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Electric Circuits (Fall 2015) Pingqiang Zhou
Lecture 13
- Frequency Response
12/1/2015
Reading: Chapter 14
1Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Outline
• Frequency response
• Resonance
• Filters
Lecture 13 2
Electric Circuits (Fall 2015) Pingqiang Zhou
Frequency Response
• When a linear, time invariant (LTI) circuit is excited by a sinusoid, it’s
output is a sinusoid at the same frequency.
Only the magnitude and phase of the output differ from the input.
• The “Frequency Response” is a characterization of the input-output
response for sinusoidal inputs at all frequencies.
Significant for applications, esp. in communications and control systems.
• Filters play critical roles in blocking or passing specific frequencies or
ranges of frequencies.
Widely used in radio, TV and phone systems.
3Lecture 13[Source: Berkeley]
Electric Circuits (Fall 2015) Pingqiang Zhou
Magic of Sinusoids
• All real-world signals are sums of sinusoids components
that have proper amplitudes, frequencies and phases.
To be found by Fourier Analysis.
Most natural signals contain thousands of components.
4Lecture 13[Allan, 6th ed.]
Electric Circuits (Fall 2015) Pingqiang Zhou
Representing a Square Wave as a Sum of Sinusoids
[Allan, 6th ed.]
Electric Circuits (Fall 2015) Pingqiang Zhou
Frequency Ranges of Selected Signals
• When we listen to music, our ears respond differently to
the various frequency components: some pleasing,
whereas others are not.
Design of signal-processing circuits for audio signals must consider
how the circuit responds to components of different frequencies.
6Lecture 13[Allan, 6th ed.]
The components can be determined by theoretical analysis or
by laboratory measurements (using Spectrum Analyzer).
Electric Circuits (Fall 2015) Pingqiang Zhou
Transfer Function – Voltage Gain
• One useful way to analyze the frequency response of a
circuit is the concept of the transfer function.
• It is a function of frequency
Complex quantity
Both magnitude and phase are function of frequency
Two Port
networkVin Vout
( )
( )
outout in
in
Vf
V
H f
out
in
VH
V
H(f)
Lecture 13 7
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• The transfer function 𝐻 𝑓 is shown below. If the input
signal is 𝑣𝑖𝑛 𝑡 = 2 cos 2000𝜋𝑡 + 40° , find an expression
as a function of time for the output.
8Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• The transfer function 𝐻 𝑓 is same as shown on previous
slide. If the input signal is
𝑣𝑖𝑛 𝑡 = 3 + 2 cos 2000𝜋𝑡 + cos 4000𝜋𝑡 − 70° ,
find an expression as a function of time for the output.
9Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
10Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Transfer Function – More General Definition
• The transfer function H(ω) is the frequency-dependent ratio
of a forced function Y(ω) to the forcing function X(ω).
YH
X
Voltage gain
Current gain
Transfer impedance
Transfer admittance
o
i
o
i
o
i
o
i
VH
V
IH
I
VH
I
IH
V
Lecture 13 11
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
12Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example: Plot
13Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Exercise
• Obtain the transfer function 𝐕𝐨/𝐕𝐬 of the RL circuit.
Assuming 𝑣𝑠 = 𝑉𝑚 cos𝜔𝑡.
14Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Zeros and Poles
• H(ω) can be expressed as the ratio of numerator N(ω)
and denominator D(ω) polynomials.
Zeros are where the transfer function goes to zero.
Poles are where it goes to infinity.
They can be related to the roots of N(ω) and D(ω).
NH
D
YH
X
Lecture 13 15
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• For the circuit below, calculate
the gain 𝐈𝐨 𝜔 /𝐈𝐢 𝜔 and its
poles and zeros.
16Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Decibel Scale
• We will soon discuss Bode plots.
• These plots are based on logarithmic scales.
• The transfer function can be seen as an expression of
gain.
Gain expressed in log form is typically expressed in bels, or more
commonly decibels2
10
1
10logdB
PG
P
Lecture 13 17
Electric Circuits (Fall 2015) Pingqiang Zhou
Bel and Decibel (dB)
• A bel (symbol B) is a unit of measure of ratios of power
levels, i.e. relative power levels.
The name was coined in the early 20th century in honor of
Alexander Graham Bell, a telecommunications pioneer.
The bel is a logarithmic measure. The number of bels for a given
ratio of power levels is calculated by taking the logarithm, to the
base 10, of the ratio.
– one bel corresponds to a ratio of 10:1.
– B = log10(P1/P2) where P1 and P2 are power levels.
• The bel is too large for everyday use, so the decibel (dB),
equal to 0.1B, is more commonly used.
dB = 10 log10(P1/P2)
dB are used to measure
– Electric power, Gain or loss of amplifiers, Insertion loss of filters.
Lecture 13 18
Electric Circuits (Fall 2015) Pingqiang Zhou
Logarithmic Measure for Power
• To express a power in terms of decibels, one starts by
choosing a reference power, Preference, and writing
• Exercise:
Express a power of 50 mW in decibels relative to 1 watt.
P (dB) =10 log10 (50 x 10-3) = - 13 dB
Express a power of 50 mW in decibels relative to 1 mW.
P (dB) =10 log10 (50) = 17 dB
• dBm to express absolute values of power relative to a
milliwatt.
100 mW = 20 dBm
10 mW = 10 dBm
dBm = 10 log10 (power in milliwatts / 1 milliwatt)
Power P in decibels = 10 log10(P/Preference)
[Source: Berkeley] Lecture 13 19
Electric Circuits (Fall 2015) Pingqiang Zhou
• From the expression for power ratios in decibels, we can
readily derive the corresponding expressions for voltage or
current ratios.
• Suppose that the voltage V (or current I) appears across (or
flows in) a resistor whose resistance is R. The corresponding
power dissipated, P, is V2/R (or I2R). We can similarly relate
the reference voltage or current to the reference power, as
Hence,
Voltage, V in decibels = 20log10(V/Vreference)
Current, I, in decibels = 20log10(I/Ireference)
Logarithmic Measures for Voltage or Current
Preference = (Vreference)2/R or Preference= (Ireference)
2R
[Source: Berkeley]
Electric Circuits (Fall 2015) Pingqiang Zhou
• Note that the voltage and current expressions are just like the
power expression except that they have 20 as the multiplier instead
of 10 because power is proportional to the square of the voltage or
current.
• The input voltage (current, or power) is taken as the reference value
of voltage (current, or power) in the decibel defining expression:
Logarithmic Measures for Voltage or Current
Voltage gain in dB = 20 log10(Voutput/Vinput)
Current gain in dB = 20log10(Ioutput/Iinput
Power gain in dB = 10log10(Poutput/Pinput)
Exercise: How many decibels larger is the voltage of a 9-volt transistor
battery than that of a 1.5-volt AA battery? Let Vreference = 1.5.
Example: The voltage gain of an amplifier whose input is 0.2 mV and
whose output is 0.5 V is 20log10(0.5/0.2x10-3) = 68 dB.
20 log10(9/1.5) = 20 log10(6) = 16 dB.
[Source: Berkeley]
Electric Circuits (Fall 2015) Pingqiang Zhou
Bode Plots
• One problem with the transfer function is that it needs to
cover a large range in frequency. So we can
Plotting the frequency response on a semilog plot
– X is in log scale
– Y scale in dB
These plots are referred to as Bode plots.
– Show magnitude (in decibels) or phase (in degrees) as a function of
frequency.
• Log Frequency Scale
Decade Ratio of higher to lower frequency = 10
Octave Ratio of higher to lower frequency = 2
Lecture 13 22
Electric Circuits (Fall 2015) Pingqiang Zhou
Bode Plot of First-Order Lowpass Filter
23Lecture 13
𝐻 𝑓 =1
1 + 𝑗 𝑓/𝑓𝐵𝑓𝐵 =
1
2𝜋𝑅𝐶
Electric Circuits (Fall 2015) Pingqiang Zhou
Magnitude Plot
Convert magnitude to decibels, we have
𝐻 𝑓 𝑑𝐵 = 20 log 𝐻 𝑓 = −10 log 1 +𝑓
𝑓𝐵
2
• When 𝑓 ≪ 𝑓𝐵, i.e., low frequency, 𝐻 𝑓 𝑑𝐵 ≅ 0;
• When 𝑓 ≫ 𝑓𝐵, 𝐻 𝑓 𝑑𝐵 ≅ −20 log𝑓
𝑓𝐵.
24Lecture 13
Corner (break) frequency
where two straight-line
asymptotes intersect
Electric Circuits (Fall 2015) Pingqiang Zhou
Phase Plot
• When 𝑓 < 𝑓𝐵/10, i.e., low frequency, ∠𝐻 𝑓 ≅ 0;
• When 𝑓 > 10𝑓𝐵, 𝐻 𝑓 𝑑𝐵 ≅ −90°.
25Lecture 13
The actual phase curve departs
from these straight-line
approximations by less than 6°.
Electric Circuits (Fall 2015) Pingqiang Zhou
Standard Form of General Transfer Function
• The transfer function may be written in terms of factors
with real and imaginary parts. For example:
• This standard form may include the following seven
factors in various combinations:
Lecture 13 26
Electric Circuits (Fall 2015) Pingqiang Zhou
Bode Plots
Lecture 13 27
Electric Circuits (Fall 2015) Pingqiang Zhou
Bode Plots
Lecture 13 28
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
29Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example - Magnitude
30Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example - Phase
31Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Resonance
• The most prominent feature of the
frequency response of a circuit may
be the sharp peak in the amplitude
characteristics.
• Resonance occurs in any system
that has a complex conjugate pair
of poles.
It enables energy storage in the form of
oscillations
It allows frequency discrimination.
It requires at least one capacitor and
inductor. [Source: Georgia State U]
Lecture 13 32
Electric Circuits (Fall 2015) Pingqiang Zhou
Series Resonance
• A series resonant circuit consists of an
inductor and capacitor in series.
Resonance occurs when the imaginary part
of Z is zero.
The value of ω that satisfies this is called the
resonant frequency.
0
1rad/s
LC
Lecture 13 33
Electric Circuits (Fall 2015) Pingqiang Zhou
Series Resonance
• At resonance:
The impedance is purely resistive
The voltage Vs and the current I are in phase
The magnitude of the transfer function is minimum.
The inductor and capacitor voltages can be much more than the
source.
Lecture 13 34
Electric Circuits (Fall 2015) Pingqiang Zhou
Half-Power Frequencies
• The frequency response of the current magnitude:
0
1rad/s
LC
Lecture 13 35
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandwidth
• Bandwidth: the difference
between the two half-
power frequencies
Lecture 13 36
𝐵 = 𝜔2 −𝜔1 =𝑅
𝐿
Electric Circuits (Fall 2015) Pingqiang Zhou
Quality Factor
• Quality factor Q: measure the
“sharpness” of the resonance.
[Source: Georgia State U]
Lecture 13 37
Electric Circuits (Fall 2015) Pingqiang Zhou
Q and B
38Lecture 13
The higher the Q,
the smaller the B.
Electric Circuits (Fall 2015) Pingqiang Zhou
Approximation of Half-Power Frequencies
• For high-Q (𝑄 ≥ 10) circuits,
half-power frequencies can
be approximated as
39Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
In the circuit, 𝑅 = 2Ω, 𝐿 =1mH and 𝐶 = 0.4𝜇F• Find resonant frequency
• Find half-power frequencies
• Calculate the quality factor and
bandwidth
• Determine the amplitude of the
current at 𝜔0,𝜔1 and 𝜔2
40Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Parallel Resonance
• The parallel RLC circuit shown here is
the dual of the series circuit.
• Resonance occurs when the imaginary
part of the admittance is zero.
This results in the same resonant frequency
as in the series circuit.
Lecture 13 41
Electric Circuits (Fall 2015) Pingqiang Zhou
𝝎𝟏, 𝝎𝟐, 𝑸 𝐚𝐧𝐝 𝑩
42Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
In the circuit, 𝑅 = 8kΩ, 𝐿 =0.2mH and 𝐶 = 8𝜇F• Find Q, B, 𝜔1 and 𝜔2
• Find half-power frequencies
• Determine the power dissipated at
𝜔0,𝜔1 and 𝜔2
43Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Summary
44Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Filters
• Circuit designed to retain a certain
frequency range but discard or
attenuate others
Low-pass: pass low frequencies and reject
high frequencies
High-pass: pass high frequencies and
reject low frequencies
Band-pass: pass some particular range of
frequencies, reject other frequencies
outside that band
Notch: reject a range of frequencies and
pass all other frequencies
Lecture 13 45
Electric Circuits (Fall 2015) Pingqiang Zhou
Common Filter Transfer Function vs. Freq
(Magnitude Plots shown)
( )H f
Frequency
High
Pass
( )H f
Frequency
Low
Pass
( )H f
Frequency
Band Pass
Frequency
Band
Reject
( )H f
Electric Circuits (Fall 2015) Pingqiang Zhou
Passive Filters
• A filter is passive if it consists only of passive elements, R,
L, and C.
• LC filters have been used in practical applications for
more than eight decades.
• They are very important circuits in that many technological
advances would not have been possible without the
development of filters.
LC filter technology feeds related areas such as equalizers,
impedance-matching networks, transformers, shaping networks,
power dividers, attenuators, and directional couplers …
Lecture 13 47
Electric Circuits (Fall 2015) Pingqiang Zhou
First-Order Lowpass Filter
• A typical lowpass filter is formed when
the output of a RC circuit is taken off
the capacitor.
• The half power frequency is:
Also referred as the cutoff frequency.
Filter is designed to pass from DC up to ωc.
1c
RC
Lecture 13 48
Electric Circuits (Fall 2015) Pingqiang Zhou
First-Order Highpass Filter
• A highpass filter is also made of a
RC circuit, with the output taken off
the resistor.
• The cutoff frequency will be the same
as the lowpass filter.
• The difference being that the
frequencies passed go from ωc to
infinity.Lecture 13 49
Electric Circuits (Fall 2015) Pingqiang Zhou
First-Order Filter Circuits
L+
–VS
C
R
Low Pass
High
Pass
HR = R / (R + jL)
HL = jL / (R + jL)
+
–VS
R
High Pass
Low
Pass
HR = R / (R + 1/jC)
HC = (1/jC) / (R + 1/jC)
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandpass Filter
• The RLC series resonant circuit
provides a bandpass filter when
the output is taken off the resistor.
• The center frequency is:
• The filter will pass frequencies
from ω1 to ω2.
• It can also be made by feeding
the output from a lowpass to a
highpass filter.
0
1
LC
Lecture 13 51
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandstop Filter
• A bandstop filter can be created from a RLC circuit by
taking the output from the LC series combination.
• The range of blocked frequencies will be the same as the
range of passed frequencies for the bandpass filter.
Lecture 13 52
Electric Circuits (Fall 2015) Pingqiang Zhou
Second-Order Filter Circuits
53Lecture 13[Source: Berkeley]
Electric Circuits (Fall 2015) Pingqiang Zhou
Example
• Determine what type of filter
is shown below. Calculate the
corner or cutoff frequency.
𝑅 = 2kΩ, 𝐿 = 2H and 𝐶 = 2𝜇F.
54Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Active Filters
• Passive filters have a few drawbacks.
They cannot create gain greater than 1.
They do not work well for frequencies below
the audio range (300 Hz < f < 3,000 Hz).
They require inductors, which tend to be
bulky and more expensive than other
components.
• It is possible, using op-amps, together
with resistors and capacitors, to create
all the common filters.
• Their ability to isolate input and output
also makes them very desirable.
Lecture 13 55
Electric Circuits (Fall 2015) Pingqiang Zhou
First Order Lowpass
• If the input and feedback elements in
an inverting amplifier are selectively
replaced with capacitors, the
amplifier can act as a filter.
If the feedback resistor is replaced with a
parallel RC element, the amplifier
becomes a lowpass filter.
Corner frequency
1c
f fR C
Lecture 13 56
Electric Circuits (Fall 2015) Pingqiang Zhou
First Order Highpass
• Placing a series RC combination in
place of the input resistor yields a
highpass filter.
• The corner frequency of the filter
will be:1
c
i iR C
Lecture 13 57
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandpass
Lecture 13 58
Electric Circuits (Fall 2015) Pingqiang Zhou
Constructing the Bode Plot of a Bandpass
59Lecture 13[Nilsson, 10th ed.]
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandpass
• To avoid the use of an inductor,
it is possible to use a cascaded
series of lowpass active filter
into a highpass active filter.
• To prevent unwanted signals
passing, their gains are set to
unity, with a final stage for
amplification.
Lecture 13 60
Electric Circuits (Fall 2015) Pingqiang Zhou
Active Bandpass Filter
61Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Constructing the Bode Plot of a Bandreject
62Lecture 13[Nilsson, 10th ed.]
Electric Circuits (Fall 2015) Pingqiang Zhou
Bandreject
• Creating a bandstop filter requires using a lowpass and
highpass filter in parallel.
Both output are fed into a summing amplifier.
It will function by amplifying the desired signals compared to the
signal to be rejected.
Lecture 13 63
Electric Circuits (Fall 2015) Pingqiang Zhou
Active Bandreject Filter
64Lecture 13
Electric Circuits (Fall 2015) Pingqiang Zhou
Further Reading
• Refer to Ch. 14.9 for Scaling
Magnitude scaling
Frequency scaling
• Read Ch. 14.11 on how to use MATLAB to do Bode Plot
• Refer to Ch. 14.12 for applications of resonant circuits and
filters
Radio receiver
Touch-tone telephone
• Refer to [Allan, Ch. 6.10] for DSP.
65Lecture 13