Chapter 2 Basic Concepts in RF Design - SJTUiwct.sjtu.edu.cn/personal/xygan/Chapter 2 Basic Concepts...
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1 Chapter 2 Basic Concepts in RF Design
Transcript of Chapter 2 Basic Concepts in RF Design - SJTUiwct.sjtu.edu.cn/personal/xygan/Chapter 2 Basic Concepts...
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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
NonlinearityNoise 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
Signals are upconverted/downconverted at TX/RX, by an oscillator controlled by a Frequency Synthesizer.
Overall transceiver
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General Considerations: Units in RF Design
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 :
Voltage gain:
Power gain:
rms value
where Vout and Vin are rms value.
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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:
where RL= 50 Ω thus,
a sinusoid signal having a peak-to-peak amplitude of Vpp an rms value of Vpp/(2√2), 0dBm is equivalent to 1mW,
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Example of Units in RF
Solution:
Notice: For a narrowband (not sinusoid) 0-dBm signal, it is still possible to approximate the (average) peak-to-peak swing as 632mV.
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.
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.
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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
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.
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.
(b) Use of fictitious buffer to visualize the signal level in “dBm”
voltage buffer
How about the power delivery?
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General Considerations: linearity
A system is linear if its output can be expressed as a linear combination (superposition) of responses to individual inputs.
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.
For arbitrary a and b, it holds that:
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Nonlinearity: Memoryless and Static System
Memoryless, linear
Memoryless, nonlinear
Memoryless or static : if its output does not depend on the past values of its input;
The input/output characteristic of a memoryless nonlinear system can be approximated with a polynomial
If the system is time variant，αj would be general functions of time.
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Nonlinearity: Memoryless and Static System
In this idealized case, the circuit displays only second-order nonlinearity.
Example of a memoryless nonlinear circuit:
Common-source stage
If M1 operates in the saturation region and can be approximated as a square-law device [Lee]
then𝐼𝐼𝐷𝐷 =
12𝜇𝜇𝑛𝑛𝐶𝐶𝑜𝑜𝑜𝑜
𝑊𝑊𝐿𝐿
𝑉𝑉𝑖𝑖𝑛𝑛 − 𝑉𝑉𝑇𝑇𝑇𝑇 2
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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) is odd symmetry if αj=0 for even j.
2 30 1 2 3( ) ( ) ( ) ( )y t x t x t x tα α α α= + + + +
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 ApproximationFor 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
Solution:
Plot the output waveform of the circuit in Fig. 1: vin1 = A1 cos ω1t vin2 = A2 cos(1.25ω1t )
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”.
A time shift in input does not result in the same time shift in output.
Fig.1
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Time Variance: Generation of Other Frequency Components
A linear system can generate frequency components that do not exist in the input signal when system is time variant.
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 sincenvelop.
T1= 2π /ω1
Multiplication in time domain
Convolution in frequency domain
Illustration:
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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
2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
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 :
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)
input:
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Effects of Nonlinearity: Harmonic Distortion
DC Fundamental Second Harmonic
ThirdHarmonic
output:
( ) cosx t A tω=
If a sinusoid is applied to a nonlinear system: the output exhibits frequency components that are integer multiplies (“harmonics”)
of the input frequency.
The term with the input frequency
Arising from second-order nonlinearity
2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
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Observations
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? An .
2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
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Example of Harmonic Distortion in Mixer
Solution:
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.
(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
Solution:
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.
(b)
Analog multiplier
Third Harmonic of x2(t)
The mixers produces two “spurious” components at ω1+3ω2 and ω1-3ω2, Cause other problems…
These are “undesired” components that are difficult to remove by filter.
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Example of Harmonics on GSM SignalThe 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
Compressive: The term α3x3 “bends” the characteristic
for sufficiently large x, decreasing the gain as the input
amplitude increases.
Most RF circuit of interest are compressive, we focus on this type.
The gain of fundamental component ω is equal to α1 + 3α3A2/4 varied as A becomes larger.
Expansive: expanding the gain as the input
amplitude increases
( ) cosx t A tω= 2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
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Gain Compression: 1-dB Compression Point
Output level, Aout, falls below its ideal value by 1 dB at the 1-dB compression point, Ain,1dB.
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.
Plotted on a log-log scale as a function of the input level
small signal gain
large signal gain
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Calculation
21 3 ,1dB 1
320log 20log 1dB4 inAα α α+ = −
large signal gain
small signal gain
1,1dB
3
0.145inA αα
=
Notice: The relationship of the input is between peak value (rather than the peak-to-peak value).
Ain and Aout are voltage quantities, but compression can also be expressed in terms of power quantities (dBm).
At the 1-dB compression point, Ain,1dB:
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Observations
• The 1-dB compression point is typically in the range of -20 to -25 dBm (63.2 to 35.6 mVpp in 50-Ω system) at the input of RF receivers front-end amplifiers.
• We use the notations A1dB and P1dB interchangeably in this book.
• Why does compression matter?– a signal is so large as to reduce the gain of a receiver
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Gain Compression: Effect on FM and AM Waveforms
Are they acceptable? FM signal carries no information in its amplitude and hence tolerates
compression. √ AM contains information in its amplitude, hence distorted by compression. X
• Gain compression on two signals?
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Gain Compression: Desensitization
According to the effect of gain compression, if a signal is too large, it will reduce the gain of a receiver.
Desensitization: when small desired signal is superimposed on the large interferer, the receiver gain is reduced even though the desired signal itself is small.
This phenomenon lowers the signal-to-noise ratio (SNR) at the receiver, even if the signal contains no amplitude information.
If there are two signals: a small desired signal + a large interferer
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Solution:
Example of Gain CompressionA 900-MHz GSM transmitter delivers a power of 1 W to the antenna. By how much must the second harmonic of the signal be suppressed (filtered) so that it does not desensitize a 1.8-GHz receiver having P1dB = -25 dBm? Assume the receiver is 1 m away and the 1.8-GHz signal is attenuated by 10 dB as it propagates across this distance.
The output power at 900 MHz is +30 dBm(1W).
With an attenuation of 10 dB, the second harmonic must not exceed -15 dBm at the transmitter antenna so that it is below P1dB of the receiver.
Thus, the second harmonic must remain at least 45 dB below the fundamental at the TX output.
In practice, this interference must be another several dB lower to ensure the RX does not compress.
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How to quantify desensitization?
The gain experienced by the desired signal is equal to α1 + 3α3A22/2,
a decreasing function of A2 if α1α3 <0. For sufficiently large A2, the gain drops to zero, and we say the signal is
“blocked”.
With the third-order characteristic,
For A1 << A2
2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
the output is:
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Observations
In RF design, the term “blocking signal” or “blocker” refers to interferers that desensitize a circuit, even if they do not reduce the gain to zero.
Some RF receivers must be able to withstand blockers that are 60 to 70 dB greater than the desired signal.
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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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Effects of Nonlinearity: Cross Modulation
When a weak signal and a strong interferer pass through a nonlinear system,
“cross modulation” will happen.
Phenomenon: modulation (or noise) on the amplitude of the interferer will
transfer to the amplitude of the weak signal.
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Effects of Nonlinearity: Cross Modulation
Desired signal at output suffers from amplitude modulation at ωm and 2ωm.
Suppose that the interferer is an amplitude-modulated signal
where m is a constant and ωm denotes the modulating frequency.
For A1 << A2
Expressing the input as:
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Solution:
Example of Cross ModulationSuppose an interferer contains phase modulation but not amplitude modulation. Does cross modulation occur in this case?
The desired signal at ω1 does not experience cross modulation. (constant can be removed.)
PM interferer (A2 is constant but Φ varies with time)
We now note that (1) the second-order term yields components at ω1 ± ω2 but not at ω1--------can be removed; (2) the third-order term expansion gives 3α3A1 cos ω1t A2
2 cos2(ω2t+Φ), according to cos2x=(1+cos2x)/2, results in a component at ω1.
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Example of Cross Modulation: observations
Phase-modulated interferers do not cause cross modulation in memoryless (static) nonlinear systems.
Cross modulation commonly arises in amplifiers that must simultaneously process many independent signal channels.
Examples include cable television transmitters and system employing “orthogonal frequency division multiplexing” (OFDM).
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Desired Signal
Desired Signal + one (large) interferer
Desired Signal + two large interferers???
Effects of Nonlinearity: Intermodulation—Recall Previous Discussion
Harmonic distortion
Desensitization + Cross Modulation
Intermodulation
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Effects of Nonlinearity: Intermodulation
If two interferers at ω1 and ω2 are applied to a nonlinear system, – the output exhibits components that are not harmonics of these
frequencies.
The phenomenon of “intermodulation” (IM) arises from “mixing” (multiplication) of the two components.
Assume
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Effects of Nonlinearity: Intermodulation
Third-order characteristic,
Intermodulation products:
Fundamental components:
Expanding the right-hand side and discarding the dc terms, harmonics, and components at ω1 ±ω2, we have:
2 31 2 3( ) ( ) ( ) ( )y t x t x t x tα α α≈ + +
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Third-order IM products
The third-order IM products at 2ω1- ω2 and 2ω2-ω1 are of particular interest. if ω1 and ω2 are close to each other, 2ω1- ω2 and 2ω2-ω1 are close to ω1 and ω2,
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Intermodulation Product Falling on Desired Channel
desired
Interferer
Suppose an antenna receives: a small desired signal at ω0
two large interferers at ω1 and ω2, Applying to a low-noise amplifier (nonlinear). Suppose
The intermodulation product at 2ω1-ω2, falls onto the desired channel, corrupting the signal.
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Solution:
Example of IntermodulationSuppose four Bluetooth users operate in a room as shown in figure below.
User 4 is in the receive mode and attempts to sense a weak signal transmitted by User 1 at 2.410 GHz;
Users 2 and 3 transmit at 2.420 GHz and 2.430 GHz, respectively. Explain what happens?
The frequencies transmitted by Users 1, 2, and 3 are equally spaced; The intermodulation in the LNA of RX4 corrupts the desired signal at 2.410 GHz.
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Example of Gain Compression and Intermodulation
A Bluetooth receiver:a) a low-noise amplifier (LNA) having a gain of 10b) input impedance of 50 Ω.
The LNA senses a desired signal level of -80 dBm at 2.410 GHz and two interferers of equal levels at 2.420 GHz and 2.430 GHz.
a) For simplicity, assume the LNA drives a 50-Ω load.a) Determine the value of α3 that yields a P1dB of -30 dBm.b) If each interferer is 10 dB below P1dB, determine the corruption experienced by
the desired signal at the LNA output.
(a) From previous equation, α3 = 14,500 V-2
(b) Each interferer has a level of -40 dBm (= 3.16 m Vp), the amplitude of the IM product at 2.410 GHz as:
If the gain is not compressed, can we say that intermodulation is negligible?
(c) The desired signal is amplified by a factor of α1=10=20dB, emerging at the output at a level of -60dBm.
(d) The IM product is as large as the signal even though the LNA does not experience significant compression.
,1dB 1 30.145inA α α= -80 dBm
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Intermodulation: Third Intercept Point
A: equal amplitude of each tone
“Third Intercept Point” (IP3): if the amplitude of each tone arises, that of the output IM products increases more sharply (∞A3). Thus, if we continue to raise A, the amplitude of the IM products eventually becomes equal to that of the fundamental tones at the output.
The input level at which this occurs is called the “input third interception point” (IIP3),where the corresponding output is represented by OIP3.
Equate the fundamental and IM amplitudes:
11dB
3
0.145A αα
=
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Example of Third Intercept Point
Solution:
A low-noise amplifier senses a -80-dBm signal at 2.410 GHz and two -20-dBm interferers at 2.420 GHz and 2.430 GHz. What IIP3 is required if the IM products must remain 20 dB below the signal? For simplicity, assume 50-Ω interfaces at the input and output.
At the LNA output:
IM amplitudeFundamental amplitude
Asig = 31.6μV; Aint = 31.6mV Extremely difficult to achieve in this case.
Usually we have:
A1dB: -20 to -25dBm
+9.6dB
IP3: -10 to -15dBm
>
>
>
3.16 VP
+20dBm
>
>
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Single Signal
Signal + one large interferer
Signal + two large interferers
Effects of Nonlinearity: Recall Previous Discussion
Harmonic distortion
Desensitization + Cross Modulation
Intermodulation
So far we have discussed the case of:
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Effects of Nonlinearity: Cascaded Nonlinear Stages
Considering only the first- and third-order terms, we have:
Consider two nonlinear stages in cascade:
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Cascaded Nonlinear Stages: Intuitive results
divide “IP3 of the second stage” by α1. Effect: the higher the gain of the first stage, the more nonlinearity is contributed by
the second stage.
α1 ↑ → ↑ → ↑→ AIP3 ↓ → nonlinearity ↑
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IM Spectra in a Cascade
Notice: The nonlinearity of the latter stages becomes increasingly more critical;
because the IP3 of each stage is equivalently scaled down by the total gain preceding that stage.
For more stages:
Most RF systems incorporate narrowband circuits the terms at ω1 ± ω2 , 2ω1, and 2ω2 are heavily attenuated at the output of
the first stage second-order nonlinearity is attenuated.
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Example of Cascaded Nonlinear Stages
Solution:
A low-noise amplifier having an input IP3 of -10 dBm and a gain of 20 dB is followed by a mixer with an input IP3 of +4 dBm. Which stage limits the IP3 of the cascade more?
α1 = 20 dB
The scaled IP3 of the second stage AIP3,2 is lower than the IP3 of the first stage AIP3,1,
The second stage limits the overall IP3 more.
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Sections to be covered
• 2.1 General Considerations• 2.2 Effects of Nonlinearity • 2.3 Noise • 2.4 Sensitivity • 2.5 Passive Impedance Transformation
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Noise
The performance of RF systems is limited by noise.
Without noise, an RF receiver would be able to detect arbitrarily small inputs,
allowing communication across arbitrarily long distances.
In this section, we review basic properties of noise and methods of calculating noise in circuits.
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Noise: Noise as a Random Process
The average current remains equal to VB/R; The instantaneous current displays random values.
“noise is random,” the instantaneous value of noise cannot be predicted.
Example: a resistor tied to a battery and carrying a current. Due to the ambient temperature, each electron carrying the current experiences
thermal agitation, moving toward the positive terminal of the battery with a random path;
Higher temperature
a) greater thermal agitation of electrons;
b) larger fluctuations in the current.
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Noise: Noise as a Random Process Noise cannot be characterized in terms of instantaneous voltages or
currents.
How do we express the concept of larger random swings for a current or voltage quantity? Average power
n(t) represents the noise waveform
Noise components in circuits have a constant average power. For example, Pn is known and constant for a resistor at a constant
temperature.
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Noise Spectrum: Power Spectral Density (PSD)
One-SidedTwo-Sided
Total area under Sx(f) represents the average power carried by x(t)
Parseval's theorem
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Example of Noise SpectrumA resistor of value R1 generates a white noise voltage whose one-sided PSD is given by
Boltzmann constant k = 1.38 × 10-23 J/K The absolute temperature T.
(a) What is the total average power carried by the noise voltage?(b) What is the dimension of Sv(f)?(c) Calculate the noise voltage for a 50-Ω resistor in 1 Hz at room temperature.
(a) Ideally, the area under Sv(f) appears to be infinite， resistor noise arises from the finite ambient heat.
In reality, Sv(f) begins to fall at f > 1 THz, exhibiting a finite total energy, i.e., thermal noise is not quite white.
(b) The dimension of Sv(f) is voltage squared per unit bandwidth (V2/Hz), we may write the PSD as
where denotes the average power of Vn in 1 Hz.
(4kTRΔf) indicates the total noise in a bandwidth of Δf .
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Example of Noise SpectrumA resistor of value R1 generates a noise voltage whose one-sided PSD is given by
where k = 1.38 × 10-23 J/K denotes the Boltzmann constant and T the absolute temperature. (c) Calculate the noise voltage for a 50-Ω resistor in 1 Hz at room temperature.
(c) For a 50-Ω resistor at T = 270 C or 3000 K
a root-mean-squared (rms) quantity:
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Noise Figure
How does the SNR degrade as the signal travels through a given circuit? If the circuit contains no noise,
• the output SNR is equal to the input SNR • (even if the circuit acts as an attenuator.)
To quantify how noisy the circuit is, we define its noise figure (NF) as:
equal to 1 for a noiseless stage.
in decibels:
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Device Noise
Thermal noise of resistors can be modeled by a series voltage source with a PSD of [Thevenin equivalent, 戴维南等效] or a parallel current source with a PSD of [Norton equivalent, 诺顿等效 ];
Polarity of the sources is unimportant but must be kept same throughout the calculations.
The choice of the model sometimes simplifies the analysis.
Model the noise by voltage and current sources:
(a) Thevenin (b) Norton models of resistor thermal noise
(a) (b)
2
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Noise in MOSFETS Thermal noise of MOS transistors
operating in the saturation regionis approximated by a current source tied between the source and drain terminals;
γ is the “excess noise coefficient” and gm the transconductance. The value of γ： 2/3 for long-channel transistors； may rise to even 2 in short-channel devices.
The actual value of γ has other dependencies and is usually obtained by measurements for each generation of CMOS technology.
(a) Current source (b) Voltage source
or can be modeled by a voltage source in series with gate.
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Other noises in MOSFETS Flicker or “1/f ” noise
– Modeled by a voltage source in series with the gate,
– 1/f dependence
– For a given device size and bias current, the 1/f noise PSD intercepts the thermal noise PSD at some frequency, called the “1/f noise corner frequency,” fc.
– It can be negligible at high frequencies.
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Effect of Transfer Function on Noise
If white noise is applied to a low-pass filter, how do we determine the PSD at the output?
If x(t) is applied to a linear, time-invariant system with a transfer function H(s), then the output spectrum is
where H(f)= H(s = j2πf ).
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Example of Device NoiseSketch the PSD of the noise voltage measured across the parallel RLC tank depicted in figure below.
Modeling the noise of R1 by a current source ,
At f0, L1 and C1 resonate, reducing the circuit to only R1. Thus, the output noise at f0 is simply equal to 4kTR1.
At lower or higher frequencies, the impedance of the tank falls ; the output noise falls.
(a) RLC
ZT
(c) output noise spectrum due to R1(b) inclusion of resistor noise
The transfer function Vn/In1 is equal to the impedance of the tank ZT ,
𝐼𝐼𝑛𝑛12
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Energy delivery of thermal noise
Available noise power
Effect: R1 continuous to draw thermal energy from its environment, converting it to noise and delivering the energy to R2.
This quantity reaches a maximum if R2 = R1 :PR2,max = kT (W/Hz) =-174 dBm/Hz
What is the energy delivered by thermal noise? R1 at room temperature; R2 is an ideal noiseless resistance (at 00K) ;
Transfer of noise from one resistor to another
The average power transferred to R2 is:
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Observations
• If a passive circuit dissipates energy, – it must contain a physical resistance – must produce thermal noise.
• “lossy circuits are noisy”.
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A Theorem about Lossy Circuit
If the real part of the impedance seen between two terminals of a passive network is equal to Re{Zout},
the PSD of the thermal noise seen between these terminals is given by 4kTRe{Zout}
passive network
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Lossy Circuit: example
A transmitter antenna dissipates energy by radiation
according to the equation V2
TX,rms/Rrad, where Rrad is the “radiation
resistance”.
(a) Transmitting antenna
As a receiving element, the antenna generates a thermal noise PSD of
(b) receiving antenna producing thermal noise
Useful Tools
• A Theorem about Lossy Circuit
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passive network
• Effect of Transfer Function on Noise
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Noise Figure
How does the SNR degrade as the signal travels through a given circuit? If the circuit contains no noise,
• then the output SNR is equal to the input SNR • even if the circuit acts as an attenuator.
To quantify how noisy the circuit is, we define its noise figure (NF) as:
equal to 1 for a noiseless stage.
in decibels:
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Noise Figure
Depends on: the circuit under consideration the SNR provided by the preceding stage.
If SNRin=∞ NF=∞ even though the circuit may have finite internal noise.
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Calculation of Noise Figure (1) A low-noise amplifier senses the signal received by an antenna;
The antenna “radiation resistance,” RS, produces thermal noise.
represents the thermal noise of the antenna;
represents the output noise of the LNA.
How to compute SNRin at the LNA input and SNRout at its output?
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Calculation of Noise Figure (2)
If the LNA exhibits an input impedance of Zin,
Both Vin and VRS experience an attenuation factor of α = Zin / (Zin + RS)
We assume a voltage gain of Avfrom the LNA input to the output;
Output signal power:
Output noise consists of two parts: the noise of the antenna amplified
by the LNA,
the output noise of the LNA,
80
Calculation of Noise Figure (3)
Calculate the output noise due to the amplifier, divide it by the gain, normalize it to 4kTRs , add 1 to the result.
81
Calculation of Noise Figure (4)
Reduce the right hand side to a simpler form: The numerator is the total noise measured at the output, denoted
as , includes both the source impedance noise and the LNA noise;
A0 =| α |Av is the voltage gain from Vin to Vout (not the gain from the LNA input to its output).
Divide total output noise by the gain from Vin to Vout
normalize the result to the noise of Rs
82
Noise Figure of Cascaded Stages
AP1 is the “available power gain” of the first stage, the “available power” at its output, Pout,av (the power that it would deliver to a
matched load) divided by the available source power, PS,av (the power that the source would deliver to a
matched load).
Called “Friis’ equation”, the noise contributed by each stage decreases as the total gain
preceding that stage increases； the first few stages in a cascade are the most critical.
83
Noise Figure of Lossy Circuits Passive circuits (filters) appear at the front end of RF transceivers; their loss proves critical.
attenuate the signal introduce noise.
Proposition: the noise figure of a passive circuit is equal to its “power loss,” defined as L=Pin/Pout, Pin is the available source power; Pout is the available power at the output.
Proof: The lossy circuit is driven by a source impedance of RS while driving a load impedance of RL.
84
Noise Figure of Lossy Circuits
The available source power is Pin =V2in/(4RS);
According to Thevenin equivalent, the available (matching) output power is Pout=V2
Thev/(4Rout).
According to the theorem of “thermal noise model of lossy circuit”, we have
Divide total output noise by the squared gain from Vin to Vout and normalize the result to the noise of Rs
VThev
85
Example of Noise Figure of Lossy CircuitsA receiver
a front-end band-pass filter (BPF) to suppress some of the interferers that may desensitize the LNA;
the filter has a loss of L; the LNA a noise figure of NFLNA;
calculate the overall noise figure.
The noise figure of the filter is NFfilt,
where NFLNA can be calculated with respect to the output resistance of the filter
According to the definition of L, available power gain Ap1=1/L
Friis’ equation:
Example:if L = 1.5 dB and NFLNA = 2 dB, then NFtot = 3.5 dB.
86
Sections to be covered
• 2.1 General Considerations• 2.2 Effects of Nonlinearity • 2.3 Noise • 2.4 Sensitivity• 2.5 Passive Impedance Transformation
87
Sensitivity
Noise Floor: the total integrated noise
The minimum signal level that a receiver can detect with “acceptable quality.” acceptable quality:
sufficient signal-to-noise ratio, It depends on the type of modulation and the corruption (e.g., bit error rate) that the
system can tolerate.
All quantities expressed in W/Hz (or dBm/Hz)
Match of receiver and antennaEq. (2.91) PRS= kT= -174 dBm/Hz
Sensitivity
88
Example of Sensitivity
Solution:
Compare the sensitivities of these two systems if both have an NF of 7 dB. A GSM receiver requires a minimum SNR of 12 dB and has a channel
bandwidth of 200 kHz. A wireless LAN receiver specifies a minimum SNR of 23 dB and has a
channel bandwidth of 20 MHz.
For the GSM receiver, Psen = -174+7+53+12= -102 dBm,
For the wireless LAN system, Psen = -174+7+73+23= -71 dBm.
Does this mean that the latter is inferior? No, the latter employs a much wider bandwidth and a more efficient
modulation to accommodate a data rate of 54 Mb/s.
The GSM system handles a data rate of only 270 kb/s.
In other words, specifying the sensitivity of a receiver without the data rate is not meaningful.
89
Source
RF Receiver
Digital Receiver
The journey of Noise
90
Sections to be covered
• 2.1 General Considerations• 2.2 Effects of Nonlinearity • 2.3 Noise • 2.4 Sensitivity• 2.5 Passive Impedance Transformation
– Thomas H. Lee– Razavi– 张肃文
91
Sections to be covered
• 2.1 General Considerations• 2.2 Effects of Nonlinearity • 2.3 Noise • 2.4 Sensitivity• 2.5 Passive Impedance Transformation
– Series RLC tank– Parallel RLC tank– Series and parallel RC/RL networks– Basic Matching Networks
92
Series RLC tank circuit
1( )Z R jX R j LC
ωω
= + = + −
2 21( )Z R LC
ωω
= + −
1LC
ωω
=
1s LC
ω =
minZ R=when
resonance frequency:
The complex impedance of this circuit is given by adding up the impedance of the components:
There is a valley in impedance at resonance.
V - the voltage of the power sourceI - the current in the circuitR - the resistance of the resistorL - the inductance of the inductorC - the capacitance of the capacitor
Resistance reactance
1s
s
LLC C
ρ ωω
= = =Characteristic impedance:
93
Attributes of impedance Z
sω
Z
ω
ω
ϕ
2π
2π
−
sω
magnitude
phase shift
1( )Z R jX R j LC
ωω
= + = + −
2 21( )Z R LC
ωω
= + −
94
Quality Factor
energy storedaverage power dissipated
Q ω=
The quality factor, Q, indicates how close to ideal an energy-storing device is.
An ideal capacitor or inductor dissipates no energy, exhibiting an infinite Q, but a series resistance or a parallel resistance reduces its Q.
The definition of Q:
11s s
sL C LQ
R R R R Cω ωρ
= = = =
As to a series RLC tank circuit:
Q is dimensionless.
95
Frequency response of the current: the magnitude
maxI
sω
22 2max
1 1
1 ( 2 )1 ( )sss
ss
II
QQ ω ωωωω ω
= =++ −
is the maximum current at resonance frequency
sω ω ω= −
2 21( )Z R LC
ωω
= + −
96
Frequency response of the current: the phase shift
ω
ϕ
2π
2π
−
sω
1Q 2Q1 2Q Q>
1 XtgR
ϕ −= −
X is the imaginary part of Z, defined as reactance
1( )Z R jX R j LC
ωω
= + = + −
VIZ
=
97
BandwidthThe resonance effect can be used for filtering;
The rapid change in impedance near resonance can be used to Pass signals block signals
A key parameter in filter design is bandwidth. The bandwidth is measured between the 3dB-points；
Half-power frequencies: the power passed through the circuit has fallen to half the value passed at resonance.
2 1s
s
BQωω ω= − =
98
Q and Bandwidth B• Q is related to bandwidth;
– low Q circuits are wide band– high Q circuits are narrow band
• Q is the inverse of fractional bandwidth( B/ωs );
• Q factor is directly proportional to selectivity, as Q factor depends inversely on bandwidth.
ssQ
Bω
=
99
Loaded Q factor
ss
s
sL s
s L
LQR R
LQ QR R R
ω
ω
=+
= <+ +
load
Unloaded Q
sV
SR
CL
LR
R
100
Parallel RLC tank circuit
1( )Y G j CL
ωω
= + −
2 21( )Y G CL
ωω
= + −
1LC
ωω
=
1p LC
ω =
min1Y GR
= =when
resonance frequency:
The complex admittance of this circuit is given by adding up the admittances of the components:
There is a valley in admittance at resonance.
inI R C L outV
As to a parallel RLC tank circuit:/p p
p
R R RQ R CL L C
ωρ ω
= = = =
Conductance susceptance
1p
p
LLC C
ρ ωω
= = =Characteristic impedance:
101
Frequency response of the voltage:the magnitude
maxVpω ω ω= −
is the maximum voltage at resonance frequency
2max 2 2
1 1
1 ( 2 )1 ( )ppp
pp
VV
QQω ωω
ωω ω
= =++ −
pω
max
VV
2 1p
p
BQω
ω ω= − =
2ω
1
0.7
1ω
-3 dB
2 21( )Y G CL
ωω
= + −
inI R C L outV
102
Loaded Q factor
1( )
11
( )
pp p s
p p s LL p
p p p p s L
QL G G
R G G GQ Q
L L L G G G
ω
ω ω ω
=+
′ + += = = <
+ +
load
Unloaded Q
sI SR C L PR LR
103
Another example
How about the property of “not-quite-parallel RLC tank circuit”?
Please refer to “高频电子线路,张肃文”
sI C
L
R
V CI
LI
• Impedance transformation– Matching Network– Series/parallel
104
105
Series and parallel RC/RL networks
pRQρ
=
Series RC circuit Equivalent parallel circuit
Series RL circuitEquivalent parallel circuit
What choice of values makes the two networks equivalent?
Here, ω ≈ωs
sQRρ
=
reforming:
Equating the impedance:
106
Series-to-Parallel Conversion (1)
11
1
pp
ss
pp
Rj C
Rj C R
j C
ωω
ω
+ =+
( )21p s p p s s p p s sR C j R C R C R C R C jω ω ω= − + +
2 1p p s sR C R C ω = s pQ Q=(1)
1
1ps s
s
p p
RCR
C
ω
ω
=Real part Equation
107
Series-to-Parallel Conversion (2)
2 1sQ
When Cp and Cs are selected, the two impedances remain equal at all frequencies? No!!! An approximation allows equivalence for a narrow frequency range.
Substitute RpCp in (1) from (2):
2p s s
p s
R Q RC C
≈
≈
Since Qs=1/CsωsRs, then
Substitute Qs into (1):
2 10
p p s s
p p s s p s
R C R CR C R C R C
ω =+ − =
(1)
(2)
For a general ω,
Simplified but with constraints, near ωs
2 2
1p s
s s
R RR C ω
= +
( )2 1p s sR Q R= +2
2 1s
p ss
QC CQ
=+
Real part equation
Image part equation
108
Parallel-to-Series Conversion
Observations: Typically, Q>4, Series-to-Parallel Conversion:
retain the value of the capacitor but raises the resistance by a factor of Qs
2. Parallel-to-Series Conversion:
reduce the resistance by a factor of QP2.
This statement applies to RL sections as well.
109
Basic Matching Networks
RL transformed down by a factor
Setting imaginary part to zero (resonate), we have
If
a load resistance must be transformed to a lower value. the capacitor in parallel with RL converts this resistance to a lower series
component. The inductance is inserted to cancel the equivalent series capacitance.
1
1
1L
p L p
p
RQ R C
C
ω
ω
= =
ω≈ ωp
ω≈ ωp
110
Solution:
Example of Basic Matching Networks
Design the matching network of the following figure so as to transform RL = 50 Ωto 25 Ω at a center frequency of 5 GHz.
Assuming QP2 >> 1, we have C1 = 0.90 pF and L1 = 1.13 nH.
however, QP = 1.41, indicating the QP
2 >> 1 approximation cannot be used.
We thus obtain C1 = 0.637 pF and L1 = 0.796 nH.
Please fill out the detail calculation by yourself!
111
Another Example of Basic Matching Networks
Determine how the circuit shown below transforms RL.
The L1-RL branch to a parallel section produces a higher resistance.
If QS2 = (L1ω/RL)2 >> 1, then the equivalent parallel resistance is
The parallel equivalent inductance is approximately equal to L1 and is cancelled by C1.
112
L-Section topologies
1. In (a), the power delivered to the input port is2. Power delivered to the load is 3. if L1 and C1 are ideal, these two powers must be equal:
a network transforming RL to a lower value “amplifies” the voltage and attenuates the current by the above factor.
C1 transforms RLto a smallerseries value and L1 cancels C1
L1 transforms RLto a smaller series value while C1resonates with L1
L1 transforms RLto a largerparallel value and C1 cancels the parallel inductance
C1 transforms RLto a largerparallel value and L1 cancels the parallel capacitance
113
Transfer a Resistance to a Higher Value
RL boosted
Viewing C2 and C1 as one capacitor, Ceq converting the resulting series section to a parallel circuit
if , the parallel combination of C1 and RL can be converted to a series network
Notice: the capacitive component must be cancelled by placing an inductor in parallel with the input.
114
Impedance Matching by Transformers
An ideal transformer having a turns ratio of n “amplifiers” the input voltage by a factor of n.
No power is lost, thus:
115
Observations
The networks studied here operate across a narrow bandwidth (near resonate frequency) transformation ratio, e.g., 1+Q2, varies with frequency;
The capacitance and inductance approximately resonate over a narrow frequency range;
Broadband matching networks can be constructed, but they typically suffer from a high loss.
116
Loss in Matching Networks (1) Analyze the effect of loss.
We define the loss as the power provided by the input divided by that delivered to RL
The loss of L1 modeled by a series resistance, Rs
The power delivered to Rin1 is entirely absorbed by RL Pin1=PL.
This network transforms RL to a lower value, Rin1=RL/(1+Q2
P), suffering from loss even if Rs appears small.
117
Loss in Matching Networks (2)The loss of L1 modeled by a parallel resistance, RP.
Pin represents the power delivered by Vin, which is entirely absorbed by RP||RL:
power delivered to the load, PL,
118
Basic matching networks
L-matchπ-matchT-matchTapped capacitor resonator as an
impedance matching networkTapped inductor match
119
L-match (refer to P112,116)
There are only two degrees of freedom (one can choose only L and C);
Once the impedance transformation ratio and resonant frequencyhave been specified, network Q is automatically determined;
How to get a different value of Q?
120
π-match
1C 2C PRinR
L
1C 2C PRinR IR
1L 2L
1CinR IR
1L 2L
2sC
IR
Two L-matches connected in cascade, one transforms down one transforms up;
The load resistance RP is transformed to a lower resistance RI at the junction of the two inductances;
RI is transformed up to a value Rin by a second L-match section.
Original circuit Cascade two L-matches
Parallel RCnetwork
Seriesequivalent
121
π-match: another example
1C
2C
PRinR L
Observations:
We have three degrees of freedom (the two capacitance and the sum of two inductances),
We can independently specify center frequency, Q, overall impedance transformation ratio;
In many instance, cascading several matching networks may be helpful.
122
T-match
1C 2CinRSR
1L 2L
Connecting up the L-sections another way leads to the dual of π-match, denoted T-match;
A single capacitor in a practical implementation has been decomposed explicitly into two separate ones;
The parallel image resistance (RI) is seen across these capacitors, either looking to the right or looking to the left.
RI
RI
123
Tapped capacitor match (refer to P107)
( )
2
21
1
1
1 1 ptot p
s tot
ptot
p
CR R
CR CC C
CC C
ω
= = +
=+
(a) capacitive matching circuit
(b) simplified circuit with parallel-to-series conversion
(c) cascading two capacitance
(d) simplified circuit with series-to-parallel conversion
The capacitive divider “boosts” the value of RP by a factor ( )2
11 pC C+
124
Tapped inductor match
a) Convert the parallel combination Lp and Rp to series circuit Ls and Rs;
b) Combining L1 and Ls into Leq;
c) Converting series circuit to parallel circuit and we have:2
1
1
1tot pp
tot p
LR RL
L L L
≈ +
= +
125
Double-tapped resonator
This circuit first boosts R2 to a larger effective parallel resistance across the whole tank than in a standard tapped capacitor network;
It then reduces this parallel resistance by the tapped inductors to the desired value Rin.
1C
2CinR2R
1L
2L
