} = 20\: ohms}\]

    Putting resistors together in parallel always gives a smaller total resistance. Note that Circuit #3 has the same current as Circuits #1 and #2.

    Voltage Divider

    Sometimes, the output of an instrument is too large for a readout device. One circuit used to reduce a voltage is a voltage divider

    VoltageDivider1.png

    Note that:

    • A representation for a voltmeter has been added to the schematic
    • The voltage is only being accessed across one of the two resistors

    Assuming that the meter resistance is much larger than R2 (i.e. no loading error occurs), then according to Ohm’s Law

    VoltageDivider2.png

    \[\mathrm{V_{in} = I(R_1 + R_2)}\]

    For a discussion of loading errors, click here.

    If the readout device (i.e. a meter) is placed across R2, than the voltage read by the meter is

    VoltageDivider3.png

    Or in other words, the divider output equals the instrument output multiplied by R2 over the total resistance

    \[\mathrm{V_{out} = V_{in} \left(\dfrac{R_2}{R_1 + R_2}\right)}\]

    In this case, the divider output is:

    VoltageDivider1.png

    \[\mathrm{V_{out} = 1.0\: V \left(\dfrac{10\, Ω}{10\, Ω + 10\, Ω}\right) = 0.5\; V}\]

    RC Voltage Dividers (Analog Filters)

    Although voltage dividers are extremely useful, they are unable to selectively filter signal voltages from noise voltages. That is:

    Voltage dividers are frequency independent.

    However, the impedance of a capacitor is frequency dependent, as shown by the following equation:

    \[\mathrm{X_C = \dfrac{1}{2πfC}}\]

    • XC is the impedance of the capacitor (impedance is the generalized form of resistance that applies to AC signals)
    • f is the frequency of the voltage source in Hertz
    • C is the capacitance in Farads

    As the frequency increases, the impedance of a capacitor decreases!

    When a capacitor is charging or discharging, the voltage across the capacitor lags behind (i.e. is out-of-phase) by 90°. If this phase difference did not exist, one could simply insert the value of the impedance into the voltage divider equation and use the same mathematics for the DC voltage divider circuit to calculate the output of the RC voltage divider circuit at a given frequency.

    However, if each impedance is treated as a phasor, the total impedance of a RC series (ZRC) is calculated using the following relationship

    \[\mathrm{Z_{RC} = \sqrt{R^2 + \left(\dfrac{1}{2πfC}\right)^2}}\]

    An excellent overview of the mathematics of AC circuits can be accessed by clicking on the following URL:

    http://www.animations.physics.unsw.edu.au/jw/AC.html

    (Accessed June 6, 2014).

    Low-Pass Filters

    • Used when the signal frequency < noise frequency
    • The relationship between Vin and Vout is analogous to a frequency independent voltage divider
    • The desired filter output is obtained across the frequency dependent component (capacitor)

    Low_Pass_Filter_2a.png

    \[\mathrm{V_{out} = V_{in}\left(\dfrac{X_C}{Z_{RC}}\right) = V_{in} \left(\dfrac{\dfrac{1}{2πfC}}{\sqrt{R^2 + \left(\dfrac{1}{2πfC}\right)^2}}\right) = V_{in} \left(\dfrac{1}{\sqrt{1+(2πfC)^2}}\right)}\]

    High-Pass Filters

    • Used when the signal frequency > noise frequency
    • The relationship between Vin and Vout is analogous to a frequency independent voltage divider
    • The desired filter output is obtained across the frequency independent component (resistor)

    High_Pass_Filter_3.png

    \[\mathrm{V_{out} = V_{in}\left(\dfrac{X_C}{Z_{RC}}\right)
    = V_{in} \left(\dfrac{R}{\sqrt{R^2 + \left(\dfrac{1}{2πfC}\right)^2}}\right)
    = V_{in} \left(\dfrac{2πfRC}{\sqrt{1 + (2πfRC)^2}}\right)}\]


    This page titled Passive Electronic Circuits is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Contributor.

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