Difference between revisions of "TF EIM Chapt2"

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Thus far we have discussed direct current circuits which were built from a battery (constant voltage supply).
 
Thus far we have discussed direct current circuits which were built from a battery (constant voltage supply).
  
Another type of circuit is one in which the current is driven in an alternating fashion.  Typically the current is increased so some positive maximum value, then decreased until it passes through zero and reverses direction until it reaches another maximum value and then it is decreased again, passing though zero once more on its way to a positive maximum value.
+
Another type of circuit is one in which the current is driven in an alternating fashion.  Typically the current is increased to some positive maximum value, then decreased until it passes through zero and reverses direction until it reaches another maximum value and then it is decreased again, passing though zero once more on its way to a positive maximum value.
  
 
==Definitions==
 
==Definitions==
Line 49: Line 49:
  
  
:: <math>\int_0^T \sin^2(x) dx = \int_0^T \frac{1}{2} \left (1 - \cos(2x) \right) dx = \frac{1}{2} + \frac{1}{T} \int_0^T \cos(t)dt = \frac{1}{2}</math> if T is infinite or an integer number of cycles
+
:: <math>\int_0^T \sin^2(x) dx = \int_0^T \frac{1}{2} \left (1 - \cos(2x) \right) dx = \frac{T}{2} + \int_0^T \cos(t)dt </math>
 +
 
 +
:<math>I_{RMS}  = \sqrt{\frac{1}{T} \int_0^T I_0^2 \sin^2(\omega t) dt}= \frac{I_0}{\sqrt{2}} </math>if T is infinite or an integer number of cycles
  
 
==== Power====
 
==== Power====
Line 55: Line 57:
 
The instantaneous power dissipated in a resister by an alternating current is given as
 
The instantaneous power dissipated in a resister by an alternating current is given as
  
:<math>P_{inst}(t) =I^2(t) R = R\sin^2(\omega t)</math>
+
:<math>P_{inst}(t) =I^2(t) R = RI^2_0\sin^2(\omega t)</math>
  
 
[[File:TF_EIM_AC_pwr_sin.png| 300 px]]
 
[[File:TF_EIM_AC_pwr_sin.png| 300 px]]
  
 
Average power
 
Average power
:<math>P_{ave} = \frac{1}{T} \int_0^T P(t) dt = \frac{1}{T} \int_0^T RI^2(t) dt = \frac{R}{T} \int_0^T \sin^2(\omega t) dt = \frac{I_0^2R}{\sqrt{2}} = I_{RMS}^2 R</math>
+
:<math>P_{ave} = \frac{1}{T} \int_0^T P(t) dt = \frac{1}{T} \int_0^T RI^2(t) dt = \frac{RI_0^2}{T} \int_0^T \sin^2(\omega t) dt = \frac{RI_0^2}{\sqrt{2}} = I_{RMS}^2 R</math>
  
===Plane waves===
+
===Current using Complex notation===
  
 
Another mathematical expression for an alternating current uses complex variables
 
Another mathematical expression for an alternating current uses complex variables
Line 76: Line 78:
 
[[File:TF_EIM_AV_VoltageSymbol.png| 200 px]]
 
[[File:TF_EIM_AV_VoltageSymbol.png| 200 px]]
  
The mathematical form is
+
A mathematical form is for an AC voltage source is:
  
 
:<math>V = V_0 \sin(\omega t)</math>
 
:<math>V = V_0 \sin(\omega t)</math>
Line 87: Line 89:
 
[[File:TF_EIM_ParallelPlatCap.jpg| 200 px]]
 
[[File:TF_EIM_ParallelPlatCap.jpg| 200 px]]
  
;Noteice: The potential is constant over the face of the capacitor
+
;Notice: The potential is constant over the face of the capacitor
 
:: <math>V_{AB} = - \int_A^B \vec{E} \cdot d \vec{l}</math>
 
:: <math>V_{AB} = - \int_A^B \vec{E} \cdot d \vec{l}</math>
  
The symbol for a capacitor in a circuit reflects this.
+
The symbol for a capacitor in a circuit is usually.
  
 
[[File:TF_EIM_CapacitorSymbol.png | 200 px]]
 
[[File:TF_EIM_CapacitorSymbol.png | 200 px]]
  
==dielectric==
+
or
  
dielectric capacitors can be shorted out.  Put the positive probe from the voltmeter on the positive side of the electrolytic capacitor, the capacitor should be good if you measure a large resistance <math>(k \Omega)</math> otherwise a shorted electrolytic capacitor (small resistance) is a bad one.
+
==electrolytic symbol==
  
 
[[File:TF_EIM_ElectrolyticCapacitorSymbol.png | 200 px]]
 
[[File:TF_EIM_ElectrolyticCapacitorSymbol.png | 200 px]]
  
 +
The electrolytic capacitor
 +
 +
 +
Electrolytic  capacitors can be shorted out.  Put the positive probe from the voltmeter on the positive side of the electrolytic capacitor, the capacitor should be good if you measure a large resistance <math>(k \Omega)</math> otherwise a shorted electrolytic capacitor (small resistance) is a bad one.
 +
 +
== Capacitor Properties==
 +
 +
# <math>C>0</math> Measures ability to store charge
 +
#<math>V=Q/V</math> : Double Q Doubles V
 +
# SI units = Farrads = Coul/Volt
 +
# Magnitude depends on the elements geometry (conductor area, separation of conductors, ...)
 +
 +
== Capacitors in Parallel and Series==
 +
 +
===Series===
 +
Capacitors in Series add like resistors in Parallel.(Capacitance decreases when hooked up in series)
 +
 +
[[File:TF_EIM_CapInSeries.jpg | 200 px]]
 +
 +
;Note:The same amount of charge is pushed onto each capacitor
 +
 +
For capacitors in series you can apply the loop theorem for the above case and the one where all the capacitors are replaced by a single capacitor.
 +
 +
 +
:<math>V_{tot} = V_1+V_2+ ...</math>
 +
:<math>Q/C_{tot} = Q/C_1+Q/C_2+ ...</math>
 +
:<math>1/C_{tot} = 1/C_1+1/C_2+ ...</math>
 +
 +
===Parallel===
 +
 +
Capacitors in parallel add like resistors in Series (Capacitance increases when hooked up in parallel)
 +
 +
[[File:TF_EIM_CapInParallel.jpg | 200 px]]
 +
 +
If you were to replace all capacitors with an equivalent capacitance (C_{eq}) then the charge driven onto the equivalent capacitance would be equal to the sum of the charge driven onto each capacitor.
 +
 +
:<math>Q_{tot} = Q_{eq} = Q_1 + Q_2 +</math> ...
 +
:<math>VC_{tot} = VQ_{eq} = VQ_1 + VQ_2 +</math> ...
 +
:<math>C_{tot} = Q_{eq} = Q_1 + Q_2 +</math> ...
  
== Capacitor circuit==
+
== Charging/Discharging Capacitor circuit==
  
 
[[File:TF_EIM_CapACCircuit.png| 200 px]]
 
[[File:TF_EIM_CapACCircuit.png| 200 px]]
Line 114: Line 155:
 
:<math>Q = VC= V_0 C \cos{\omega t)}</math>
 
:<math>Q = VC= V_0 C \cos{\omega t)}</math>
  
:<math>I = \frac{dQ}{dt} = -V_0C \omega \sin{\omega t)}= V_0 C \cos{\omega t + \pi/2)}</math>
+
:<math>I = \frac{dQ}{dt} = -V_0C \omega \sin(\omega t)= V_0 C \omega \cos(\omega t + \pi/2)</math>
  
  
 
;Notice that the current is 90 degrees out of phase with the driving voltage.
 
;Notice that the current is 90 degrees out of phase with the driving voltage.
  
 +
[[File:TF_EIM_CapChargeScopePic_PhaseShift.png|200 px|thumb|Phase shift using 562nF capacitor and a 200 mV sine wave voltage source at 65 Hz ]]
  
;Note:  The current increases faster than the voltage.  Below are a few scope pictures measuring the voltage coming out of the voltage source (V_{in}) and the voltage on the ground side of the capacitor (the capacitor is grounded through the scope).
+
;Note:  The current increases faster than the voltage.  Below are a few scope pictures measuring the voltage coming out of the voltage source <math>(V_{in})</math> and the voltage on the ground side of the capacitor <math>(V_{out})</math> (the capacitor is grounded through the scope).  This is actually a high pass filter because of the internal resistance of the scope used to measure <math>V_{out}</math>.
  
[
 
  
  
 
{| border="1"  |cellpadding="20" cellspacing="0  
 
{| border="1"  |cellpadding="20" cellspacing="0  
 
|-
 
|-
| 1 Hz || 10kHz || MHz
+
| 1 Hz || 16Hz || kHz
 
|-
 
|-
| [[File:TF_EIM_CapChargeScopePic_1Hz.png|200 px|thumb|Capacitor response at low frequency]] || [[File:TF_EIM_CapChargeScopePic_kHz.png|200px|thumb|kHz frequency]] || [[File:TF_EIM_CapChargeScopePic_MHz.png|200px|thumb|MHz frequency]]
+
| [[File:TF_EIM_CapChargeScopePic_1Hz.png|200 px|thumb|Capacitor response at low frequency]] || [[File:TF_EIM_CapChargeScopePic_16Hz.png|200px|thumb|16Hz frequency]] || [[File:TF_EIM_CapChargeScopePic_7kHz.png|200px|thumb|kHz frequency]]
 
|}
 
|}
  
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===Resistive nature of capacitance===
+
===Reactance: Resistive nature of capacitance===
  
  
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:<math>V=V_0 cos(\omega t) = V_0 Re \left [ e^{\omega t} \right ]</math>
+
:<math>V=V_0 \cos(\omega t) = V_0 Re \left [ e^{i\omega t} \right ]</math>
  
 
The notation
 
The notation
  
:<math>  e^{\omega t}  = \cos(\omega t) + i \sin(\omega t)</math> Euler's equation
+
:<math>  e^{\omega t}  = \cos(\omega t) + i \sin(\omega t)</math>: Euler's equation
 +
 
 +
:<math>  Re \left [ e^{i\omega t} \right ]=Re \left [ \cos(\omega t) + i \sin(\omega t) \right ]= \cos(\omega t) </math> The real part of a complex variable.
 +
 
 +
 
 +
The current is written as
 +
 
 +
 
 +
:<math>I=\omega CV_0  Re \left [ e^{i \left (\omega t + \pi/2 \right ) } \right ]</math>  (The current is 90 degrees<math> \pi/2</math> out of phase with the voltage)
 +
::<math> =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )} e^{i \left ( \pi/2 \right )} \right ]</math>
 +
::<math> =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )}  \left ( \cos(\pi/2) + i sin(\pi/2) \right ) \right ]</math>
 +
::<math> =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )}  \left ( 0 + i  \right )\right ]</math>
 +
::<math> =\omega C V_0 Re \left [ i e^{i \left ( \omega t \right )}  \right ]</math>
 +
::<math>  =Re \left [ V_0 \frac{e^{i \left ( \omega t \right )}}{\frac{1}{i\omega C}} \right ]=Re \left [ V_0 \frac{e^{i \left ( \omega t \right )}}{X_C} \right ]</math>
 +
 
 +
where
 +
 
 +
:<math>X_C = \frac{1}{i \omega C}</math> Effective impedance of a Capacitor
  
:<math> Re \left [ e^{\omega t} \right ]=Re \left [ \cos(\omega t) + i \sin(\omega t) \right ]= \cos(\omega t) </math>
+
;Observe
 +
: At high frequencies <math>(\omega)</math> the effective impedance (X_C) is small so the driving voltage should pass right through. 
 +
: At low frequencies <math>(\omega)</math> the effective impedance (X_C) is large so the driving voltage will be severely attenuated.
 +
:: Capacitors block Alternating Currents so you can reduce ripples on DC voltages.
 +
: You could measure the capacitance by measuring <math>(I/V)^2</math> for several frequencies
  
 
== RC circuit==
 
== RC circuit==
  
Now add a resistor in series with the capacitor in the above capacitor charging circuit.
+
=== The Low pass Fiter===
 +
 
 +
Now add a resistor in series with the capacitor in the above capacitor charging circuit so it resembles the circuit below
  
 
[[File:TF_EIM_Lab3.png | 400 px]]
 
[[File:TF_EIM_Lab3.png | 400 px]]
  
 +
=== Quick Solution===
 +
 +
Applying  the loop theorem using the reactance<math> X_C</math> of the capacitor instead of Q/C
 +
 +
 +
<math>V - IR -X_CI = =V -(X_R +X_C) I = 0</math>
 +
 +
:<math>\Rightarrow</math>  the capacitor can be added in series with the other resistor
 +
 +
:<math>X_{tot} = X_R + X_C</math>
 +
 +
It looks like the voltage divider from the resistance section
 +
 +
:<math>V_{out}= Re\left [\frac{X_C}{R+X_C} V_{in} \right ]=  Re \left [\frac{X_C}{X_R+X_C} V_{in} \right ]</math>
 +
 +
To evaluate <math>Re[Z] = \sqrt{Z Z^*}</math>  where<math> Z = x+iy</math> and <math>Z^* = x-iy</math>
 +
 +
 +
:<math>\frac{V_{out}}{V_{in}} = Re \left [ \frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \right ] =  \sqrt{\frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \frac{\frac{1}{-i \omega C}}{R+\frac{1}{-i \omega C}} } = \frac{1}{\sqrt{1 + \omega^2 R^2 C^2}}</math>
 +
 +
Let
 +
:<math>\omega_b = \frac{1}{RC} =</math> break point (cut off ) frequency
 +
 +
then
 +
 +
:<math>\frac{V_{out}}{V_{in}} =  \frac{1}{\sqrt{1 + \left ( \frac{\omega}{\omega_b}\right )^2}}= \frac{1}{\sqrt 2}</math>
 +
 +
[[File:TF_EIM_LowPassVoutVin-vs-omega.png|200px]]
 +
 +
 +
;Note: The RC circuit is another way to measure the capacitance by measureing <math>V_{out}/V_{in}</math> -vs- <math>\omega</math> and do a linear extrapolation to determine the break point frequency.
 +
 +
===Phasor Diagram===
 +
 +
 +
Phasor diagrams are a technique used to illustrate the phase difference between the input and output alternating voltages/currents.
 +
 +
 +
Returning to the loop theorem for the low pass RC filter
 +
 +
:<math>V_{in} = V_R + V_{out}</math>
 +
:<math>V_{in} = IX_R + IX_C</math>
 +
:<math>V_{in} = IR + I\frac{-i}{\omega C}</math>
 +
 +
We can write the voltage as a complex variable using a coordinate system in which the "x-axis" represents real numbers and the "y-axis" represents the complex components.  This is analogous to expressing 2-D vectors.
 +
 +
Below is a figure representing the complex plane.
 +
 +
[[File:TF_EIM_RC_PhasorDiag.png| 200 px]]
 +
 +
The angle <math>\phi</math> is the angle between<math> V_{out} = \frac{-i}{\omega C}</math> and <math>V_{in} = IR + I\frac{-i}{\omega C}</math>
 +
Based on the above Phasor diagram
 +
 +
:<math>\tan(\phi) = \left (\frac{R}{\frac{-1}{\omega C}} \right )=  \left(-\omega RC \right )</math>
 +
 +
=== High Pass Filter===
 +
 +
A high pass filter switches the order of the resistor and the capacitor.
 +
 +
[[File:TF_EIM_Lab4.png | 400 px]]
 +
 +
 +
Using the same reactance based analysis as above
 +
 +
:<math>V_{in} - IX_C - IR = 0</math>
 +
 +
:<math>V_{out} = IR</math>
 +
 +
:<math>\frac{V_{out}}{V_{in}} = \frac{IR}{I \left ( X_C + R \right ) }= \frac{R}{\frac{1}{i \omega C} + R} = \frac{ i \omega RC}{1 + i \omega RC}</math>
 +
 +
: <math>\left |\frac{V_{out}}{V_{in}} \right | = \sqrt{\left ( \frac{ i \omega RC}{1 + i \omega RC}\right ) \left ( \frac{- i \omega RC}{1 - i \omega RC}\right )} = \frac{\omega RC}{\sqrt{(1 + \omega^2 R^2C^2}}</math>
 +
 +
 +
;Phasor diagram
 +
 +
The phasor diagram is very similar except this time V_{out} is along the x-axis
 +
 +
 +
Returning to the loop theorem for the low pass RC filter
 +
 +
:<math>V_{in} = V_R + V_{out}</math>
 +
:<math>V_{in} = IX_C + IX_R </math>
 +
:<math>V_{in} =  I\frac{-i}{\omega C}+IR</math>
 +
 +
We can write the voltage as a complex variable using a coordinate system in which the "x-axis" represents real numbers and the "y-axis" represents the complex components.  This is analogous to expressing 2-D vectors.
 +
 +
Below is a figure representing the complex plane.
 +
 +
[[File:TF_EIM_RC_PhasorDiagHigh.png| 200 px]]
 +
 +
The angle <math>\phi</math> is the angle between <math>V_{out} =IR</math> and <math>V_{in} = I\frac{-i}{\omega C} +  IR </math>
 +
Based on the above Phasor diagram
 +
 +
:<math>\tan(\phi) = \left (\frac{\frac{-1}{\omega C}}{R} \right ) = (\frac{-1}{\omega RC})</math>
 +
 +
=== High Pass -vs- Low Pass graphs===
 +
if
 +
:<math>\omega_B=\frac{1}{RC}=10</math>
 +
 +
Then
 +
 +
:<math>\phi_{\mbox{Low Pass}} = \tan^{-1} \left ( -\frac{\omega}{\omega_B}\right )</math>
 +
:<math>\phi_{\mbox{High Pass}} = \tan^{-1} \left ( -\frac{\omega_B}{\omega}\right )</math>
 +
 +
{| border="1"  |cellpadding="20" cellspacing="0
 +
|-
 +
| Low pass || High Pass
 +
|-
 +
| [[File:TF_EIM_LowPassVoutVin-vs-omega.png|200 px|thumb|Gain -vs- <math>\omega</math>]] || [[File:TF_EIM_HighPassVoutVin-vs-omega.png|200px|thumb|Gain -vs- <math>\omega</math>]]
 +
|-
 +
| [[File:TF_EIM_LowPassPhaseShift-vs-omega.png|200 px|thumb|Phase Shift -vs- <math>\omega</math> ]] || [[File:TF_EIM_HighPassPhaseShift-vs-omega.png|200px|thumb|Phase Shift -vs- <math>\omega</math>]]
 +
|}
 +
 +
=== exact solution===
 
Again apply the loop theorem
 
Again apply the loop theorem
  
Line 161: Line 339:
 
or
 
or
  
: <math>R \frac{dQ}{dt} + \frac{1}{C} Q - V = 0</math> First order linear differential equation
+
: <math>R \frac{dQ}{dt} + \frac{1}{C} Q =V(t)</math> First order non-homogeneous linear differential equation
 +
 
 +
 
 +
==== First solve homogeneous part====
 +
 
 +
: <math>R \frac{dQ}{dt} + \frac{1}{C} Q =0</math> First order homogeneous linear differential equation
 +
 
 +
The above is integratable
 +
 
 +
: <math> \frac{dQ}{dt} = -\frac{1}{RC} Q </math>
 +
: <math>\int \frac{dQ}{Q} = -\int \frac{1}{RC} dt </math>
 +
: <math>\ln Q = -\frac{t}{RC}  -Constant</math>
 +
: <math> Q(t) = e^{-\frac{t}{RC}} e^{Constant} = Q_0e^{-\frac{t}{RC}}</math>
 +
:<math> I(t)= \frac{d Q}{d t} = \frac{d}{dt} Q_0 e^{-\frac{t}{RC}} = \frac{-Q_0}{RC}e^{-\frac{t}{RC}}</math>
 +
 
 +
This gives you the general solution for the homogeneous problem
 +
 
 +
In this case it represents the solution for a discharging capacitor.
 +
 
 +
 
 +
====Trial solution method for Non-Homogeneous Solution====
 +
 
 +
 
 +
A common and more general technique to solving such a differential equation is to use the solution from the discharging equation (the homogeneous diff. eq.)  as a trial solution to the charging (non-homogeneous) problem by constructing the linear combination below.
 +
 
 +
:<math>Q_{trial} = A + Be^{-t/RC}</math>
 +
 
 +
Where A & B are constants to e determined
 +
 
 +
You then substitute this trial solution into the differential equation and apply boundary conditions to determine the coefficients.
 +
 
 +
: <math>\frac{d}{dt} Q+ \frac{1}{RC}Q = \frac{V}{R}</math>
 +
: <math>\frac{d}{dt} \left ( A + Be^{-t/RC} \right ) + \frac{1}{RC} \left ( A + Be^{-t/RC} \right ) = \frac{V}{R}</math>
 +
: <math> \left (  B \frac{-1}{RC}e^{-t/RC} \right ) + \frac{1}{RC} \left ( A + Be^{-t/RC} \right ) = \frac{V}{R}</math>
 +
: <math>\Rightarrow \frac{1}{RC} \left ( A  \right ) = \frac{V}{R}</math>
 +
: <math>A = C V</math>
 +
 
 +
To determine the other coefficient you apply the boundary condition that q = 0 when t=0
 +
: <math>0= A + Be^{-0/RC} = A+B</math>
 +
: <math>A=-B</math>
 +
 
 +
:<math>Q=CV (1 - e^{-t/RC}) =Q_0 (1 - e^{-t/RC})</math>
 +
 
 +
= Inductance=
 +
 
 +
== Inductor/Choke==
 +
 
 +
An inductor is basically a loop of wire wrapped around a core (or air) to form a coil.
 +
 
 +
 
 +
[[File:TF_EIM_BasicIncuctorCoil.jpg]]
 +
 
 +
Consider what happens in the above circuit when you first connect a voltage source (or close the switch).
 +
 
 +
1.) A current starts to flow through the wire such that I=V/R  (R can be the internal resistance of the battery which changes with time).
 +
 
 +
2.) A current produces a magnetic field
 +
 
 +
; Ampere's Law
 +
:<math>\oint \vec{B} \cdot d \vec{l} = \mu_0 I</math>
 +
 
 +
If the current distribution isn't symmetric enough to make the line integral easy
 +
 
 +
then you may need to use
 +
 
 +
;Biot-Savart Law
 +
: <math>d\vec{B} = \frac{\mu_0 I}{4 \pi} \frac{\vec{l} \times \vec{r}}{r^2}</math>
 +
 
 +
 
 +
Most inductors take the geometrical form of a solenoid.
 +
 
 +
[[File:TF_EIM_SolenoidAmperslaw.jpg | 400 px]]
 +
 
 +
 
 +
:<math>\oint \vec{B} \cdot \vec{l} = \int_a^b \vec{B} \cdot d\vec{l} + \int_b^c \vec{B} \cdot d\vec{l} + \int_c^d \vec{B} \cdot d\vec{l} + \int_d^a \vec{B} \cdot d\vec{l} = \mu_0 I_{tot}</math>
 +
:: <math>\int_a^b \vec{B} \cdot d\vec{l} = B\ell</math>
 +
:: <math>\int_b^c \vec{B} \cdot d\vec{l} = \int_d^a \vec{B} \cdot d\vec{l} =0  : \vec{B} \perp d\vec{l}</math>
 +
:: <math>\int_c^d \vec{B} \cdot d\vec{l} =0 : B \approx 0</math>
 +
 
 +
::<math> I_{tot} = NI</math> = total current enclosed by the loop is I times the number of loops N
 +
 
 +
:<math>B = \mu_0 \frac{NI}{\ell} = \mu_0 nI</math>
 +
 
 +
where
 +
 
 +
: <math>n \equiv \frac{N}{\ell} =</math> number of turns per unit length
 +
 
 +
3.) When using an AC source the current is changing with time and as a result the B-field is also changing in time
 +
 
 +
4.) The changing current induces and emf (<math>\mathcal{E}</math>)
 +
 
 +
Faraday's law of induction says
 +
 
 +
:<math>\mathcal{E} = - \frac{d \Phi_B}{dt}</math> = self induced emf
 +
 
 +
where
 +
 
 +
:<math>\Phi_B = \int \vec{B} \cdot d \vec{A}</math>
 +
 
 +
:<math>\mathcal{E} = - \frac{d }{dt} \int \vec{B} \cdot d \vec{A}</math>: There are 3 ways to induce an emf (B and A change with time or the angle between them)
 +
 
 +
The B-field is usually perpendicular to the area of the induction and mostly uniform.
 +
 
 +
 
 +
:<math>\mathcal{E} = - \frac{d }{dt} BA = - \frac{d }{dt} \mu_o nIA  = - \mu_0nA \frac{dI}{dt} </math>
 +
 
 +
The above is the emf induced in a single loop
 +
 
 +
;If you assume there are N loop in the inductor
 +
:Then
 +
::<math>\mathcal{E} = - N\frac{d }{dt} \mu_o nIA  = - N\mu_0nA \frac{dI}{dt} \equiv -L \frac{dI}{dt}</math>
 +
 
 +
 
 +
: <math>L = N\mu_0 n A = \mu_0 \frac{N^2}{\ell} \pi R^2</math> = Inductance
 +
 
 +
where
 +
:<math>\ell=</math> length of loop
 +
:<math>R =</math> radius of coil
 +
 
 +
Lenz's law says that the induced current will appear in a direction that opposes the change producing it.
 +
 
 +
So in essence the inductor acts "like" a resistor which resists the current flow according to the amount of current is changing. 
 +
 
 +
There is an effective reactance <math>X_{L}</math> for inductors as there was one for capacitors
 +
 
 +
== Inductor Reactance==
 +
 
 +
Consider an AC voltage source with an inductor attached.
 +
 
 +
 
 +
 
 +
Loop theorem
 +
:<math>\Rightarrow V - L \frac{dI}{dt} = 0</math>
 +
 
 +
:<math>V_0 cos(\omega t) dt = L dI</math>
 +
 
 +
:<math>\int_0^t V_0 cos(\omega t) dt =\int_0^t  L dI</math>
 +
: <math>\left . V_0 \frac{sin(\omega t)}{\omega} \right |_o^t = L \left .  I(t)\right |_0^t</math>
 +
: <math>V_0 \frac{sin(\omega t)}{\omega}  -0  = L \left ( I(t) - I(0)\right )</math>
 +
: <math>\Rightarrow I(t) = V_0 \frac{sin(\omega t)}{\omega L}  + I(0)</math>
 +
 
 +
Assume I(0) =0 and rewrite <math>\sin</math> in terms of a phase shifted <math>\cos</math>
 +
 
 +
: <math>\Rightarrow I(t) = \frac{V_0}{\omega L} cos(\omega t - \pi/2) </math> : V and I are 90 degrees out of phase
 +
::<math>= Re \left ( \frac{V_0}{\omega L} e^{i(\omega t - \pi/2)}\right )</math>
 +
::<math>= Re \left ( \frac{V_0}{\omega L} e^{i(\omega t)} (-i)\right )</math>
 +
::<math>= Re \left ( \frac{V_0}{i\omega L} e^{i(\omega t)}\right )</math>
 +
 
 +
::<math>X_L \equiv i\omega L</math>
 +
 
 +
;Note
 +
:Large L makes it hard to change current
 +
:Small L makes it easy to change current
 +
 
 +
==parallel LC circuit (Band Stop/Notch  Filter/ tank circuit)==
 +
 
 +
[[File:TF_EIM_Parallel_RLC.png| 200 px]]
 +
 
 +
The above circuit has a resistor in series with an inductor and capacitor in parallel.
 +
 
 +
Using the complex analysis techniques for AC circuits you can consider the Inductor and Capacitor as impedances in parallel.
 +
 
 +
so
 +
 
 +
:<math>\frac{1}{X_{tot}} = \frac{1}{X_C} + \frac{1}{X_L} = \frac{i \omega C}{1} + \frac{1}{i \omega L} = \frac{1- \omega^2 LC}{i \omega L}</math>
 +
:<math> X_{tot} = \frac{i \omega L}{1- \omega^2 LC} = \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}}</math>
 +
 
 +
::<math>\omega_{LC} \equiv \frac{1}{\sqrt{LC}}</math>
 +
 
 +
or
 +
 
 +
::<math>\nu_{LC} =  \frac{\omega_{LC}}{2\pi} = \frac{1}{2 \pi \sqrt{LC}}</math>
 +
 
 +
 
 +
 
 +
 
 +
{| border="1"  |cellpadding="20" cellspacing="0
 +
|-
 +
|<math>L = 33 \mu H</math> and  <math>C = 1 \mu F </math>
 +
|-
 +
|[[File:TF_EIM_Impedance_LC_Parallel.png| 400 px]]
 +
|-
 +
|<math>\Rightarrow \omega = 174077.66</math> rad/s
 +
|}
 +
 
 +
 
 +
=== Phase differences===
 +
 
 +
 
 +
;Notice
 +
: <math>X_C = \frac{1}{i \omega C}</math>  & <math>X_L = i \omega L</math>
 +
:<math>I = \frac{V}{X}</math>
 +
 
 +
:: <math>I_C = \frac{V}{\frac{1}{i \omega C}} = i \omega CV</math>
 +
::<math>I_L = \frac{V}{i \omega L} = \frac{-iV}{\omega L }</math>
 +
 
 +
The current through the inductor side of the circuit is 180 degrees out of phase with the capacitor side.
 +
 
 +
=== gain===
 +
Loop Theorem
 +
 
 +
:<math>\Rightarrow V= I(R+X_{tot}) = I \left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )</math>
 +
 
 +
or
 +
 
 +
:<math> I= \frac{V_0 e^{i \omega t}}{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )}</math>
 +
 
 +
;Notice
 +
:When <math>\omega \approx \omega_{LC} = \sqrt{\frac{1}{LC}}</math> then the AC signal is attenuated.
 +
 
 +
Looking at the Voltage divider aspect of the circuit
 +
 
 +
:<math>V_{AB}=V_{out} = \frac{X_{tot} }{R + X_{tot}}V_{in}</math>
 +
 
 +
:<math>\left |\frac{ V_{out}} {V_{in}}\right | = \sqrt{ \left [ \frac{X_{tot} }{R + X_{tot}} \right ]  \left [ \frac{X_{tot} }{R + X_{tot}} \right ]^*}</math>
 +
 
 +
::<math> = \sqrt{ \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]  \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*}</math>
 +
::<math> = \sqrt{ \frac{ \omega^2 L^2 \omega_{LC}^4}{R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2L^2 \omega_{LC}^4}}</math>
 +
::<math> = \sqrt{ \frac{ \omega^2 }{C^2R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}</math>
 +
::<math> = \sqrt{ \frac{ \omega^2 }{\frac{1}{\omega_{RC}^2}(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}</math>
 +
 
 +
 
 +
 
 +
 
 +
{| border="1"  |cellpadding="20" cellspacing="0
 +
|-
 +
|<math>L = 33 \mu H</math> and  <math>C = 1 \mu F </math> and R=200 <math>\Omega</math>
 +
|-
 +
|[[File:TF_EIM_LC_paral_Gain.png | 400 px]]
 +
|-
 +
|<math>\Rightarrow \omega = 174077.66</math> rad/s  or <math>\nu = \frac{\omega}{2 \pi} = 27705</math> Hz
 +
|}
 +
 
 +
==== Q and Bandwidth====
 +
 
 +
 
 +
In the above circuit
 +
 
 +
: <math>\nu = \omega/2 \pi = \frac{1}{2 \pi \sqrt{LC}}= 27705 Hz</math>
 +
 
 +
The inductors reactance at this resonance frequency is
 +
 
 +
:<math>X_{L} = \omega L = 2 \pi \nu L = \frac{L}{\sqrt{LC}} = \sqrt{\frac{L}{C}} = \sqrt{\frac{33 \times 10^{-6} H}{1 \times 10^{-6}F}} = 5.7 \Omega</math>
 +
 
 +
 
 +
;Bandwith: The most common definition for the Bandwidth of this circuit is the frequency range over which the output decreases by 3 dB. 
 +
 
 +
;Decibels
 +
:The decibel system is a logarithmic scale for the ratio of two quantities
 +
 
 +
;For power
 +
:dB = <math>10 \log_{10} \left ( \frac{P_{in}}{P_{out}}\right )</math>
 +
 
 +
;For amplitude (Voltage)
 +
:dB = <math>20 \log_{10} \left ( \frac{V_{in}}{V_{out}}\right )</math>
 +
 
 +
 
 +
A 3dB drop corresponds to the frequency at which the circuits power is cut in half from the resonance frequency.
 +
 
 +
:<math>P = I^2 R</math>
 +
:<math>P_{1/2} = (\frac{1}{\sqrt{2}}I)^2 R= \frac{(\frac{1}{\sqrt{2}} V)^2}{R}</math>
 +
:<math>\Rightarrow V = \frac{1}{\sqrt{2}} V = 0.71 V</math>
 +
 
 +
:The Band Width is a measurement of the two frequencies which will decrease the resonance voltage by 30 %.
 +
 
 +
[[File:TF_EIM_BandWidthDef_LC.gif | 400 px]]
 +
 
 +
;Quality Factor
 +
:The Quality factor for this circuit may be expressed in term of the bandwidth such that
 +
::<math>Q=\frac{\omega_0}{BW}=\frac{\omega_0}{\Delta \omega}</math>
 +
 
 +
 
 +
;Notice
 +
:The Bandwidth can be changed in the above circuit by changing the step down resistor (R) without changing the resonance frequency.
 +
 
 +
=== Quality Factor (Q)===
 +
 
 +
The Quality factor of a circuit is defined in terms of a ratio between the energy stored to the energy lost.
 +
 
 +
:<math>Q \equiv 2 \pi \frac{\mbox{Energy Stored per cycle}}{\mbox{Energy lost per cycle}} = 2 \pi \frac{W_S}{W_L}</math>
 +
 
 +
;How much energy does an inductor store?
 +
 
 +
: <math>W = \mbox{Work} = \int \vec{F} \cdot d\vec{r} = \Delta U = \int V dQ = \int L \frac{d I}{dt} (I dt) = L \int_0^I I dI = \frac{1}{2} LI^2</math>
 +
::<math>W_S = \frac{1}{2} LI^2</math>
 +
 
 +
 
 +
;How much energy does an inductor lose?
 +
The system loses energy through heat via power dissipation so in general
 +
::<math>W_L = \int p(t) dt = \int_0^T I^2(t) R dt </math> where R is the effective resistance of the circuit
 +
 
 +
 
 +
;Note
 +
:A circuit with high Q will have amplitude (or blocking) at the desired resonance frequency and the least amplitude (or blocking) at non-resonant frequencies.
 +
 
 +
An Inductor in series with a capacitor is known as a Band Pass filter.  Since wires are used to hook everything up and real capacitors have leakage resistance so an RLC circuit is a realistic version of the series LC circuit.
 +
 
 +
==RLC circuit==
 +
 
 +
An RLC circuit is a Resistor, an Inductors, and a Capacitor in series with an electromotive force.
 +
 
 +
[[File:TF_EIM_Lab5_RLC.png| 200 px]]
 +
 
 +
 
 +
===Effective impedance===
 +
 
 +
:<math>X_{out} = R_L + X_C + X_L = R_L + \frac{1}{i \omega C} + i \omega L</math>
 +
 
 +
:<math>\left | X_{out} \right | =  \sqrt{\left [ R_L + i \left (\frac{-1}{\omega C} +  \omega L\right ) \right ]\left [ R - i \left (\frac{-1}{\omega C} + \omega L\right ) \right ]^*}</math>
 +
:: <math>=  \sqrt{ R_L^2 +  \left ( \omega L -  \frac{1}{\omega C} \right )^2}</math>
 +
 
 +
=== Gain===
 +
Loop Theorem
 +
 
 +
: <math>V_{in} = I (R+ X_{out})</math>
 +
 
 +
 
 +
Voltage Divider
 +
 
 +
:<math>V_{AB}=V_{out} = \frac{X_{out}}{R + X_{out}}V_{in}</math>
 +
 
 +
 
 +
:<math>\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{X_{out}}{R + X_{out}}\right ]\left [ \frac{X_{out}}{R + X_{out}}\right ]^*}</math>
 +
<math>R_L + i \left ( \omega L - \frac{1}{\omega C}\right)</math>
 +
:<math>\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]^*}</math>
 +
 
 +
::<math> = \sqrt{\frac{R_L^2 + \left ( \omega L - \frac{1}{\omega C}\right)^2}{(R + R_L)^2 +  \left ( \omega L - \frac{1}{\omega C}\right)^2}}</math>
 +
 
 +
::<math> = \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}{(R + R_L)^2 +  \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}}</math>
 +
 
 +
Let
 +
 
 +
:<math>\omega_0 = \sqrt{\frac{1}{LC}}</math>
 +
 
 +
Then
 +
:<math>\left | \frac{V_{out}}{V_{in}}\right |  = \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 - \omega_0^2}{\omega_0^2 \omega C}\right)^2}{(R + R_L)^2 +  \left ( \frac{\omega^2- \omega_0^2}{\omega_o^2 \omega C}\right)^2}}</math>
 +
 
 +
When<math> \omega = \sqrt{\frac{1}{LC}} = \omega_0</math>
 +
 
 +
Then
 +
:<math>\left | \frac{V_{out}}{V_{in}}\right |  =\frac{R}{R+R_L}</math>
 +
 
 +
[[File:TF_EIM_RLC.png | 200 px]]
 +
 
 +
=== Phase shift===
 +
 
 +
<math>X_C = \frac{-i}{\omega C} \Rightarrow</math> Voltage lags current for a capacitor
 +
 
 +
<math>X_L = i \omega L \Rightarrow</math> Voltage leads current for Inductor
 +
 
 +
[[File:TF_EIM_RLC_Phasor.png | 300 px]]
 +
 
 +
 
 +
:<math>\tan(\phi) = \frac{X_{out}}{R_L} = \frac{\omega L - \frac{1}{\omega C}}{R}= \frac{\omega^2 - \omega_o^2}{\omega_0^2R}</math>
 +
 
 +
 
 +
If <math>\omega < \omega_0</math> then the phase shift is negative
 +
 
 +
If <math>\omega > \omega_0</math> then the phase shift is positive
  
  
 
[[Forest_Electronic_Instrumentation_and_Measurement]]
 
[[Forest_Electronic_Instrumentation_and_Measurement]]

Latest revision as of 16:51, 5 February 2015

Alternating Current (AC)

Thus far we have discussed direct current circuits which were built from a battery (constant voltage supply).

Another type of circuit is one in which the current is driven in an alternating fashion. Typically the current is increased to some positive maximum value, then decreased until it passes through zero and reverses direction until it reaches another maximum value and then it is decreased again, passing though zero once more on its way to a positive maximum value.

Definitions

frequency ([math]f[/math], [math]\nu[/math] )
The frequency is number of complete cycles which occur in 1 sec (cycles/sec = Hz)
Angular frequency =[math] \omega = 2\pi f[/math] = radians/sec
Period
The period is the time to complete one cycle = 1/[math]f[/math]
Amplitude
The change in the current from zero to its most positive value
peak-to-peak
The amount the current changes from its largest positive value to its most negative.


phase
The time a current takes to reaches its maximum value compared to another alternating current


Note
The above definitions can be used for voltage as easily as current.

Mathematical description

Trigonometric

[math]I(t) = \sin(\omega t + \phi) = \cos(\omega t + \phi - \pi/2)[/math]

CosinePhaseShiftPiover2.gif

Note
A 180 degree ([math]\pi[/math] radian) phase shift will put the above cosine wave completely out of sync with the original one such that if we add the two signals together the net current would be zero.

I_{RMS}

With this functional form for the alternating current we can now calculate another property, the RMS or root mean square.

The RMS quantifies an average fluctuation of the alternating current such that

[math]I_{RMS} \equiv \sqrt{\frac{1}{T} \int_0^T I^2(t) dt} = \sqrt{\frac{1}{T} \int_0^T I_0^2 \sin^2(\omega t) dt}= \frac{I_0}{\sqrt{2}} [/math]

where T represent the time interval over which the average is calculated ([math]\infty[/math])


[math]\int_0^T \sin^2(x) dx = \int_0^T \frac{1}{2} \left (1 - \cos(2x) \right) dx = \frac{T}{2} + \int_0^T \cos(t)dt [/math]
[math]I_{RMS} = \sqrt{\frac{1}{T} \int_0^T I_0^2 \sin^2(\omega t) dt}= \frac{I_0}{\sqrt{2}} [/math]if T is infinite or an integer number of cycles

Power

The instantaneous power dissipated in a resister by an alternating current is given as

[math]P_{inst}(t) =I^2(t) R = RI^2_0\sin^2(\omega t)[/math]

TF EIM AC pwr sin.png

Average power

[math]P_{ave} = \frac{1}{T} \int_0^T P(t) dt = \frac{1}{T} \int_0^T RI^2(t) dt = \frac{RI_0^2}{T} \int_0^T \sin^2(\omega t) dt = \frac{RI_0^2}{\sqrt{2}} = I_{RMS}^2 R[/math]

Current using Complex notation

Another mathematical expression for an alternating current uses complex variables

[math]I\left ( \omega t \right )=I_0 \exp^{i \left ( \omega t \right )} = I_0 \left ( \cos \left ( \omega t \right) + i \sin \left ( \omega t \right) \right )[/math]

Expressing the current in this form will be useful later

Voltage sources of AC current

The circuit symbol for an emf used to drive alternating currents is given below

TF EIM AV VoltageSymbol.png

A mathematical form is for an AC voltage source is:

[math]V = V_0 \sin(\omega t)[/math]

Capacitors

A capacitor is a device used to store charge in a circuit. The simplest capacitor is built from two conducting plates placed close to each other.


TF EIM ParallelPlatCap.jpg

Notice
The potential is constant over the face of the capacitor
[math]V_{AB} = - \int_A^B \vec{E} \cdot d \vec{l}[/math]

The symbol for a capacitor in a circuit is usually.

TF EIM CapacitorSymbol.png

or

electrolytic symbol

TF EIM ElectrolyticCapacitorSymbol.png

The electrolytic capacitor


Electrolytic capacitors can be shorted out. Put the positive probe from the voltmeter on the positive side of the electrolytic capacitor, the capacitor should be good if you measure a large resistance [math](k \Omega)[/math] otherwise a shorted electrolytic capacitor (small resistance) is a bad one.

Capacitor Properties

  1. [math]C\gt 0[/math] Measures ability to store charge
  2. [math]V=Q/V[/math] : Double Q Doubles V
  3. SI units = Farrads = Coul/Volt
  4. Magnitude depends on the elements geometry (conductor area, separation of conductors, ...)

Capacitors in Parallel and Series

Series

Capacitors in Series add like resistors in Parallel.(Capacitance decreases when hooked up in series)

TF EIM CapInSeries.jpg

Note
The same amount of charge is pushed onto each capacitor

For capacitors in series you can apply the loop theorem for the above case and the one where all the capacitors are replaced by a single capacitor.


[math]V_{tot} = V_1+V_2+ ...[/math]
[math]Q/C_{tot} = Q/C_1+Q/C_2+ ...[/math]
[math]1/C_{tot} = 1/C_1+1/C_2+ ...[/math]

Parallel

Capacitors in parallel add like resistors in Series (Capacitance increases when hooked up in parallel)

TF EIM CapInParallel.jpg

If you were to replace all capacitors with an equivalent capacitance (C_{eq}) then the charge driven onto the equivalent capacitance would be equal to the sum of the charge driven onto each capacitor.

[math]Q_{tot} = Q_{eq} = Q_1 + Q_2 +[/math] ...
[math]VC_{tot} = VQ_{eq} = VQ_1 + VQ_2 +[/math] ...
[math]C_{tot} = Q_{eq} = Q_1 + Q_2 +[/math] ...

Charging/Discharging Capacitor circuit

TF EIM CapACCircuit.png


If

[math]V=V_0 cos(\omega t)[/math]

then applying loop theorem lead to

[math]V-\frac{Q}{C} = 0[/math]
[math]Q = VC= V_0 C \cos{\omega t)}[/math]
[math]I = \frac{dQ}{dt} = -V_0C \omega \sin(\omega t)= V_0 C \omega \cos(\omega t + \pi/2)[/math]


Notice that the current is 90 degrees out of phase with the driving voltage.
Phase shift using 562nF capacitor and a 200 mV sine wave voltage source at 65 Hz
Note
The current increases faster than the voltage. Below are a few scope pictures measuring the voltage coming out of the voltage source [math](V_{in})[/math] and the voltage on the ground side of the capacitor [math](V_{out})[/math] (the capacitor is grounded through the scope). This is actually a high pass filter because of the internal resistance of the scope used to measure [math]V_{out}[/math].


1 Hz 16Hz kHz
Capacitor response at low frequency
16Hz frequency
kHz frequency



Reactance: Resistive nature of capacitance

To illustrate the resistive nature of capacitors, let's write the driving voltage in using complex notation


[math]V=V_0 \cos(\omega t) = V_0 Re \left [ e^{i\omega t} \right ][/math]

The notation

[math] e^{\omega t} = \cos(\omega t) + i \sin(\omega t)[/math]: Euler's equation
[math] Re \left [ e^{i\omega t} \right ]=Re \left [ \cos(\omega t) + i \sin(\omega t) \right ]= \cos(\omega t) [/math] The real part of a complex variable.


The current is written as


[math]I=\omega CV_0 Re \left [ e^{i \left (\omega t + \pi/2 \right ) } \right ][/math] (The current is 90 degrees[math] \pi/2[/math] out of phase with the voltage)
[math] =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )} e^{i \left ( \pi/2 \right )} \right ][/math]
[math] =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )} \left ( \cos(\pi/2) + i sin(\pi/2) \right ) \right ][/math]
[math] =\omega C V_0 Re \left [ e^{i \left ( \omega t \right )} \left ( 0 + i \right )\right ][/math]
[math] =\omega C V_0 Re \left [ i e^{i \left ( \omega t \right )} \right ][/math]
[math] =Re \left [ V_0 \frac{e^{i \left ( \omega t \right )}}{\frac{1}{i\omega C}} \right ]=Re \left [ V_0 \frac{e^{i \left ( \omega t \right )}}{X_C} \right ][/math]

where

[math]X_C = \frac{1}{i \omega C}[/math] Effective impedance of a Capacitor
Observe
At high frequencies [math](\omega)[/math] the effective impedance (X_C) is small so the driving voltage should pass right through.
At low frequencies [math](\omega)[/math] the effective impedance (X_C) is large so the driving voltage will be severely attenuated.
Capacitors block Alternating Currents so you can reduce ripples on DC voltages.
You could measure the capacitance by measuring [math](I/V)^2[/math] for several frequencies

RC circuit

The Low pass Fiter

Now add a resistor in series with the capacitor in the above capacitor charging circuit so it resembles the circuit below

TF EIM Lab3.png

Quick Solution

Applying the loop theorem using the reactance[math] X_C[/math] of the capacitor instead of Q/C


[math]V - IR -X_CI = =V -(X_R +X_C) I = 0[/math]

[math]\Rightarrow[/math] the capacitor can be added in series with the other resistor
[math]X_{tot} = X_R + X_C[/math]

It looks like the voltage divider from the resistance section

[math]V_{out}= Re\left [\frac{X_C}{R+X_C} V_{in} \right ]= Re \left [\frac{X_C}{X_R+X_C} V_{in} \right ][/math]

To evaluate [math]Re[Z] = \sqrt{Z Z^*}[/math] where[math] Z = x+iy[/math] and [math]Z^* = x-iy[/math]


[math]\frac{V_{out}}{V_{in}} = Re \left [ \frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \right ] = \sqrt{\frac{\frac{1}{i \omega C}}{R+\frac{1}{i \omega C}} \frac{\frac{1}{-i \omega C}}{R+\frac{1}{-i \omega C}} } = \frac{1}{\sqrt{1 + \omega^2 R^2 C^2}}[/math]

Let

[math]\omega_b = \frac{1}{RC} =[/math] break point (cut off ) frequency

then

[math]\frac{V_{out}}{V_{in}} = \frac{1}{\sqrt{1 + \left ( \frac{\omega}{\omega_b}\right )^2}}= \frac{1}{\sqrt 2}[/math]

TF EIM LowPassVoutVin-vs-omega.png


Note
The RC circuit is another way to measure the capacitance by measureing [math]V_{out}/V_{in}[/math] -vs- [math]\omega[/math] and do a linear extrapolation to determine the break point frequency.

Phasor Diagram

Phasor diagrams are a technique used to illustrate the phase difference between the input and output alternating voltages/currents.


Returning to the loop theorem for the low pass RC filter

[math]V_{in} = V_R + V_{out}[/math]
[math]V_{in} = IX_R + IX_C[/math]
[math]V_{in} = IR + I\frac{-i}{\omega C}[/math]

We can write the voltage as a complex variable using a coordinate system in which the "x-axis" represents real numbers and the "y-axis" represents the complex components. This is analogous to expressing 2-D vectors.

Below is a figure representing the complex plane.

TF EIM RC PhasorDiag.png

The angle [math]\phi[/math] is the angle between[math] V_{out} = \frac{-i}{\omega C}[/math] and [math]V_{in} = IR + I\frac{-i}{\omega C}[/math] Based on the above Phasor diagram

[math]\tan(\phi) = \left (\frac{R}{\frac{-1}{\omega C}} \right )= \left(-\omega RC \right )[/math]

High Pass Filter

A high pass filter switches the order of the resistor and the capacitor.

TF EIM Lab4.png


Using the same reactance based analysis as above

[math]V_{in} - IX_C - IR = 0[/math]
[math]V_{out} = IR[/math]
[math]\frac{V_{out}}{V_{in}} = \frac{IR}{I \left ( X_C + R \right ) }= \frac{R}{\frac{1}{i \omega C} + R} = \frac{ i \omega RC}{1 + i \omega RC}[/math]
[math]\left |\frac{V_{out}}{V_{in}} \right | = \sqrt{\left ( \frac{ i \omega RC}{1 + i \omega RC}\right ) \left ( \frac{- i \omega RC}{1 - i \omega RC}\right )} = \frac{\omega RC}{\sqrt{(1 + \omega^2 R^2C^2}}[/math]


Phasor diagram

The phasor diagram is very similar except this time V_{out} is along the x-axis


Returning to the loop theorem for the low pass RC filter

[math]V_{in} = V_R + V_{out}[/math]
[math]V_{in} = IX_C + IX_R [/math]
[math]V_{in} = I\frac{-i}{\omega C}+IR[/math]

We can write the voltage as a complex variable using a coordinate system in which the "x-axis" represents real numbers and the "y-axis" represents the complex components. This is analogous to expressing 2-D vectors.

Below is a figure representing the complex plane.

TF EIM RC PhasorDiagHigh.png

The angle [math]\phi[/math] is the angle between [math]V_{out} =IR[/math] and [math]V_{in} = I\frac{-i}{\omega C} + IR [/math] Based on the above Phasor diagram

[math]\tan(\phi) = \left (\frac{\frac{-1}{\omega C}}{R} \right ) = (\frac{-1}{\omega RC})[/math]

High Pass -vs- Low Pass graphs

if

[math]\omega_B=\frac{1}{RC}=10[/math]

Then

[math]\phi_{\mbox{Low Pass}} = \tan^{-1} \left ( -\frac{\omega}{\omega_B}\right )[/math]
[math]\phi_{\mbox{High Pass}} = \tan^{-1} \left ( -\frac{\omega_B}{\omega}\right )[/math]
Low pass High Pass
Gain -vs- [math]\omega[/math]
Gain -vs- [math]\omega[/math]
Phase Shift -vs- [math]\omega[/math]
Phase Shift -vs- [math]\omega[/math]

exact solution

Again apply the loop theorem

[math]V -IR -\frac{Q}{C} =0[/math]

or

[math]R \frac{dQ}{dt} + \frac{1}{C} Q =V(t)[/math] First order non-homogeneous linear differential equation


First solve homogeneous part

[math]R \frac{dQ}{dt} + \frac{1}{C} Q =0[/math] First order homogeneous linear differential equation

The above is integratable

[math] \frac{dQ}{dt} = -\frac{1}{RC} Q [/math]
[math]\int \frac{dQ}{Q} = -\int \frac{1}{RC} dt [/math]
[math]\ln Q = -\frac{t}{RC} -Constant[/math]
[math] Q(t) = e^{-\frac{t}{RC}} e^{Constant} = Q_0e^{-\frac{t}{RC}}[/math]
[math] I(t)= \frac{d Q}{d t} = \frac{d}{dt} Q_0 e^{-\frac{t}{RC}} = \frac{-Q_0}{RC}e^{-\frac{t}{RC}}[/math]

This gives you the general solution for the homogeneous problem

In this case it represents the solution for a discharging capacitor.


Trial solution method for Non-Homogeneous Solution

A common and more general technique to solving such a differential equation is to use the solution from the discharging equation (the homogeneous diff. eq.) as a trial solution to the charging (non-homogeneous) problem by constructing the linear combination below.

[math]Q_{trial} = A + Be^{-t/RC}[/math]

Where A & B are constants to e determined

You then substitute this trial solution into the differential equation and apply boundary conditions to determine the coefficients.

[math]\frac{d}{dt} Q+ \frac{1}{RC}Q = \frac{V}{R}[/math]
[math]\frac{d}{dt} \left ( A + Be^{-t/RC} \right ) + \frac{1}{RC} \left ( A + Be^{-t/RC} \right ) = \frac{V}{R}[/math]
[math] \left ( B \frac{-1}{RC}e^{-t/RC} \right ) + \frac{1}{RC} \left ( A + Be^{-t/RC} \right ) = \frac{V}{R}[/math]
[math]\Rightarrow \frac{1}{RC} \left ( A \right ) = \frac{V}{R}[/math]
[math]A = C V[/math]

To determine the other coefficient you apply the boundary condition that q = 0 when t=0

[math]0= A + Be^{-0/RC} = A+B[/math]
[math]A=-B[/math]
[math]Q=CV (1 - e^{-t/RC}) =Q_0 (1 - e^{-t/RC})[/math]

Inductance

Inductor/Choke

An inductor is basically a loop of wire wrapped around a core (or air) to form a coil.


TF EIM BasicIncuctorCoil.jpg

Consider what happens in the above circuit when you first connect a voltage source (or close the switch).

1.) A current starts to flow through the wire such that I=V/R (R can be the internal resistance of the battery which changes with time).

2.) A current produces a magnetic field

Ampere's Law
[math]\oint \vec{B} \cdot d \vec{l} = \mu_0 I[/math]

If the current distribution isn't symmetric enough to make the line integral easy

then you may need to use

Biot-Savart Law
[math]d\vec{B} = \frac{\mu_0 I}{4 \pi} \frac{\vec{l} \times \vec{r}}{r^2}[/math]


Most inductors take the geometrical form of a solenoid.

TF EIM SolenoidAmperslaw.jpg


[math]\oint \vec{B} \cdot \vec{l} = \int_a^b \vec{B} \cdot d\vec{l} + \int_b^c \vec{B} \cdot d\vec{l} + \int_c^d \vec{B} \cdot d\vec{l} + \int_d^a \vec{B} \cdot d\vec{l} = \mu_0 I_{tot}[/math]
[math]\int_a^b \vec{B} \cdot d\vec{l} = B\ell[/math]
[math]\int_b^c \vec{B} \cdot d\vec{l} = \int_d^a \vec{B} \cdot d\vec{l} =0 : \vec{B} \perp d\vec{l}[/math]
[math]\int_c^d \vec{B} \cdot d\vec{l} =0 : B \approx 0[/math]
[math] I_{tot} = NI[/math] = total current enclosed by the loop is I times the number of loops N
[math]B = \mu_0 \frac{NI}{\ell} = \mu_0 nI[/math]

where

[math]n \equiv \frac{N}{\ell} =[/math] number of turns per unit length

3.) When using an AC source the current is changing with time and as a result the B-field is also changing in time

4.) The changing current induces and emf ([math]\mathcal{E}[/math])

Faraday's law of induction says

[math]\mathcal{E} = - \frac{d \Phi_B}{dt}[/math] = self induced emf

where

[math]\Phi_B = \int \vec{B} \cdot d \vec{A}[/math]
[math]\mathcal{E} = - \frac{d }{dt} \int \vec{B} \cdot d \vec{A}[/math]: There are 3 ways to induce an emf (B and A change with time or the angle between them)

The B-field is usually perpendicular to the area of the induction and mostly uniform.


[math]\mathcal{E} = - \frac{d }{dt} BA = - \frac{d }{dt} \mu_o nIA = - \mu_0nA \frac{dI}{dt} [/math]

The above is the emf induced in a single loop

If you assume there are N loop in the inductor
Then
[math]\mathcal{E} = - N\frac{d }{dt} \mu_o nIA = - N\mu_0nA \frac{dI}{dt} \equiv -L \frac{dI}{dt}[/math]


[math]L = N\mu_0 n A = \mu_0 \frac{N^2}{\ell} \pi R^2[/math] = Inductance

where

[math]\ell=[/math] length of loop
[math]R =[/math] radius of coil

Lenz's law says that the induced current will appear in a direction that opposes the change producing it.

So in essence the inductor acts "like" a resistor which resists the current flow according to the amount of current is changing.

There is an effective reactance [math]X_{L}[/math] for inductors as there was one for capacitors

Inductor Reactance

Consider an AC voltage source with an inductor attached.


Loop theorem

[math]\Rightarrow V - L \frac{dI}{dt} = 0[/math]
[math]V_0 cos(\omega t) dt = L dI[/math]
[math]\int_0^t V_0 cos(\omega t) dt =\int_0^t L dI[/math]
[math]\left . V_0 \frac{sin(\omega t)}{\omega} \right |_o^t = L \left . I(t)\right |_0^t[/math]
[math]V_0 \frac{sin(\omega t)}{\omega} -0 = L \left ( I(t) - I(0)\right )[/math]
[math]\Rightarrow I(t) = V_0 \frac{sin(\omega t)}{\omega L} + I(0)[/math]

Assume I(0) =0 and rewrite [math]\sin[/math] in terms of a phase shifted [math]\cos[/math]

[math]\Rightarrow I(t) = \frac{V_0}{\omega L} cos(\omega t - \pi/2) [/math] : V and I are 90 degrees out of phase
[math]= Re \left ( \frac{V_0}{\omega L} e^{i(\omega t - \pi/2)}\right )[/math]
[math]= Re \left ( \frac{V_0}{\omega L} e^{i(\omega t)} (-i)\right )[/math]
[math]= Re \left ( \frac{V_0}{i\omega L} e^{i(\omega t)}\right )[/math]
[math]X_L \equiv i\omega L[/math]
Note
Large L makes it hard to change current
Small L makes it easy to change current

parallel LC circuit (Band Stop/Notch Filter/ tank circuit)

TF EIM Parallel RLC.png

The above circuit has a resistor in series with an inductor and capacitor in parallel.

Using the complex analysis techniques for AC circuits you can consider the Inductor and Capacitor as impedances in parallel.

so

[math]\frac{1}{X_{tot}} = \frac{1}{X_C} + \frac{1}{X_L} = \frac{i \omega C}{1} + \frac{1}{i \omega L} = \frac{1- \omega^2 LC}{i \omega L}[/math]
[math] X_{tot} = \frac{i \omega L}{1- \omega^2 LC} = \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}}[/math]
[math]\omega_{LC} \equiv \frac{1}{\sqrt{LC}}[/math]

or

[math]\nu_{LC} = \frac{\omega_{LC}}{2\pi} = \frac{1}{2 \pi \sqrt{LC}}[/math]



[math]L = 33 \mu H[/math] and [math]C = 1 \mu F [/math]
TF EIM Impedance LC Parallel.png
[math]\Rightarrow \omega = 174077.66[/math] rad/s


Phase differences

Notice
[math]X_C = \frac{1}{i \omega C}[/math] & [math]X_L = i \omega L[/math]
[math]I = \frac{V}{X}[/math]
[math]I_C = \frac{V}{\frac{1}{i \omega C}} = i \omega CV[/math]
[math]I_L = \frac{V}{i \omega L} = \frac{-iV}{\omega L }[/math]

The current through the inductor side of the circuit is 180 degrees out of phase with the capacitor side.

gain

Loop Theorem

[math]\Rightarrow V= I(R+X_{tot}) = I \left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )[/math]

or

[math] I= \frac{V_0 e^{i \omega t}}{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )}[/math]
Notice
When [math]\omega \approx \omega_{LC} = \sqrt{\frac{1}{LC}}[/math] then the AC signal is attenuated.

Looking at the Voltage divider aspect of the circuit

[math]V_{AB}=V_{out} = \frac{X_{tot} }{R + X_{tot}}V_{in}[/math]
[math]\left |\frac{ V_{out}} {V_{in}}\right | = \sqrt{ \left [ \frac{X_{tot} }{R + X_{tot}} \right ] \left [ \frac{X_{tot} }{R + X_{tot}} \right ]^*}[/math]
[math] = \sqrt{ \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ] \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*}[/math]
[math] = \sqrt{ \frac{ \omega^2 L^2 \omega_{LC}^4}{R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2L^2 \omega_{LC}^4}}[/math]
[math] = \sqrt{ \frac{ \omega^2 }{C^2R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}[/math]
[math] = \sqrt{ \frac{ \omega^2 }{\frac{1}{\omega_{RC}^2}(\omega_{LC}^2 - \omega^2)^2 - \omega^2}}[/math]



[math]L = 33 \mu H[/math] and [math]C = 1 \mu F [/math] and R=200 [math]\Omega[/math]
TF EIM LC paral Gain.png
[math]\Rightarrow \omega = 174077.66[/math] rad/s or [math]\nu = \frac{\omega}{2 \pi} = 27705[/math] Hz

Q and Bandwidth

In the above circuit

[math]\nu = \omega/2 \pi = \frac{1}{2 \pi \sqrt{LC}}= 27705 Hz[/math]

The inductors reactance at this resonance frequency is

[math]X_{L} = \omega L = 2 \pi \nu L = \frac{L}{\sqrt{LC}} = \sqrt{\frac{L}{C}} = \sqrt{\frac{33 \times 10^{-6} H}{1 \times 10^{-6}F}} = 5.7 \Omega[/math]


Bandwith
The most common definition for the Bandwidth of this circuit is the frequency range over which the output decreases by 3 dB.
Decibels
The decibel system is a logarithmic scale for the ratio of two quantities
For power
dB = [math]10 \log_{10} \left ( \frac{P_{in}}{P_{out}}\right )[/math]
For amplitude (Voltage)
dB = [math]20 \log_{10} \left ( \frac{V_{in}}{V_{out}}\right )[/math]


A 3dB drop corresponds to the frequency at which the circuits power is cut in half from the resonance frequency.

[math]P = I^2 R[/math]
[math]P_{1/2} = (\frac{1}{\sqrt{2}}I)^2 R= \frac{(\frac{1}{\sqrt{2}} V)^2}{R}[/math]
[math]\Rightarrow V = \frac{1}{\sqrt{2}} V = 0.71 V[/math]
The Band Width is a measurement of the two frequencies which will decrease the resonance voltage by 30 %.

TF EIM BandWidthDef LC.gif

Quality Factor
The Quality factor for this circuit may be expressed in term of the bandwidth such that
[math]Q=\frac{\omega_0}{BW}=\frac{\omega_0}{\Delta \omega}[/math]


Notice
The Bandwidth can be changed in the above circuit by changing the step down resistor (R) without changing the resonance frequency.

Quality Factor (Q)

The Quality factor of a circuit is defined in terms of a ratio between the energy stored to the energy lost.

[math]Q \equiv 2 \pi \frac{\mbox{Energy Stored per cycle}}{\mbox{Energy lost per cycle}} = 2 \pi \frac{W_S}{W_L}[/math]
How much energy does an inductor store?
[math]W = \mbox{Work} = \int \vec{F} \cdot d\vec{r} = \Delta U = \int V dQ = \int L \frac{d I}{dt} (I dt) = L \int_0^I I dI = \frac{1}{2} LI^2[/math]
[math]W_S = \frac{1}{2} LI^2[/math]


How much energy does an inductor lose?

The system loses energy through heat via power dissipation so in general

[math]W_L = \int p(t) dt = \int_0^T I^2(t) R dt [/math] where R is the effective resistance of the circuit


Note
A circuit with high Q will have amplitude (or blocking) at the desired resonance frequency and the least amplitude (or blocking) at non-resonant frequencies.

An Inductor in series with a capacitor is known as a Band Pass filter. Since wires are used to hook everything up and real capacitors have leakage resistance so an RLC circuit is a realistic version of the series LC circuit.

RLC circuit

An RLC circuit is a Resistor, an Inductors, and a Capacitor in series with an electromotive force.

TF EIM Lab5 RLC.png


Effective impedance

[math]X_{out} = R_L + X_C + X_L = R_L + \frac{1}{i \omega C} + i \omega L[/math]
[math]\left | X_{out} \right | = \sqrt{\left [ R_L + i \left (\frac{-1}{\omega C} + \omega L\right ) \right ]\left [ R - i \left (\frac{-1}{\omega C} + \omega L\right ) \right ]^*}[/math]
[math]= \sqrt{ R_L^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2}[/math]

Gain

Loop Theorem

[math]V_{in} = I (R+ X_{out})[/math]


Voltage Divider

[math]V_{AB}=V_{out} = \frac{X_{out}}{R + X_{out}}V_{in}[/math]


[math]\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{X_{out}}{R + X_{out}}\right ]\left [ \frac{X_{out}}{R + X_{out}}\right ]^*}[/math]

[math]R_L + i \left ( \omega L - \frac{1}{\omega C}\right)[/math]

[math]\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]^*}[/math]
[math] = \sqrt{\frac{R_L^2 + \left ( \omega L - \frac{1}{\omega C}\right)^2}{(R + R_L)^2 + \left ( \omega L - \frac{1}{\omega C}\right)^2}}[/math]
[math] = \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}{(R + R_L)^2 + \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}}[/math]

Let

[math]\omega_0 = \sqrt{\frac{1}{LC}}[/math]

Then

[math]\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 - \omega_0^2}{\omega_0^2 \omega C}\right)^2}{(R + R_L)^2 + \left ( \frac{\omega^2- \omega_0^2}{\omega_o^2 \omega C}\right)^2}}[/math]

When[math] \omega = \sqrt{\frac{1}{LC}} = \omega_0[/math]

Then

[math]\left | \frac{V_{out}}{V_{in}}\right | =\frac{R}{R+R_L}[/math]

TF EIM RLC.png

Phase shift

[math]X_C = \frac{-i}{\omega C} \Rightarrow[/math] Voltage lags current for a capacitor

[math]X_L = i \omega L \Rightarrow[/math] Voltage leads current for Inductor

TF EIM RLC Phasor.png


[math]\tan(\phi) = \frac{X_{out}}{R_L} = \frac{\omega L - \frac{1}{\omega C}}{R}= \frac{\omega^2 - \omega_o^2}{\omega_0^2R}[/math]


If [math]\omega \lt \omega_0[/math] then the phase shift is negative

If [math]\omega \gt \omega_0[/math] then the phase shift is positive


Forest_Electronic_Instrumentation_and_Measurement