Difference between revisions of "Impedance Analysis"

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For analyzing the frequency response of circuits with various components, it is convenient to think of a generalized form of resistance, or opposition to current flow, known as impedance (<math>Z</math>). Clearly, we know that a resistor's resistance to current flow is known by its resistance. Although capacitors and inductors don't technically have a "resistance" value, they do resist current flow in their own ways and we can call those their respective impedances. Thus, Ohm's law applies not only to resistances, but to impedances as well:  
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For analyzing the frequency response of circuits with various components, it is convenient to think of a generalized form of resistance, or opposition to current flow, known as '''impedance''' (<math>Z</math>). Clearly, we know that a resistor's resistance to current flow is known by its resistance. Although capacitors and inductors don't technically have a "resistance" value, they do resist current flow in their own ways and we can call those their respective impedances. Thus, Ohm's law applies not only to resistances, but to impedances as well:  
 
{| class="wikitable" style="margin: 1em auto 1em auto;"
 
{| class="wikitable" style="margin: 1em auto 1em auto;"
 
|align="center"|<math>V = I Z</math>  
 
|align="center"|<math>V = I Z</math>  

Revision as of 21:20, 21 July 2016

20.309: Biological Instrumentation and Measurement

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Overview

When a circuit contains capacitors and inductors, the behavior of the circuit can change with time. Furthermore, often while using these components, we are interested in time-varying input signals, such as a sinusoidal voltage input. Since the current-voltage relationships for capacitors and inductors involve derivatives or integrals, analyzing such circuits often involve differential equations. To simplify our calculations, we can perform the steady-state analysis of circuits with time-varying signals using the frequency domain instead of the time domain. For example, a sine wave constantly varies over time, but only contains one frequency. Here we will discuss how to approach circuits using this mind set.


For analyzing the frequency response of circuits with various components, it is convenient to think of a generalized form of resistance, or opposition to current flow, known as impedance ($ Z $). Clearly, we know that a resistor's resistance to current flow is known by its resistance. Although capacitors and inductors don't technically have a "resistance" value, they do resist current flow in their own ways and we can call those their respective impedances. Thus, Ohm's law applies not only to resistances, but to impedances as well:

$ V = I Z $

where $ V $ is the voltage drop across the components of interest, $ I $ is the current, and $ Z $ is the impedance of those components.


The table below indicates the impedance values for different circuit elements. Here, $ j $ is the imaginary number and $ \omega $ is frequency in units of radians. Recall that $ \omega = 2 \pi f $, where $ f $ is frequency in units of Hz.

ImpedanceChart.png

Derivation

Below is a short demonstration of how we can derive the impedances for the circuit elements listed above. Let's use the capacitor as an example. Since we are interested in transient (time varying) signals, we will let voltage $ V $ be a time varying signal $ Ae^{j \omega t}. $ (Click here for a refresher on complex numbers.)

Since we know that $ I = C {dV\over dt} $, we can take the derivative of V and plug it in:
$ I = C A j \omega e^{j \omega t} $
With some reorganizing, we can see that $ I = j \omega C V $
To get it into the form $ V = I Z $,
$ Z = {1\over j \omega C} $

This impedance value follows what we already know about capacitors--that they act as an open circuit with a constant DC voltage ($ Z \rarr \infin $ when $ \omega = 0 $, also $ I=0 $ because $ {dV\over dt}=0 $). At very high frequencies, as $ \omega \rarr \infin $, the capacitor acts as a short circuit, like a wire without resistance ($ Z \rarr 0 $).

The same exercise can be done for the inductor. It will be left as an exercise for the reader.

Impedances in series and in parallell

When combining the effect of multiple circuit components in series or parallel, their impedances can be treated as resistances. Thus, for elements in series, their impedances add. Meanwhile, for elements in parallel, the inverses of their impedances add. This is summarized in the following table.

AddImpedanceTable.png

Example circuit problems