Chapter 2 Basic Concepts in RF Design

Chapter 2 Basic Concepts in RF Design 1Sections 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 2Chapter Outline Nonlinearity Noise Impedance Transformation  Harmonic Distortion  Compression  Intermodulation  Noise Spectrum  Device Noise  Noise in Circuits  Series-Parallel Conversion  Matching Networks 3The Big Picture: Generic RF Transceiver Overall transceiver  Signals are upconverted/downconverted at TX/RX, by an oscillator controlled by a Frequency Synthesizer. 4General 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 : 5where 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 6Example 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, <ul style="display: flex;"><li style="flex:1">where RL= 50 Ω </li><li style="flex:1">thus, </li></ul>7Example 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 8approximate 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. 9dBm 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” 10 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. 11 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. 12 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] 1ꢇ ꢈ2ꢀꢁ = ꢂꢃꢄꢅꢆ ꢉ − ꢉꢋꢌ ꢊꢃ 2then Common-source stage  In this idealized case, the circuit displays only second-order nonlinearity. 13 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. 14 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: <ul style="display: flex;"><li style="flex:1">Differential pair </li><li style="flex:1">Input/output characteristic </li></ul>15 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. 16 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)] <ul style="display: flex;"><li style="flex:1">Time Variance </li><li style="flex:1">Nonlinearity </li></ul>Do not be confused by these two attributes. 17 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”. 18 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) 20 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. 21 Effects of nonlinearity • Harmonic distortion (谐波失真) • Gain compression (增益压缩) • Cross modulation (互调) • Intermodulation (交叉调制) 22 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) <ul style="display: flex;"><li style="flex:1">DC </li><li style="flex:1">Fundamental </li><li style="flex:1">Second </li><li style="flex:1">Third </li></ul><ul style="display: flex;"><li style="flex:1">Harmonic </li><li style="flex:1">Harmonic </li></ul>Arising from second-order nonlinearity The term with the input frequency 23 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 . 24 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. 25 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, 26  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 27 Effects of nonlinearity • Harmonic distortion (谐波) • Gain compression (增益压缩) • Cross modulation (互调) • Intermodulation (交叉调制) 28 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 29  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, 30 Ain,1dB .

Chapter 2 Basic Concepts in RF Design

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