RF Fundamental Concepts and erformance arameters CCE 50 RF and Microwave System Design Dr. Owen Casha B. Eng. (Hons.) h.d. 09/0/0
Overview Introduction Nonlinearity and Time Variance System Noise Thermal Noise Measurement of noise temperature by the γ-factor method Noise Factor and Noise Figure Noise in two-port networks Noise figure of cascaded systems Minimum Detectible Signal (MDS) Harmonic Distortion Intermodulation Distortion Gain Compression Second Harmonic Distortion Intermodulation Distortion Ratio Intercept oint Sensitivity and Dynamic Range 09/0/0 Owen Casha 0
Introduction RF/Microwave designers use many concepts and ideas that originate from the theory of signals and systems. In this chapter, these concepts will be described and the terminology used will be clearly defined. These are the fundamental concepts which are used in any RF/Microwave design environment. 09/0/0 Owen Casha 0
Nonlinearity and Time Variance A system is linear if its output can be expressed as a linear combination of responses to individual inputs (superposition theorem). More accurately, if for inputs x (t) and x (t) operation of the system x ax ( t) y ( t), x ( t) y ( t) ( t) bx ( t) ay ( t) by ( t) constants Non-linearities: non-zero initial conditions finite offsets non-linear functions such as x, trig functions, log(x) etc 09/0/0 Owen Casha 0 4
Example: Linearity vs Non-linearity Linear System Non-Linear System -6 V - V 0.6 V 0.66 V 0.68 V -4 V 09/0/0 Owen Casha 0 5
Example: Linearity vs Non-linearity Diode Circuit Logarithmic Function (Non-linear) For small values it may be assumed as a linear response Inverting Amplifier roportional Function (Linear) 09/0/0 Owen Casha 0 6
Nonlinearity and Time Variance A system is time invariant if a time shift in its input results in the same time shift in its output. That is, x x ( t) y( t) ( t τ) y( t τ) While nonlinearity and time variance are intuitively obvious concepts, they may be confused with each other in some cases. Sine Wave V in switch V out Linear Time Variant Nonlinear Time Variant V c Square Wave 09/0/0 Owen Casha 0 7
Nonlinearity and Time Variance A linear system can generate frequency components that do not exist in the input signal. This is possible if the system is time variant. A system is called memoryless if its output does not depend on the past values of its input. For a memoryless lienar system: ( t) x( t) y α where α is a function of time if the system is time variant. For a memoryless nonlinear system, the input-output relationship can be approximated with a polynomial, y ( ) ( ) ( ) ( ) t α αxt α xt α xt... 0 09/0/0 Owen Casha 0 8
System Noise Random movement of charges or charge carriers in an electronic device generates currents and voltages that vary randomly with time. Their amplitude can be therefore expressed only in terms of the probability density functions. For most applications, it suffices to know the mean square (noise power) or the root-mean square value of noise. Noise power is normally a function of frequency and the power-per-unit frequency is defined as the power spectral density of noise. 09/0/0 Owen Casha 0 9
Types of Noise Thermal Noise: This is the most basic type of noise which is caused by thermal vibration of bound charges. Shot Noise: This is due to random fluctuations of charge carriers that pass through the potential barrier in an electronic device. For example, charge carriers in Schottky diodes produce a current that fluctuates about the average value I. The mean-square current due to shot noise is generally given by: i sh eib e is the electronic charge (.6 x 0-9 C) B is the bandwidth in Hz 09/0/0 Owen Casha 0 0
Types of Noise Flicker noise: This occurs in solid-state devices at low frequency. Its magnitude decreases inversely proportional with frequency. It is generally attributed to chaos in the dynamics of the system. It is often called /f noise or pink noise. It occurs in almost all electronic devices, and results from a variety of effects, such as impurities in a conductive channel, generation and recombination noise in a transistor due to base current, and so on. It is always related to a direct current. 09/0/0 Owen Casha 0
Thermal Noise Consider a resistor R that is a temperature T (in kelvin). Electrons in this resistor are in random motion with a kinetic energy that is proportional to the temperature T. This random motion produces small, random voltage fluctuations across its terminals. This voltage has a zero average value, but a non-zero mean square value. Via lanck s black body radiation law: 09/0/0 v n centre frequency in Bandwidth exp 4hfRB ( hf / kt) lanck s constant (6.546 x 0-4 Js) Bandwidth (in Hz) Boltzmann s Constant.8 x 0 - J/K Owen Casha 0
Thermal Noise Now for a temperature above 0 K and a for a frequency below 00 GHz: hf < 6.546 x 0 - J and kt >.8 x 0 - J such that the exponential term can be approximated to hf/kt 4hfRB v n 4kTBR hf / kt <v n > ~ R Thevenin s Equivalent Circuit (R is noiseless) 09/0/0 Owen Casha 0
Thermal Noise This source will supply maximum power to a load of resistance R. The power delivered is: vn 4 R n ktb Conversely, if an arbitrary white noise source with its driving point impedance R delivers a noise power s to a load R then it can be represented by a noisy resistor of value R that is at temperature T e. T s e kb 09/0/0 Owen Casha 0 4
Thermal Noise Consider a noisy amplifier. Its gain is G over a bandwidth B. Let the amplifier be matched to the noiseless source resistor R S and load resistor R L. 0 K T e K R Noisy s R L R Noiseless s Amplifier Amplifier R L If the source resistor is at a hypothetical temperature of 0 K, then the power input to the amplifier i will be zero and the output noise power o will be only due to the noise generated by the amplifier. 09/0/0 Owen Casha 0 5
Thermal Noise We can obtain the same noise power at the output of the ideal noiseless amplifier by raising the temperature of the source resistor from 0 K to T e K, where T e is the equivalent noise temperature of the amplifier. T o e GkB R s Noiseless Amplifier R L Output noise referred to the input of the amplifier 09/0/0 Owen Casha 0 6
Measurement of Noise Temperature The γ-factor method According to definition, the noise of any two port network (such as an amplifier) can be determined by setting the source resistance at 0 K and then measuring the output noise power. However a temperature of 0 K cannot be achieved in practice. We can circumvent this problem by repeating the experiment at different temperatures. Consider an amplifier with power gain G over a frequency band B. 09/0/0 Owen Casha 0 7
Measurement of Noise Temperature Further, its equivalent noise temperature is T e K. The input port of the amplifier is terminated by a matched resistor R while a matched power meter is connected at the output. T k G T c GkT k B GkT B GkTB GkTB c e e γ T T k c T T e e T e Tk γt γ c 09/0/0 Owen Casha 0 8
Example An amplifier has a power gain of 0 db in the 500 MHz.5 GHz frequency band. The following was obtained using the γ-factor method: T k 90 K T c 77 K -70 dbm -75 dbm Determine the equivalent noise temperature. If this amplifier is used with a source that has an equivalent noise temperature at 450 K, find the output noise power in dbm. Ans: -7 dbm 09/0/0 Owen Casha 0 9
Noise Factor and Noise Figure Noise factor of a -port network is obtained by dividing the SNR at its input port by the SNR at its output. F S S i o i o S Signal N Noise If the -port network is noise-free, the SNR out SNR in such that F. The network will add its own noise while the input signal and noise will be altered by the same factor (gain or loss). It will lower the output SNR resulting in a higher F. Therefore F>. 09/0/0 Owen Casha 0 0
Noise Factor and Noise Figure By definition the input noise power is assumed to be the noise power resulting from a matched resistor at T o 90K, that is, N i kt o B. ~ R Noisy -port network G, B, Te R i S i N i o S o N o 09/0/0 Owen Casha 0
Noise Factor and Noise Figure Using the circuit arrangement illustrated in the previous slide, the noise factor of a noisy -port network can be defined as: F Total output noise in B when input source temperature is Output noise of source (only) at 90K 90K F ktobg ktbg o int int ktbg o int represents output noise power that is generated by the -port network internally. 09/0/0 Owen Casha 0
Noise Factor and Noise Figure int ( F ) TBG ktbg k o e where T e is the equivalent noise temperature of a -port network. When expressed in decibels noise factor is called noise figure. F 0log0F ( db) 09/0/0 Owen Casha 0
Example ower gain of an amplifier is 0dB in the frequency band of 0GHz-GHz. If its noise figure is.5db, find the output noise power in dbm (T o 90K). o kt o BG int FkT o BG F 0.5/0.4 G 0 0/0 00 Boltzmann s Constant (k).8 x 0 - J/K o 57 dbm 09/0/0 Owen Casha 0 4
Noise in two-port networks Consider a noisy two-port network: I I I s Y s V Noisy -port network V where Y s G s jb s is the source admittance that is connected at port of the network. The noise it generates is represented by i s with rms value I S. 09/0/0 Owen Casha 0 5
Noise in two-port networks This noisy network can be replaced by a noise free network with current source i n and a voltage source v n connected at its input. I n and V n represent rms values. It is assumed that the noise represented by I s is uncorrelated with i n and v n. However, a part of i n, i nc, is assumed to be correlated with v n via the correlation admittance Y c G c jx c while the remaining part i nu is uncorrelated. 09/0/0 Owen Casha 0 6
Noise in two-port networks V n I I s Y s I n Noiseless two-port network V I s i s 4kTBG s V n v n 4kTBR n Now we find a Norton equivalent for the circuit that is connected at the input of the noise-free -port network. I nu i nu 4kTBG nu 09/0/0 Owen Casha 0 7
Noise Figure of a Cascaded System Consider a two-port network with gain G, noise factor F and equivalent noise temperature T E connected in cascade with another two-port network having gain G, noise factor F and noise temp T E. N in T in G F T E N G F T E N out Assume that the noise power input to the first network is N in. Its equivalent noise temperature T in. 09/0/0 Owen Casha 0 8
Noise Figure of a Cascaded System Output noise power of the first system is N, whereas it is N out after the second system. Hence, GkTB GkT i E B out G GkT E B out out ( GkTB GkT B) G i GGkBT ( T ) i E E GkT GkT E B E B out GGkB Ti T E TE G 09/0/0 Owen Casha 0 9
Owen Casha 0 0 09/0/0 Noise Figure of a Cascaded System ( ) e i E E i out T GGkBT G T T T GGkB Noise temperature of a cascaded system (T e ) ( ) ( ) ( ) i e i E i E T F T T F T T F T ( ) ( ) ( ) ( ) G F F F G T F T F T F G T T T i i i E E e Noise Figure of a Cascaded System
Noise Figure of a Cascaded System Thus from the above derivation we can generalize to: F F ( F ) ( F ) ( F ) 4 G GG GGG... T e T T T T E E E 4 E G GG GGG... 09/0/0 Owen Casha 0
Example Two amplifiers, each with 0dB gain are connected in cascade as shown below. The noise figure of amplifier A is db while that of A is 5dB. Calculate the overall gain and noise figure for this arrangement. If the order is changed in the system find the resulting noise figure. A A NF db NF5 db 09/0/0 Owen Casha 0
Example A receiving antenna is connected to an amplifier through a transmission line that has an attenuation of db. The gain of the amplifier is 5dB and its noise temperature is 50K over a 00MHz bandwidth. All the components are at a temperature of 00K. (a) Find the noise figure of the cascaded system (b) What would be the noise figure if the amplifier were placed before the transmission line? Antenna Transmission Line 5dB Amplifier 09/0/0 Owen Casha 0
Minimum Detectible Signal (MDS) Consider a receiver circuit with gain G over a bandwidth B. Assume that its noise factor is F. N I I Receiver N o o o kt o FBG This constitutes the noise floor of the receiver. A signal weaker than this will be lost in the noise. The signal is generally taken as db higher than this level: ( kt ) F 0log B 0 G 0 log0 log o db o 0 0 09/0/0 Owen Casha 0 4
Minimum Detectible Signal (MDS) The corresponding signal power IMDS at the input is: IMDS IMDS db dbm 0log 0 0log ( kt ) 0 o ( kt ) o F F 0log 0 0log 0 B db B 0 Room Temperature 90K IMDS dbm 0log 0 (.8 0 90) F 0log 0 B 0 IMDS dbm 04 F 0log 0 B 7 F 0log 0 B 09/0/0 Owen Casha 0 5
Example The noise figure of a comms receiver is found at 0dB at room temperature. Determine the minimum detectable signal power if bandwidth is MHz, GHz, 0GHz. IMDS dbm 7 F 0log0B 0 dbm at MHz 7 dbm at GHz The receiver can detect a relatively weak signal when its bandwidth is narrow 6 dbm at 0GHz 09/0/0 Owen Casha 0 6
Sensitivity The sensitivity of an RF receiver is defined as the minimum signal level that the system can detect with acceptable signal-to-noise ratio. To calculate the sensitivity, we write: S R in F S R out sig S R RS out where sig denotes the input signal power and RS the source resistance noise power, both per unit bandwidth. It follows that: F sig RS S R out 09/0/0 Owen Casha 0 7
Sensitivity Since the overall signal power is distributed across the channel bandwidth, B, sig must be integrated over the bandwidth to obtain the total mean square power. Thus, for a flat channel, sig ( tot ) RS F S Rout B The above equation predicts the sensitivity as the minimum input signal that yields a given value for the output SNR. sig in F S R 0logB, mindbm RSdbm / Hz min db db Independent of the system gain 09/0/0 Owen Casha 0 8 B
Sensitivity k.80 6504 0 J/K Assuming conjugate matching at the input, we obtain RS as the noise power that R S delivers to the receiver: Boltzmann s Constant Temperature (Kelvin) Mean square noise voltage RS v n 4kTR ( R ) in Rs 4Rin s v n R s Receiver s Input Impedance R in Receiver RS 0log0kT 74 dbm/hz Room Temperature 09/0/0 Owen Casha 0 9
Sensitivity in 74dBm / Hz F S R 0log, mindbm min db db B The sum of the first three terms is the total integrated noise of the system and is sometimes called the noise floor. Since in,min is a function of the bandwidth, a receiver may appear very sensitive simply because it employs a narrowband channel but this comes at the cost of a low information rate. 09/0/0 Owen Casha 0 40
Intermodulation Distortion The electrical noise of a system determines the minimum signal level that it can detect. On the other hand, the signal will get distorted if its level is too high. This occurs because of the nonlinear characteristics of the system or device. 09/0/0 Owen Casha 0 4
Intermodulation Distortion V i Nonlinear System V out Consider a nonlinear system. Assume that its nonlinearity is frequency independent and can be represented by the following power series: V o kv i kvi kvi For simplicity, we assume that k i are real and the first terms are sufficient to represent its output signal. Assume that: ( ω t) b ( ωt) V i a cos cos -tone signal... 09/0/0 Owen Casha 0 4
Owen Casha 0 4 09/0/0 Intermodulation Distortion Therefore, the corresponding output signal can be written as: ( ) ( ) [ ] ( ) { } ( ) { } ( ) ( ) { } ( ) ( ) ( ) { } ( ) { } ( ) ( ) ( ) ( ) { } ( ) { } ( ) t ab t b a k t b t b t b a t b a k t ab t a t ab t a k t t ab t b t a k t b t a k V o cos 4 cos 4... cos 4 cos 4 cos cos 4... cos 4 cos 4 cos cos 4... cos cos cos cos...... cos cos ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω
Intermodulation Distortion Therefore, the output signal has several frequency components in its spectrum as shown below: 0 X: 50 Y: 9.69 ω 8 ω Amplitude (V) 6 4 X: 0 Y: 6.5 ω -ω X: 0 Y:.60 ω ω X: 70 Y:.599 ω -ω ω X: 00 Y:.79 09/0/0 X: 0 Y: 0.05 X: 40 Y: 0.5056 X: 80 Y: 0.4788 X: 0 Y: 0.4787 X: 50 Y: 0.55 0 0 50 00 50 00 ω -ω ω ω Frequency (Hz) ω ω ω Owen Casha 0 44
Intermodulation Distortion The figure below illustrates the input-output characteristic of an amplifier. If the input signal is too low then it may be submerged under the noise. Output power rises linearly above the noise as the input is increased. However it deviates from the linear characteristic after a certain level of input power. out O ( dbm) ( dbm) GdB ( ) out in db Dynamic Range Noise level db compression point IMDS D in 09/0/0 Owen Casha 0 45
db Compression oint The input power for which the output deviates by db below its linear characteristic is known as the db compression point. O D ( dbm) ( dbm) GdB ( ) ( dbm) ( dbm) GdB ( ) O D The difference between the input power at db compression point and the minimum detectable signal defines the dynamic range (DR). Hence, DR D ( dbm) ( dbm) IMDS DR ( dbm) GdB ( ) F 0 ( B ) O log 0 MHZ 09/0/0 Owen Casha 0 46
Gain Compression Nonlinear characteristics of a system or circuit (amplifier, mixer, LNA) compress its gain. If there is only one input signal, i.e. b is zero, then the amplitude a of cosω t in its output is found to be: a ka k a 4 The first term represents the linear case (ideal) while the second results from the nonlinearity. Generally, k is a negative constant, and therefore tends to reduce a. The single tone gain compression factor can be defined as: a A C ka ka 4k 09/0/0 Owen Casha 0 47
Gain Compression 0log 0 a ka 0log 0 ka 4k db ka 4k 0 0 a 4k 0 db 0 k k 0.45 k Let us now consider the case when both of the input signals are present. The amplitude of cosω t in the output becomes: ka k 4 a ab If b is large in comparison with a then the term k may denominate over the first one (undesired). 09/0/0 Owen Casha 0 48
Second Harmonic Distortion Second harmonic distortion occurs due to k. If b is zero, then the amplitude of the second harmonic will be k (a /). Since power is proportional to the square of the voltage, the desired term in the output can be expressed as: a proportionality constant 0log0 ck 0log0( a) C Similarly, the power in the second harmonic component: a 0log 0 ck 40log0( a) C 09/0/0 Owen Casha 0 49
Second Harmonic Distortion While the input power is: a in 0log c 0 a C 0 log 0 ( ) Intercept oint I in C C C C in Fundamental and second harmonic signals are linearly related with the input power. However the second harmonic power increases at twice the rate of the fundamental. 09/0/0 Owen Casha 0 50
Intermodulation Distortion Ratio We found that the cubic term produces intermodulation frequencies ω ±ω and ω ±ω. If ω and ω are very close than ω ω and ω ω will be far away from the desired signal and can be easily filtered out. However the other terms will be so close to ω and ω that these components may be within the pass-band of the system, thus distorting the output. This characteristic of a nonlinear system is specified via the intermodulation distortion. It is obtained after dividing the amplitude of one of the intermodulation terms by the desired output signal. For example an input signal with both ω and ω ( tone input), k 4 k 4 IMR ab or k a a b 4k k kb ab k ab 4k 09/0/0 Owen Casha 0 5
rd order Intercept oint Since power is proportional to the square of the voltage, intermodulation distortion power may be defined as: IMD c ( / 4ka ) b If the two input signals are equal in amplitude then ab, and the expression simplifies to: IMD c ( / 4 ka ) 09/0/0 Owen Casha 0 5
rd order Intercept oint Similarly, the power in one of the input signal components can be expressed as: in IMD ca α in α is a constant The ratio of the intermodulation distortion power IMD to the desired output power o ( o k in ) for the case where two input signal amplitudes are the same is known as the intermodulation distortion ratio: IMR IMD O α in 09/0/0 Owen Casha 0 5
rd order Intercept oint (I ) Note that IMD increases with the cube of in while the desired signal power o is linearly related with in. 09/0/0 Owen Casha 0 54
Example The intercept point in the transfer characteristic of a nonlinear system is found to be 5dBm. If a -5dBm signal is applied to the system, find the intermodulation ratio. At intercept point IMR α II α / II /(x0 - x0 5/0 ) 0 IMR 0log 0 (0) 0log 0 ( in )0(-5-0) -80 db 09/0/0 Owen Casha 0 55
Dynamic Range Noise at one end and distortion on the other limit the range of detectible signals of a system. The amount of distortion that can be tolerated depends on the time of application. If we set the upper limit that a system can detect as the signal level at which the intermodulation distortion is equal to the minimum detectable signal then we can formulate an expression for its dynamic range. 09/0/0 Owen Casha 0 56
Dynamic Range Thus, the ratio of the signal power that causes distortion power (in one frequency component) to be equal to the noise floor to that of the minimum detectable signal is the dynamic range (DR) of the system (amplifier, mixer, receiver). DR( db) { ( dbm) ( dbm) } I f 09/0/0 Owen Casha 0 57