ECE 3336 Introduction to Circuits & Electronics

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ECE 3336 Introduction to Circuits & Electronics. Note Set #11 Frequency Response . Fall 2013, TUE&TH 4 : 0 0-5: 3 0 pm Dr. Wanda Wosik. Frequency Dependence is Important in Electronics.

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ECE 3336 Introduction to Circuits & Electronics

Note Set #11

Frequency Response

Fall 2013,

TUE&TH 4:00-5:30 pm

Dr. Wanda Wosik

Frequency Dependence is Important in Electronics

Frequency dependence is seen in many non-electrical phenomena (resonant vibrations, mechanical oscillations, damping, rotations and many, many others).

Frequency dependence is also “frequently” present and very important in electronic circuits.

Electrical signals of various frequencies used for various applications are being processed by circuits designed specifically to operate at these frequencies.

Examples include dependences between voice frequency and tone , frequency and color of light

We will study behavior of circuits when signal frequency changes.

We will use phasors and impedances to analyze acsignals that are delivered to the circuits and propagate there.

Circuits will perform various functions, and finally the signals will appear (or not i.e. will disappear) at the output.

We would like to monitor these ac signals in circuits that operate in the steady-state conditions.

We have to develop a tool that would let us predict the behavior of the output signalfor any input signal that is supplied to the input.

This means that, at the beginning, we will be rather analyzing circuits not the specific signals.

Clear? Probably NOT yet.

Signals of various frequencies and amplitudes applied to the circuit input may appear at the output terminals with different amplitudes and with their phases shifted compared with the input signal.

• Depending on a circuit, some input signals may appear at the output with smaller amplitudes or not appear at all. We say that these signals are attenuated.
Frequency Response

Attenuation and distortion are the consequences of capacitive and inductive behavior of various circuit elements.

They can be just simple capacitors and/or inductors in the circuit but more frequently there are parasitics that result in such effects.

(Fig. 6.15 from Rizzoni’s book).

How to Deal with Frequency Dependence in Circuits?
• If multiple frequencies, such as those from a CD player, are included in the input signal, we may have a distortion at the output since some frequencies can be lost by attenuation.

Source

So, it is wise to determine a Frequency Response or a Transfer Function Hv(j)that will work for a given circuit. This transfer function will affect any ac steady state signal applied to the input.

Use Phasors to Find the Frequency Response

To predict the frequency behavior of a circuit, we define the transfer function that relates the output (load) voltage to the input voltage in the frequency domain.

Th. Eq.

Now, to find VL we will first find Thevenin equivalent

Transfer Function Characterizes

a Circuit not a Signal

The transfer function HV(j) is a complex number that has a magnitude andphase

Once we derive HV(j) for a circuit

(NOT FOR A SIGNAL!)

we will find an output signal VL(j) for any input signal Vs(j) .

The output voltage (load) VL(j)is calculated from the source voltage Vs(j) multiplied by the transfer function using the complex numbers rules.

where

Is amplitude

Is phase

Periodic Signals Represented by Fourier Series

The goal is to find the Transfer Function for periodic signals.

A periodic signal is shown below.

n=1, 2, 3, ….; T=period

We will use Fourier theorem, which allows for representation of periodic signals as a superposition of various sinusoidal components of various frequency and different amplitudes.

Fourier Analysis

Any periodic signal will be represented by superposition of infinitive number of sinusoidal signals expressed by a Fourier Series

Sine-cosine representation

Magnitude and phase representation

In Fourier Series:

Fundamental frequency 0=2f0=2/T

Harmonics:

20,30, 40

Equivalent coefficients

For any signal, the coefficients a0, an and bn have to be calculated in order to use Fourier series to represent this signal.

Square Wave Expanded in Fourier Series

Example: For a square wave, which is odd function an=0 while bn≠0.

So the signal will be represented by a sum of sine waves:

Each consecutive harmonic has a decreasing amplitude and increasing frequency.

Increasing number of harmonics allows for improved representation of the non-sinusoidal signals. Infinitive number of harmonics will give a perfect match.

You may want to see More Examples of Signals Represented by a Fourier Series http://www.stat.ucla.edu/~dinov/courses_students.dir/04/Spring/Stat233.dir/Stat233_notes.dir/JavaApplet.html to play with harmonics

Other Examples of Signals Represented by a Fourier Series

Sawtooth function

From the calculated coefficients an=0 and

we have v(t)

Notice how the amplitude of harmonicsdecreases

We can calculate the amplitudes cn and the phases n

More Examples of Signals Represented by a Fourier Series

Coefficients

Pulse train signal

Amplitude of harmonics

Matching will improve with the number of harmonics included in the Fourier Series

Phase of harmonics

You may want to see More Examples of Signals Represented by a Fourier Series http://www.stat.ucla.edu/~dinov/courses_students.dir/04/Spring/Stat233.dir/Stat233_notes.dir/JavaApplet.html to play with harmonics

Response of Linear Systems to Periodic Input Signals

Periodicsignals applied to the input of a circuit, which has capacitors or inductors (real or parasitic) will be modified by this circuit.

The output signal will have modified amplitude and phase compared to the original signal at the input.

We will use the Fourier series to represent such signals.

If we take a sinusoidal signal at the input of a linear system, represented by a finite number of components

we can trace the Qout(j) output signal if we know both the input signal Qin(j) and the transfer function H(j).

Filters
• Filters are very important circuits that allow for selective attenuation of signals of a specific frequency or of specific frequency ranges. All other signals with other frequencies will pass without attenuation.
• Magnitudeof the transfer function |H(j)| affects the amplitude of the output signal.
• Phaseof thetransfer function |H(j)| corresponds to the phase shift between the output and input signals.

A two port circuit

where

H(jw)

Is amplitude

Is phase

Filters
• The frequency blocks marked in red in the figure below, indicate the passing frequencies. Signals with all other remaining frequencies will be stopped by the circuit so they will not appear at the output.

|H(j)|

|H(j)|

low

stopband

high

stopband

pass

pass

A low-pass filter passes

Only low frequency signals

A high-pass filter passes only high frequencies

Other filters

A very important family of filters include resonant circuits, which have quite significant applications of electronics.

They select specific frequencies to be either passed or frequencies to be attenuated.

They usually require all three elements: capacitors, inductors, and resistors. We will call them second order filters.

|H(j)|

|H(j)|

band

band

rejection

pass

|H(j)|

|H(j)|

Very narrow bands

passing

rejection

Resonance filters

Which Filters Will We Cover Here?
• First order filters operating as low-pass and high-pass filters. These are made either as RC and RL filters. That means that capacitors and inductors will not be included simultaneously.
• Bandpass and resonant filters. They will have (usually, but not always) all three elements included: an inductor, capacitor and resistor.
• Active filters. They have very similar operation but they also can amplify the amplitude of the signals that pass through the circuit. They are built using operational amplifiers. This will be done later in the course (Notes #13&14).
Low-pass filter

Output

Slow will pass

Fast will be stopped

Low-pass Filter

To monitor the frequency dependence of this filter we need to derive H(j)

To find H(j) we have to find V0(j) as a a function of the input signal Vi(j).

So we will use a voltage divider

Now we can estimate the magnitude of H(j) at very low and very high frequency:

For =0 (DC conditions) H(j)=1

Here the capacitor acts as an open circuit

For ∞ |H(jw)|  0

The capacitor acts as a short (Z=1/jC)

Therefore the transfer function is:

Low-pass filter

Our transfer function will be now expressed using the capacitance and resistance.

AND

We will find its magnitude and phase.

Phase

Magnitude

• We will select a new parameter called: cutoff frequency=1/RC.
• This will be a reference pointfrom which we will monitor amplitude and phase changes in out output signal.

Frequency Response for the Low-pass Filter

We will now find the magnitude and phase of H(j) usingthe expression for the breakpoint frequency=1/RC.

.

We will calculate the

Magnitude of the transfer function

0.707

At 0

0

We will then calculate the

Phase of the transfer function

-45°

At ~0 Phase=0°

At 0 Phase=-45°

At 0 Phase=-90°

0

High-pass Filters

Output

Fast will pass

Slow will be stopped

High-pass Filter

To monitor the frequency dependence of this filter we need to derive the Transfer Function H(j). We will use the same approach as we used for the low-pass filter.

To find H(j) we have to find V0(j) as a function of Vi(j).

So again we will use a voltage divider

Now we can estimate the magnitude of H(j) at very low and very high frequency:

For =0 (DC conditions) H(j)=0

Here the capacitor acts as an open circuit

For ∞ |H(jw)|  1

The capacitor acts as a short (Z=1/jC)

Therefore the transfer function is:

We will find the magnitude and phase of H(j).

Frequency Behavior of a High-pass Filter

phase

magnitude

• As for the low-pass filter
• We will again select a new parameter called: cutoff frequency=1/RC.
• This will be a reference pointfrom which we will monitor amplitude and phase changes in our output signal.

We will first rewrite the expression for the transfer function by introducing 0=1/RC

Magnitude and Phase of the Transfer Function

Calculate the

Phaseof the transfer function

Calculate the

Magnitudeof the transfer function which at 0 is:

At ~0 Phase=90°

At 0 Phase=45°

At 0 Phase=0°

That's why 0 is also called half-power frequency. Here |Vo(j)|~|H(j)| and the power delivered by V0 is divided by

Notice: The phase shift coincides with the change in the magnitude. No change in the magnitude no change in the phase.

45°

0.707

Amplitude

phase

Bode Plots

Bode Plots will be used to simplify graphical representation of the magnitude and phase of the transfer functions. We will use them instead of plotting directly the results from calculation of the |H(j)| and H(j).

Bode Plots will give us an approximation of the transfer function in the broad range of frequency changes. Accuracy will be acceptable even at breakpoint frequencies.

These are straight line approximations both for the magnitude and phase of the Transfer Function. That allows us to easily predict its frequency dependence.

Magnitude

note the linear scale

Phase

Towards Derivation of Bode Plots

Back to a low-pass filter composed only of one capacitor and one resistor.

Earlier we have derived its transfer function:

We also calculated and plotted its magnitude |H(j)| and phaseH(j).

0.707

magnitude

-45°

phase

Now we will use new approximations for the magnitude and the phase instead of calculating step by step the transfer characteristics.

To do that we will now use decibels (dB) defined as:

Derivation of Bode Plots

These are approximations for the magnitude and the phase – accurate ones!

using decibels (dB), which are defined as:

Since the transfer function H(j) is a ratio of the two voltages and it is a complex number, we will calculate the magnitude of this number |H(j)|

using decibels. We will obtain |H(j)|dB

Plotting Bode Plots of the Transfer Function (low pass)

We will plot the MAGNITUDE at selected multiples of 0.

Then we will plot the PHASE

For <<0 |H(j)|dB=-20log10(1)=0 dB

At =0 |H(j)|dB=-20log10√(1+1)=-3 dB

=0/10

H(j-tan-1(1/10)=0°

For >> 0

=0

H(j-tan-1(1)=-45°

=100 |H(j)|dB=-20log(10)=-20 dB

=100

H(j j-tan-1(10)=-90°

|H(j)|dB=-20log(100)=-40 dB

=1000

=10000

etc.

-

-

The influence of 0 is seen in |H(j)| for all subsequent frequencies.

The 0 affects the phase only locally: within two decades only

-

-20dB/dec

-45°

-45°/dec

-

-

-

0

0

-

-90°

-

also known as 3dB frequency

-

Plotting Bode Plots of the Transfer Function (high pass)

Plot the MAGNITUDE at selected fractions of 0.

Plot the PHASE

At =0 |H(j)|dB=-20log10√(1+1)=-3 dB

H(j90°-tan-1(1/10)=0°

=0/10

=0/10|H(j)|dB=-20log(10-1)=-20 dB

H(j90°-tan-1(1)=45°

=0

H(j j90°-tan-1(10)=0°

|H(j)|dB=-20log(10-2)=-40 dB

=100

=0/100

>>0

|H(j)|dB=0

The influence of 0 is seen in |H(j)| for all frequencies <<0.

The 0 affects the phase only locally: within two decades only

+20dB/dec

0

aka 3dB frequency

Example: High pass filter

We start with the current, which is the same through the capacitor and resistor

Which for low frequencies i.e. when <<otherefore vi(t)>>vo(t), becomes

So the high-pass filter acts then as a differentiator.

RC

Short time constant RC gives distortion; a square wave with a very small amplitude appears at the output.

Increasing RC will make both the signals increasingly the same

Increasing time constant RC

or equivalently: decreasing

=1/RC breakpoint frequency of the circuit, makes the output signal lessdistorted compared to the input signal of a specific frequency. Exponential decay.

RC

Differentiator
• Frequency of the input signal changes
• from low
• through medium
• to high

http://www.electronics-tutorials.ws

Example: Low pass filter

We start again with the current, which is the same through the resistor and capacitor

Which for high frequencies i.e. when >>o therefore vi(t)>>vo(t), becomes

Long RCtime constant gives distortion i.e. a triangular wave with a very small amplitude appears at the output

But decreasing RC will make both the input and output signals increasingly the same.

So the high-pass filter acts then as a integrator.

Decreasing time constant RC

or equivalently: increasing

=1/RC breakpoint frequency of the circuit, makes the output signal less distorted compared to the input signal of a specific frequency.

RC

RC

Integrator
• Frequency of the input signal changes
• from low
• through medium
• to high

http://www.electronics-tutorials.ws