Cornell University - Visit www.cornell.edu Kirby Research Group at Cornell: Microfluidics and Nanofluidics : - Home College of Engineering - visit www.engr.cornell.edu Cornell University - Visit www.cornell.edu
Cornell University, College of Engineering Search Cornell
News Contact Info Login

Donations keep this resource free! Give here:

Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

[Return to Table of Contents]


Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

5.4 Electrical circuits [electrodynamics top]

In electrical circuits, we consider idealized elements and simple relations that relate voltage and current. Below, we list circuit components, discuss complex representations of real properties, discuss notational concepts such as impedance and phasors, and summarize circuit relations. This discussion is a description of electrostatics and electrodynamics for discretized elements with well-defined electrical properties.

5.4.1 Components and properties

Key idealized circuit components include voltage sources, wires, resistors, capacitors, and inductors. Voltage sources are objects that specify a voltage V [V] at a given point. Ideal wires link together regions of space that have the same voltage V or electrical potential. Ideal resistors(Figure 5.12) are objects with a finite resistance to the motion of charge, defined by theresistance R [Ω] and a voltage-current relation given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.61)


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.12: Symbol and voltage-current relation for a resistor.


Idealcapacitors (Figure 5.13) are physical systems with a finite capacity to carry charge, denoted by acapacitance C [F] and a voltage-current relation given by I = Cmicrofluidics textbook nanofluidics textbook Brian Kirby CornellΔV . An ideal capacitor consists of two conductors separated by a dielectric material–the two conductors each are at a specific voltage and the polarization of the dielectric causes a net charge on the surface of the conductors.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.13: Symbol and voltage-current relation for a capacitor.


Idealinductors (Figure 5.14) havean inductance L [H] and a voltage-current relation given by ΔV = Lmicrofluidics textbook nanofluidics textbook Brian Kirby CornellI.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.14: Symbol and voltage-current relation for an inductor.


We construct electrical circuits by wiring together voltage sources with these resistors, capacitors, and inductors.

5.4.2 Electrical impedance

The impedanceof a circuit element is a complex quantity that extendsOhm’s law (V = IR) to AC circuits. If the voltages and currents are sinusoidal and written as follows (note Re means the real part of):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.62)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.63)

then for circuit elements we define an impedance microfluidics textbook nanofluidics textbook Brian Kirby Cornell, which describes the voltage-current relationship through the equation microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell or microfluidics textbook nanofluidics textbook Brian Kirby Cornell0 = microfluidics textbook nanofluidics textbook Brian Kirby Cornell0microfluidics textbook nanofluidics textbook Brian Kirby Cornell. Each circuit element has a complex impedance corresponding to its circuit properties:

  • resistors:
    microfluidics textbook nanofluidics textbook Brian Kirby Cornell

  • capacitors:
    microfluidics textbook nanofluidics textbook Brian Kirby Cornell

  • inductors:
    microfluidics textbook nanofluidics textbook Brian Kirby Cornell

5.4.3 Circuit relations

Ohm’s law and the relations for impedance describe the current through an element or the voltage drop across that element. For a circuit that is composed of a network of theseelements, Kirchoff’s current law links these network elements using theconservation of current relation (Figure 5.15):

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where I in this case is defined positive into the node. Circuit networks can be solved as systems of algebraic equations constructed fromOhm’s law and circuit element impedance relations.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.15: Implementation of Kirchoff’s law at a node linking four circuit elements.


5.4.4 Series and parallel component rules

The results of Kirchoff’s law for parallel and series circuits are shown below.

Series circuit rules The resistance of two resistors in series is equal to the sum of the resistances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

the reciprocal of the capacitance of two capacitors in series is equal to the sum of the reciprocals of the capacitances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

the inductance of two inductors in series is equal to the sum of the inductances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and the impedance of two impedances in series is equal to the sum of the impedances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Series component relations are depicted in Figure 5.16.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.16: Series component relations for circuit elements.


Parallel circuit rules The reciprocal of the resistance of two resistors in parallel is equal to the sum of the reciprocals of the resistances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

the capacitance of two capacitors in parallel is equal to the sum of the capacitances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

the reciprocal of the inductance of two inductors in parallel is equal to the sum of the reciprocals of the inductances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and the reciprocal of the impedance of two impedances in parallel is equal to the sum of the reciprocals of the impedances:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Parallel component relations are depicted in Figure 5.17.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.17: Parallel component relations for circuit elements.


[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not constitute an endorsement by the Kirby Research Group or Cornell University.

Donations keep this resource free! Give here: