Chapter 2 Basic Concepts in
RF Design
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Sections to be covered
• 2.1 General Considerations • 2.2 Effects of Nonlinearity
• 2.3 Noise • 2.4 Sensitivity and Dynamic Range • 2.5 Passive Impedance Transformation
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Chapter Outline
Nonlinearity
Noise
Impedance
Transformation
Harmonic Distortion Compression Intermodulation
Noise Spectrum Device Noise Noise in Circuits
Series-Parallel
Conversion
Matching Networks
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The Big Picture: Generic RF
Transceiver
Overall transceiver
Signals are upconverted/downconverted at TX/RX, by an oscillator controlled by a Frequency Synthesizer.
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General Considerations: Units in RF Design
Voltage gain:
rms value
Power gain:
These two quantities are equal (in dB) only if the input and output
impedance are equal .
Example:
an amplifier having an input resistance of R0 (e.g., 50 Ω) and driving a load resistance of R0 :
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where Vout and Vin are rms value.
General Considerations: Units in RF Design
“dBm”
The absolute signal levels are often expressed in dBm (not in watts or volts);
Used for power quantities, the unit dBm refers to “dB’s above
1mW”.
To express the signal power, Psig, in dBm, we write
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Example of Units in RF
An amplifier senses a sinusoidal signal and delivers a power of 0 dBm to a load resistance of 50 Ω. Determine the peak-to-peak voltage swing across the load.
Solution:
a sinusoid signal having a peak-to-peak amplitude of Vpp
an rms value of Vpp/(2√2),
0dBm is equivalent to 1mW,
- where RL= 50 Ω
- thus,
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Example of Units in RF
A GSM receiver senses a narrowband (modulated) signal having a level of -100 dBm. If the front-end amplifier provides a voltage gain of 15 dB, calculate the peak-to-peak voltage swing at the output of the amplifier.
Solution:
suppose the input and output impedance are equal. convert the received signal level to voltage:
-100 dBm
is 100 dB below 632 mVpp.
100 dB for voltage quantities is equivalent to 105.
-100 dBm is equivalent to 6.32 μVpp.
This input level is amplified by 15 dB (≈ 5.62),
The output swing is 35.5 μVpp.
Notice: For a narrowband (not sinusoid) 0-dBm signal, it is still possible to
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approximate the (average) peak-to-peak swing as 632mV.
Voltage vs Power
Why the output voltage of the amplifier is of interest in this example?
– If the circuit following the amplifier does not present a 50-Ω input impedance, the power gain and voltage gain are not equal in dB.
– Mostly, the next stage may exhibit a purely capacitive input impedance, thereby requiring no signal “power”.
• one stage drives the gate of the transistor in the next stage.
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dBm Used at Interfaces Without Power
Transfer
We use “dBm” at interfaces that do not necessarily entail power transfer.
(a) LNA driving a pure-capacitive impedance with a 632-mVpp swing, delivering no average power.
How about the power delivery?
Assumption: attaching an ideal voltage buffer to node X and drive a 50-Ω load.
the signal at node X has a level of 0 dBm, means that if this signal were applied to a 50-Ω load, then it would deliver 1 mW.
(b) Use of fictitious buffer to visualize the
signal level in “dBm”
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voltage buffer
General Considerations: linearity
A system is linear if its output can be expressed as a linear combination
(superposition) of responses to individual inputs.
For arbitrary a and b, it holds that: Any system that does not satisfy this condition is nonlinear. Example:
nonzero initial conditions or dc offsets cause nonlinearity;
However, we often relax the rule ---------- accommodate these two effects.
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Nonlinearity: Memoryless and Static System
Memoryless or static : if its output does not depend on the past values of its input;
Memoryless, linear
The input/output characteristic of a memoryless nonlinear system can be approximated with a polynomial
Memoryless, nonlinear
If the system is time variant,αj would be general functions of time.
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Nonlinearity: Memoryless and Static System
Example of a memoryless nonlinear circuit:
If M1 operates in the saturation region and can be approximated as a square-law device [Lee]
1
ꢇ
ꢈ
2
ꢀꢁ = ꢂꢃꢄꢅꢆ
ꢉ − ꢉꢋꢌ
ꢊꢃ
2
then
Common-source stage
In this idealized case, the circuit displays only second-order nonlinearity.
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Nonlinearity: Odd symmetry
A system has “odd symmetry” if y(t) is an odd function of x(t), i.e., if the response to –x(t) is the negative of that to x(t).
y(t) = α0 +α1x(t) +α2 x2 (t) +α3x3 (t) +
y(t) is odd symmetry if αj=0 for even j.
Such a system is sometimes called “balanced”, as exemplified by the differential pair shown in the next page.
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Example of Polynomial Approximation
For square-law MOS transistors operating in saturation, the characteristic “differential pair circuit” can be expressed as
If the differential input is small, approximate the characteristic by a polynomial.
Assuming Approximation (Taylor Expansions) gives us:
- Differential pair
- Input/output characteristic
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Example of Polynomial Approximation
Observations
The first term represents linear operation,
the small-signal voltage gain of the circuit (-gmRD);
Due to symmetry, even-order nonlinear terms are absent;
Notice: square-law devices yield a third-order characteristic in
this case.
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General Considerations: Time Variance
A system is time-invariant if a time shift in its input results in the same time shift in its output.
If
y(t) = f [x(t)]
then
y(t-τ) = f [x(t-τ)]
- Time Variance
- Nonlinearity
Do not be confused by these two attributes.
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Example of Time Variance
Plot the output waveform of the circuit in Fig. 1:
vin1 = A1 cos ω1t
vin2 = A2 cos(1.25ω1t )
Solution:
Fig.1
Switch: vout tracks vin2 if vin1 > 0 and is pulled down to zero by R1 if vin1 < 0. vout is equal to the product of vin2 and a square wave toggling between 0 and 1. This is an example of RF “mixers”.
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A time shift in input does not result in the same time shift in output.
Time Variance: Generation of Other Frequency
Components
S(t) denotes a square wave toggling between 0 and 1 with a frequency of
f1=ω1/(2π)
The spectrum of square wave: a train of impulses whose amplitude follow a sinc envelop.
Illustration:
T1= 2π /ω1
Multiplication in time domain
Convolution in frequency domain
A linear system can generate frequency components that do not exist in t1h9e input signal when system is time variant.
Effects of nonlinearity
• ? Frequency • ? Amplitude • Harmonic distortion (谐波失真) • Gain compression (增益压缩) • Cross modulation (互调) • Intermodulation (交叉调制)
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Notice
Analog and RF circuits can be approximated with a linear model for
small-signal operation.
In general, we have
memoryless time-variant systems with input/output characteristic :
y(t) ≈ α1x(t) +α2 x2 (t) +α3x3 (t)
α1 is considered as the small-signal gain.
The nonlinearity effects primarily arise from the third-order term α3.
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Effects of nonlinearity
• Harmonic distortion (谐波失真) • Gain compression (增益压缩) • Cross modulation (互调) • Intermodulation (交叉调制)
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Effects of Nonlinearity: Harmonic Distortion
If a sinusoid is applied to a nonlinear system: the output exhibits frequency components that are integer multiplies (“harmonics”) of the input frequency.
input:
output:
x(t) = Acosωt
y(t) ≈ α1x(t) +α2 x2 (t) +α3x3 (t)
- DC
- Fundamental
- Second
- Third
- Harmonic
- Harmonic
Arising from second-order nonlinearity
The term with the input frequency
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Observations
y(t) ≈ α1x(t) +α2 x2 (t) +α3x3 (t)
Even-order harmonics result from αj with even j and vanish if the system has odd symmetry,
If mismatches corrupt the symmetry, what will happen?
The amplitude of the nth harmonic grows in proportion to?
n
A .
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Example of Harmonic Distortion in Mixer
An analog multiplier “mixes” its two inputs, producing y(t) = kx1(t)x2(t), where k is a constant. Assume x1(t) = A1 cos ω1t and x2(t) = A2 cos ω2t. Question:
(a) If the mixer is ideal, determine the output frequency components.
Solution:
(a)
Analog multiplier
The output contains the sum and difference frequencies. These are “desired” components.
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Example of Harmonic Distortion in Mixer
An analog multiplier “mixes” its two inputs below, producing y(t) = kx1(t)x2(t), where k is a constant. Assume x1(t) = A1 cos ω1t and x2(t) = A2 cos ω2t. (a)If the mixer is ideal, determine the output frequency components.
(b) If the input port sensing x2(t) suffers from third-order nonlinearity, determine the output frequency components.
Solution:
Third Harmonic of x2(t)
(b)
Analog multiplier
The mixers produces two “spurious” components at ω1+3ω2 and ω1-3ω2,
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Cause other problems…
These are “undesired” components that are difficult to remove by filter.
Example of Harmonics on GSM Signal
The transmitter in a 900-MHz GSM cellphone delivers 1 W of power to the antenna. Explain the effect of the harmonics of this signal.
The second harmonic?
falls within another GSM cellphone band around 1800 MHz; Must be sufficiently small to impact the other users in that band.
The third, fourth, and fifth harmonics?
do not coincide with any popular bands; but must still remain below a certain level imposed by regulatory organizations in each country. (中国工信部无线电管理局/US FCC)
The sixth harmonic?
falls in the 5-GHz band used in wireless local area networks (WLANs).
fundamental
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Effects of nonlinearity
• Harmonic distortion (谐波)
• Gain compression (增益压缩)
• Cross modulation (互调) • Intermodulation (交叉调制)
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Gain Compression– Sign of α1, α3
y(t) α1x(t) +α2 x2 (t) +α3x3 (t)
≈
x(t) = Acosωt
The gain of fundamental component ω is equal to α1 + 3α3A2/4 varied as A becomes larger.
Compressive:
The term α3x3 “bends” the characteristic for sufficiently large x,
decreasing the gain as the input amplitude increases.
Expansive:
expanding the gain as the input amplitude increases
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Most RF circuit of interest are compressive, we focus on this type.
Gain Compression: 1-dB Compression Point
With α1α3 <0, the fundamental gain is equal to α1 + 3α3A2/4 and falls as A rises.
1-dB compression point: defined as the input signal level that causes the small signal gain to drop by 1dB.
small signal gain large signal gain
Plotted on a log-log scale as a function of the input level
Output level, Aout, falls below its ideal value by 1 dB at the 1-dB compression point,
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Ain,1dB
.