Bài giảng Radio communications circuits - Chapter 4: RF oscillators and frequency synthesizers - Trịnh Xuân Dũng
Dung Trinh, PhD
HCMUT / 2019
Chapter 4
RF Oscillators and Frequency
Synthesizers
Trinh Xuan Dung, PhD
Department of Telecommunications
Faculty of Electrical and Electronics Engineering
Ho Chi Minh city University of Technology
1
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
Contents
1. Introduction
2. LC Oscillators
3. RC Oscillators
4. Crystal Oscillator
5. Voltage Controlled Oscillator and Phase Locked Loop
6. Exercises
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction
Oscillators convert a DC input (the supply voltage) into an AC
output (the waveform), which can have a wide range of different
wave shapes and frequencies.
RF
Oscillators are used in a number of applications in which a reference tone is required.
For instance, they can be used as the clock for digital circuits or as the source of the
LO signal in transmitters.
In receivers, oscillator waveforms are used as the reference frequency to mix down
the received RF to an IF or to baseband. In most RF applications, sinusoidal
references with a high degree of spectral purity (low phase noise) are required.
Communications systems, digital systems (including computers), and test equipment
make use of oscillators.
Waveform generated by oscillators
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Introduction
Classification of sinusoidal Oscillators:
. RC Oscillators: Wien Oscillator, Phase Shift Oscillator.
• Use a resistance-capacitance network to determine the oscillator frequency.
• Suitable for low and moderate frequency applications (5Hz to 1MHz).
. LC Oscillators: Colpitts, Hartley Oscillator.
• Use a inductance-capacitance network to determine the oscillator frequency.
• Suitable for radio frequency (1 to 500MHz) applications.
. Crystal Oscillators:
• Use piezoelectric crystal (or quartz) which has very high degree of stability
and accuracy. Suitable for radio frequency applications.
. Microwave Oscillators: Gunn diode, cavity tuned, dielectric.
Example of LC oscillators
Example of crystal oscillators
Example of RC oscillators
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
An oscillator can be seen as unstable feedback systems. An unstable system is one in
which an initially small excitation or disturbance produces an output that grows in
time due to constructive, or positive, feedback.
We have:
푉 (푠) = 푉 (푠) + 훽(푠)푉 (푠)
휀
푖
표
푉 (푠) = 퐴(푠)푉 (푠)
0
휀
Then:
푉 (푠)
퐴(푠)
표
퐴푓 푠 =
=
푉 (푠) 1 − 훽 푠 퐴(푠)
푖
where T(s) = 훽 푠 퐴(푠) is known as open loop gain or sometimes shortened to loop gain.
Replace 푠 with 푗휔:
퐴(푗휔)
퐴푓 푗휔 =
1 − 푇(푗휔)
Block diagram of a feedback system
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
Note that the loop gain is the gain obtained by opening the feedback loop and taking
the output at the point where the loop was opened (푉 푉 ).
푓
푖
Suppose that the loop gain is equal to 1 at some frequency 휔0, 푇 휔0 = 1. Then the
voltage transfer function is singular (infinite) which can be interpreted as finite output
for zero input.
In other words, the circuit is a potential source of radio frequency energy at the
frequency where the loop gain is 1, even in the absence of any input excitation 푉 .
푖
Condition for the feedback loop to provide sinusoidal oscillations of frequency 휔0
(Barkhausen Criterion) is
• At 휔0 the phase of the loop gain should be zero
arg 푇(휔0) = 0.
푇(휔0) = 1.
• At 휔0 the magnitude of the loop gain should be unity
Visualization of open loop gain
6
Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
A negative-feedback oscillatory system can be built using “LC-tuned” amplifier
stages.
At high frequencies, C1
At
the
resonant
At low frequencies, L1
dominates the load
frequency
dominates the load
푉
푉
표푢푡
푉
표푢푡
표푢푡
= −푔푚/푗휔퐶1
= −푔푚푅푃
= −푔푚푗휔퐿1
푉
푉
푖푛
푉
푖푛
푖푛
푉
표푢푡
푉
표푢푡
• The phase shift from the
•
dimished and
≃ 900
•
is very small and
푉
푖푛
푉
표푢푡
푉
푖푛
input to the output is
푉
표푢푡
∠
≃ −900
∠
푉
표푢푡
∠
= 1800
푉
푖푛
푉
푖푛
푉
푖푛
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
The circuit provides a phase shift of 1800 with possibly adequate gain 푔푚푅푃 at 휔0.
We simply need to increase the phase shift to 3600.
Wave form is shown above (right). A unique attribute of inductive loads is that they
can provide peak voltages above the supply. The growth of VX and VY ceases when
M1 and M2 enter the triode region for part of the period, reducing the loop gain.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
In practice the condition arg 푇(휔0) = 0 is usually applied to solve for the potential
frequency of oscillation, 휔0. Then the condition 푇(휔0) = 1 is applied to determine
the gain of amplifier.
For practical designs, the amplifier gain is set to a value somewhat higher than the
value determined by above equation to ensure that oscillations will reliably start and
grow when the circuit is powered up.
Circuits with the topology shown in following figure are commonly employed as
oscillators. The active device could be a BJT or an FET.
Topology for one class of oscillator circuits
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
1. Conditions for Oscillation
The passive elements of the model (e.g., 푟 , 퐶휋, 퐶휇 etc.) can be lumped into the
휋
external impedances 푍1, 푍2, 푍3.
To find the open loop gain of this circuit, we break the loop at a convenient point and
terminate that point in the impedance that it sees when the loop is closed.
The termination is required so that the open-loop circuit operates under the same
conditions as when the loop is closed. The loop gain is computed by exciting the
circuit at the input to the opened loop and computing the output across 푍1:
푉
푍 푍 +푍
2 1 3
표
퐴 = 푉 = −푔푚 푍2 푍1 + 푍3 = −푔푚
푍 +푍 +푍
3
1
2
푖푛
푉
푍
푓
1
훽 = = 푍 +푍
푉
표
1
3
푉
푉
푍 푍
1 2
푓
표푢푡
푇 = 푉
=
= 퐴훽 = −푔푚 푍 +푍 +푍
1 2 3
푖푛
푉
푖푛
Feedback loop terminated with Z1.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using Loop Gain
We assume for the moment that 푍1 and 푍2 are purely reactive, i.e., 푍1 = 푗푋1, 푍2 =
푗푋2. We allow 푍3 to have a non-zero (positive) real part. Then:
푔푚푋1푋2
푇 =
푗(푋1 + 푋2) + 푍3
For oscillation to occur, the phase angle of 푇 must be zero at some frequency. Now
there are two possibilities for the signs of 푋1 and 푋2:
• 푋1 > 0 (inductor), 푋2 > 0 (inductor): Hartley Oscillator
• 푋1 < 0 (capacitor), 푋2 < 0 (capacitor): Colpitts Oscillator
The frequency of oscillation is easily found by finding the frequency where the phase
angle of 푍푠 = 푗 푋1 + 푋2 + 푍3 = 0. This leads to
1
1
• Colpitts: 휔0 =
Hartley: 휔0 =
퐶 퐶
1 2
퐶 +퐶
1
퐶(퐿 +퐿 )
1
2
퐿
2
Colpitts Oscillator
Hartley Oscillator
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using Loop Gain
In fact, if 푍3 was lossless, then 푍1, 푍2 and 푍3 would constitute a lossless resonant
circuit. A lossless circuit will oscillate indefinitely at the resonant frequency once it is
initially excited.
When 푍3 is lossy, then the oscillations cannot be maintained unless the transistor is
added to make up for the losses.
So far we have considered only half of the Barkhausen criterion (arg[푇] = 0). The
results yielded the potential frequency of oscillation. For oscillations to occur, the
gain of the transistor must be large enough to make 푇 = 1
0. This leads to:
휔=휔
푔
푚
1
2
• Colpitts:
• Hartley:
= 1 → 푔푚 = 푅휔02퐶1퐶2 = 푅 퐶 +퐶
.
2
푅휔 퐶 퐶
퐿
1 2
0
2
푔 휔 퐿 퐿
푅
푅퐶 퐿 +퐿
1 2
푚
1 2
0
= 1 → 푔푚 = 휔 퐿 퐿
=
.
2
0
푅
퐿 퐿
1 2
1 2
The values of 푔푚 obtained in above equations are the values necessary for the circuit
to support steady-state oscillations, denoted by 푔푚,푠푠. In practical applications the
transistor is biased to set the transconductance to a value somewhat larger, e.g. a
factor of 2 to 5 larger, than 푔푚,푠푠. One reason for doing this is to ensure that
oscillations start reliably even if component values change slightly.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using Loop Gain
Setting 푇 > 1 means that the oscillations will not be maintained at a steady state;
rather, they will grow in amplitude. Growth will proceed until the active device is no
longer operating in the “small-signal” mode.
As the oscillation grows, eventually the amplitude of the oscillation will be limited by
nonlinear effects. To first order, the onset of nonlinear operation coincides with a
reduction in the gain of the active device.
This is called gain saturation, and the effect can be modeled as a decrease in the
transconductance, 푔푚, and hence a decrease in the loop gain.
As the amplitude of the oscillation grows, the transconductance is decreased to the
point where the magnitude of the loop gain is 1. At this point steady-state oscillation
will be maintained.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using KCL
Applying KCL at the output, we have:
푉 푉
표
푖푛
푔푚푉푖푛 + +
푍2 푍1
= 0
푍 +푍
1
3
where 푉 = 푉
. Replace 푉 in the above equation, then:
표
푖푛
표
푍
1
푉 푍1 + 푍3
푉
푖푛
푖푛
푔푚푉푖푛 +
+
= 0
= 0
푍2
푍1
푍1
푍3
1
1
Oscillator circuit
→ 푉 푔푚 + + +
푖푛
푍1 푍2 푍1푍2
For oscillation, 푉 must not be zero, therefore it enforces:
푖푛
1
1
푍3
푔푚 + + +
= 0
푍1 푍2 푍1푍2
1
1
For Colpitts oscillator, 푍1 = 푗휔퐶 , 푍2 = 푗휔퐶 , and 푍3 = 푅 + 푗휔퐿3, then:
1
2
1
휔0 =
퐶 퐶
1 2
퐶 +퐶
1
퐿
2
푅 퐶1 + 퐶2
퐿
푔푚 = 푅휔02퐶1퐶2 =
Colpitts Oscillator
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using KCL
1
For Hartley oscillator, 푍1 = 푗휔퐿1, 푍2 = 푗휔퐿2, and 푍3 = 푅 + 푗휔퐶 , then:
3
1
휔0 =
퐶(퐿 +퐿 )
1
2
푅
푅퐶 퐿1 + 퐿2
퐿1퐿2
푔푚 =
=
휔02퐿1퐿2
Colpitts Oscillator
Hartley Oscillator
Example 1: For the Colpitts oscillator in following figure, assume parameters of 퐿 =
1휇퐻, 퐶1 = 퐶2 = 1푛퐹 and 푅 = 4푘Ω. Determine the oscillator frequency and the
required value of 푔푚. Is this value of 푔푚 reasonable for a MOSFET? Why?
Solution:
1
1
푓 =
0
=
= 7.12푀퐻푧.
−9
−9
−9
퐶 퐶
1 2
10 ×10
−6
2휋 퐿
2휋 10 ×
−9
퐶 +퐶
1
2
10 +10
−9
퐶
10
3
2
푔푚 = 푅퐶
=
= 0.25푚퐴/푉.
−9
4×10 ×10
1
Equivalent circuit
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using Negative Resistance
The requirements for oscillation in a circuit can be derived using the so-called
negative resistance concept.
Consider a circuit consisting of arbitrary impedances 푍퐴 and 푍퐵, as shown in the
following figure. The Kirchoff’s loop equation is I 푍퐴 + 푍퐵 = 0. Hence, finite
current I is allowed if 푍퐴 + 푍퐵 = 0.
It is instructive to analyze the oscillator circuit using this approach. Suppose that the
circuit is divided into two parts, by defining 푍퐴 = 푍1.
The impedance 푍퐵 is the impedance seen looking into the port defined by the base
and emitter terminals of the transistor, with 푍1 removed from the circuit.
Applying KCL at the input and output, we have:
푣푥 − 푣표
푣표
푍3
푖푥 =
= 푔푚푣푥 +
푍3
푣푥
푍2 + 푍3
→ 푍퐵 = =
푖푥 1 + 푔푚푍2
Enforcing 푍퐴 + 푍퐵 = 0 yields
푍1 +
푍2 + 푍3
Impedance of oscillator circuit
= 0
1 + 푔푚푍2
푍1푍2
→ −푔
= 1
푚 푍1 + 푍2 + 푍3
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. LC Oscillator Analysis using KCL
Example 2: A Hartley oscillator circuit having two individual inductors of 0.5mH each
(퐿1 = 퐿2 = 0.5푚퐻), are design to resonate in parallel with a variable capacitor that can
be varied from 100pF to 500pF. Determine the upper and lower frequencies of
oscillation and also the Hartley oscillators bandwidth.
Solution:
The upper frequency of oscillation:
1
1
푓퐻 =
=
= 503 (푘퐻푧)
= 225 (푘퐻푧)
2휋 0.5 × 10−3 + 0.5 × 10−3 100 × 10−12
2휋 퐿1 + 퐿2 퐶1
The lower frequency of oscillation:
1
1
푓 =
퐿
=
2휋 0.5 × 10−3 + 0.5 × 10−3 500 × 10−12
2휋 퐿1 + 퐿2 퐶2
The bandwidth of oscillation:
퐵푊 = 푓퐻 − 푓 = 503 − 225 = 278 (푘퐻푧)
퐿
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Hartley and Colpitts Oscillator
The total capacitance of the tank circuit is split into two parts, 퐶1 and 퐶2 and
connected in series so that the net capacitance of the tank circuit is reduced. And
1
퐿
thereby quality factor 푄 = 푅
of the tank circuit increases hence Colpitts has better
퐶
frequency stability when compared to Hartley oscillator.
Hartley Oscillator
Colpitts Oscillator
Hartley oscillator uses a tapped coil for It is easier to tap 2 capacitors in series as
the feedback for oscillation, and these well as being cheaper than to use a
tend to be more difficult to make.
Capacitive tuning hence offers very low Circuit is economical and small in size due
wear and tear problem. to requirement of one inductor.
Circuit is bulky due to the present of two Inductive tuning offers very high wear and
tapped coil for feedback purpose.
inductors.
tear problem.
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Common-Collector Colpitts Oscillator
Practical circuits for Colpitts-type oscillators are shown in following figures. We will
analyze the common-collector configuration.
The small-signal equivalent circuit is shown where the transistor has been replaced
with its hybrid-pi model, and the finite Q of the inductor is modeled with a series
resistance, r.
CC Colpitts Oscillator
CB Colpitts Oscillator
CE Colpitts Oscillator
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Dept. of Telecoms Engineering
Dung Trinh, PhD
HCMUT / 2019
2. Common-Collector Colpitts Oscillator
The small-signal equivalent circuit can be further simplified by combining some
elements as shown in following figure where
1
푍1 =
푟
휋
푗휔 퐶1 + 퐶휋
1
푍2 =
푅푒
푗휔 퐶2 + 퐶표
1
푍3 = 푅1 푅2 푟 + 푗휔퐿
푗휔퐶휇
In practice it is useful to choose 퐶1 ≫ 퐶휋, 퐶2 ≫ 퐶표. This ensures that the external
components swamp the internal capacitances of the transistor, thereby minimizing the
circuit’s dependence on variations in the internal transistor capacitances.
Simplified small signal equivalent circuit.
Small signal equivalent circuit of CC Colpitts Oscillator
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