Transcript Chapter 17

Frequency Characteristics of AC Circuits
Chapter 17
 Introduction
 A High-Pass RC Network
 A Low-Pass RC Network
 A Low-Pass RL Network
 A High-Pass RL Network
 A Comparison of RC and RL Networks
 Bode Diagrams
 Combining the Effects of Several Stages
 RLC Circuits and Resonance
 Filters
 Stray Capacitance and Inductance
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Introduction
17.1
 Earlier we looked at the bandwidth and frequency
response of amplifiers
 Having now looked at the AC behaviour of
components we can consider these in more detail
 The reactance of both inductors and capacitance is
frequency dependent and we know that
X L  L
1
XC 
C
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 We will start by considering very simple circuits
 Consider the potential divider shown here
– from our earlier consideration of the
circuit
Z2
vo  vi 
Z1  Z 2
– rearranging, the gain of the circuit is
vo
Z2

vi
Z1  Z 2
– this is also called the
transfer function of the circuit
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A High-Pass RC Network
17.2
 Consider the following circuit
– which is shown re-drawn in a more usual form
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 Clearly the transfer function is
vo
ZR
R
1



v i Z R  ZC R  j 1 1  j 1
C
CR
 At high frequencies
–  is large, voltage gain  1
 At low frequencies
–  is small, voltage gain  0
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 Since the denominator has
real and imaginary parts, the
magnitude of the voltage gain is
Voltage gain 
1
 1 
1 

 CR 
2
 When 1/CR = 1
Voltage gain 
2
1
1

 0.707
1 1
2
 This is a halving of power, or a fall in gain of 3 dB
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 The half power point is the cut-off frequency of the
circuit
– the angular frequency C at which this occurs is given by
1
1
cCR
1 1
c 
 rad/s
CR 
– where  is the time constant of the CR network. Also
fc 
c
1

Hz
2 2CR
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 Substituting  =2f and CR = 1/ 2fC in the earlier
equation gives
vo
1


1
v i 1 j
1 j
CR
1
1
 1 
(2f )

 2fc 

1
1 j
fc
f
 This is the general form of the gain of the circuit
 It is clear that both the magnitude of the gain and the
phase angle vary with frequency
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 Consider the behaviour of the circuit at different
frequencies:
 When f >> fc
– fc/f << 1, the voltage gain  1
 When f = fc
vo
1
1
1 (1  j)
(1  j)




 0 .5  0 .5 j
f
v i 1  j c 1  j 1  j  (1  j)
2
f
 When f << fc
vo
1
1
f


j
v i 1  j fc  j fc
fc
f
f
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 The behaviour in these three regions can be
illustrated using phasor diagrams
 At low frequencies the gain is linearly related to
frequency. It falls at -6dB/octave (-20dB/decade)
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 Frequency response of
the high-pass network
– the gain response has
two asymptotes that
meet at the cut-off
frequency
– figures of this form are
called Bode diagrams
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A Low-Pass RC Network
17.3
 Transposing the C and R gives
1
vo
ZC
1
C 


v i Z R  ZC R  j 1 1  jCR
C
j
 At high frequencies
–  is large, voltage gain  0
 At low frequencies
–  is small, voltage gain  1
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A Low-Pass RC Network
17.3
 A similar analysis to before
gives
Voltage gain 
1
1  CR 2
 Therefore when, when CR = 1
Voltage gain 
1
1

 0.707
1 1
2
 Which is the cut-off frequency
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 Therefore
– the angular frequency C at which this occurs is given by
cCR  1
1 1
c 
 rad/s
CR 
– where  is the time constant of the CR network, and as
before

1
fc  c 
Hz
2 2CR
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 Substituting  =2f and CR = 1/ 2fC in the earlier
equation gives
vo
1


v i 1  jCR
1
1 j

c

1
1 j
f
fc
 This is similar, but not the same, as the transfer
function for the high-pass network
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 Consider the behaviour of this circuit at different
frequencies:
 When f << fc
– f/fc << 1, the voltage gain  1
 When f = fc
vo
1  j1  j  1  j  0.5  0.5 j
1


1  j
v i 1 j f
2
fc
 When f >> fc
vo
f
1
1


 j c
f
v i 1 j f
f
j
fc
fc
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 The behaviour in these three regions can again be
illustrated using phasor diagrams
 At high frequencies the gain is linearly related to
frequency. It falls at 6dB/octave (20dB/decade)
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 Frequency response of
the low-pass network
– the gain response has
two asymptotes that
meet at the cut-off
frequency
– you might like to
compare this with
the Bode Diagram
for a high-pass
network
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A Low-Pass RL Network
17.4
 Low-pass networks can also
be produced using RL circuits
– these behave similarly to the
corresponding CR circuit
– the voltage gain is
vo
ZR
R
1



v i Z R  Z L R  jL 1  j L
R
– the cut-off frequency is
c 
R 1
 rad/s
L 
fc 
c
R

Hz
2 2L
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A High-Pass RL Network
17.5
 High-pass networks can also
be produced using RL circuits
– these behave similarly to the
corresponding CR circuit
– the voltage gain is
vo
ZL
jL
1
1




v i Z R  Z L R  jL 1  R 1  j R
jL
L
– the cut-off frequency is
c 
R 1
 rad/s
L 
fc 
c
R

Hz
2 2L
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A Comparison of RC and RL Networks
17.6
 Circuits using RC and RL
techniques have similar
characteristics
– for a more detailed
comparison, see
Figure 17.10 in the
course text
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Bode Diagrams
17.7
 Straight-line approximations
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 Creating more detailed Bode diagrams
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Combining the Effects of Several Stages
17.8
 The effects of several stages ‘add’ in bode diagrams
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 Multiple high- and low-pass
elements may also be combined
– this is illustrated in Figure 17.14
in the course text
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RLC Circuits and Resonance
17.9
 Series RLC circuits
– the impedance is given by
Z  R  jL 
1
1
 R  j(L 
)
jC
C
– if the magnitude of the reactance
of the inductor and capacitor are
equal, the imaginary part is zero,
and the impedance is simply R
– this occurs when
1
L 
C
1
 
LC
2

1
LC
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 This situation is referred to as resonance
– the frequency at which is occurs is the
resonant frequency
o 
1
LC
fo 
1
2 LC
– in the series resonant
circuit, the impedance is
at a minimum at resonance
– the current is at a maximum
at resonance
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 The resonant effect can be quantified by the
quality factor, Q
– this is the ratio of the energy dissipated to the energy
stored in each cycle
– it can be shown that
Quality factor Q 
– and
Q
X L XC

R
R
1 L
 
R C 
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 The series RLC circuit is an acceptor circuit
– the narrowness of bandwidth is determined by the Q
Quality factor Q 
Resonant frequency fo

Bandwidth
B
– combining this equation with the earlier one gives
B
R
Hz
2L
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 Parallel RLC circuits
– as before
o 
1
LC
fo 
1
2 LC
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 The parallel arrangement is a rejector circuit
– in the parallel resonant
circuit, the impedance is
at a maximum at resonance
– the current is at a minimum
at resonance
– in this circuit
C 
QR  
L
B
1
Hz
2RC
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Filters
17.10
 RC Filters
 The RC networks considered earlier are first-order
or single-pole filters
– these have a maximum roll-off of 6 dB/octave
– they also produce a maximum of 90 phase shift
 Combining multiple stages can produce filters with a
greater ultimate roll-off rates (12 dB, 18 dB, etc.) but
such filters have a very soft ‘knee’
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 An ideal filter would have
constant gain and zero phase
shift for frequencies within its
pass band, and zero gain for
frequencies outside this range
(its stop band)
 Real filters do not have these
idealised characteristics
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 LC Filters
 Simple LC filters can
be produced using
series or parallel tuned
circuits
– these produce narrowband filters with a
centre frequency fo
fo 
1
2 LC
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 Active filters
– combining an op-amp
with suitable resistors
and capacitors can
produce a range of filter
characteristics
– these are termed active
filters
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 Common forms include:
 Butterworth
– optimised for a flat response
 Chebyshev
– optimised for a sharp ‘knee’
 Bessel
– optimised for its phase response
see Section 17.10.3 of the course
text for more information on these
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Stray Capacitance and Inductance
17.11
 All circuits have stray capacitance
and stray inductance
– these unintended elements can
dramatically affect circuit operation
– for example:
 (a) Cs adds an unintended low-pass filter
 (b) Ls adds an unintended low-pass filter
 (c) Cs produces an unintended resonant
circuit and can produce instability
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Key Points
 The reactance of capacitors and inductors is dependent on
frequency
 Single RC or RL networks can produce an arrangement
with a single upper or lower cut-off frequency.
 In each case the angular cut-off frequency o is given by
the reciprocal of the time constant 
 For an RC circuit  = CR, for an RL circuit  = L/R
 Resonance occurs when the reactance of the capacitive
element cancels that of the inductive element
 Simple RC or RL networks represent single-pole filters
 Active filters produce high performance without inductors
 Stray capacitance and inductance are found in all circuits
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