Instantaneous frequency Up to now, we have defined the frequency as the speed of rotation of a phasor (constant frequency phasor) φ( t) = A exp

Transcription

1 Exponential modulation Instantaneous requency Up to now, we have deined the requency as the speed o rotation o a phasor (constant requency phasor) φ( t) = A exp j( ω t + θ ). We are going to generalize this deinition to general complex unctions o the real variable t. Consider such unction: z( t) r( t)exp ( jθ ( t) ) =. It is obvious that it is a generalization o the constant requency phasor. It can be represented graphically as a vector with modulus (amplitude) r(t) and argument θ(t). r( t ) θ ( t) The argument θ(t) o z(t) is called the instantaneous phase. Since this phase varies, the generalized phasor is going to rotate. However, it is not going to rotate at a constant speed. We can thus deine an instantaneous speed o rotation or this unction. It is the instantaneous requency: dθ ω ( t) = rd/s dt 31

2 We can measure this requency in Hertz: ω( t) 1 dθ ( t) = = π π dt In this section, we are interested in constant amplitude phasors. Furthermore, we assume that the instantaneous requency has an average value with a deviation around it d(t): ( t) = + d( t) The average value o d(t) is zero. This means that the instantaneous phase can be expressed as: θ ( t) = π t + φ( t) = ω t + φ( t) φ ( t) is called the instantaneous phase deviation and we have: d( t) = 1 dφ π dt To this generalized phasor, we can associate the ollowing signal: [ ] ( ω φ ) x( t) = Re w( t) = r( t)cos t + ( t) This signal has the general shape o a bandpass signal. In order or x(t) to be bandpass, its quadrature components must be bandlimited to a requency W <. The quadrature components are: a( t) = r( t)cos φ( t) and b( t) = r( t)sin φ( t). This condition is not always satisied. However, in practical exponential modulation, the carrier is usually very high (hundreds o MHz). So, we can consider that the obtained modulated signals are bandpass signals. 3

3 Frequency and Phase Modulation (FM & PM) For both phase and requency modulation, the modulated signal must have constant amplitude. The inormation is carried in the phase deviation. These modulations are called "exponential Modulation" because the signal has always the ollowing shape: ( φ ) ( ω ) x( t) = Re A exp j ( t) exp j t Phase modulation is a modulation process that makes the phase deviation φ(t) proportional to the baseband signal s(t). φ( t) = k sɶ ( t) = ( φ) s( t) φ The constant φ = k sɶ ( t) is called the maximum phase deviation. φ max In order to avoid phase ambiguity, this constant cannot exceed π. φ π The expression o a real phase modulated signal is: ( ω φ ) x( t) = A cos t + ( ) s( t) Frequency modulation, on the other hand, is a modulation system where the requency deviation is made proportional to the inormation signal. The constant d( t) = k sɶ ( t) = ( ) s( t) max = k sɶ ( t) is called the maximum requency deviation. The instantaneous requency is: ( t) = + ( ) s( t) 33

4 In order to have always a positive requency, we must have. t The instantaneous phase deviation is: φ( t) = π ( ) s( λ) dλ. The lower bound o the integral is not indicated to take into account any initial phase. Sometimes, the lower bound is assumed to be. So, the expression o a requency modulated signal is: ( ) ω π λ λ t x( t) A cos t ( ) s( ) d = + I we look at the relations that exist between the phase and the requency, we remark that the two modulations are related. In act, we can built a requency modulator using a phase modulator, a requency demodulator using also a phase demodulator and vice versa. s( t ) Integrator Frequency Modulator Phase Modulator x( t ) s( t ) Dierentiator Phase Modulator Frequency Modulator x( t ) 34

5 x( t ) Phase Demodulator Frequency Demodulator Dierentiator s( t ) x( t ) Frequency Demodulator Phase Demodulator Integrator s( t ) The above our igures show how we can build one type o modulator or demodulator using the other. Exponential modulation is a highly nonlinear modulation. This means that it is very hard to relate the spectrum o the modulated waveorm with the one o the baseband as we did with the linear modulations. So, an analysis o the modulated signal in the requency domain is quite diicult in the general case. There are two special cases where this analysis is not very complicated: the narrowband phase and requency modulation where the phase deviation is very small and the sinusoidal modulation where the baseband signal is sinusoidal. Narrowband phase and requency modulation The modulated signal in this case has the general shape o: 35

6 ( ω φ ) x( t) A cos t ( t) = + along with t max Developing the cosine, x(t) becomes: Using the act that φ ( ) << 1 x( t) = A cos φ( t)cosω t A sin φ( t)sinω t φ ( t) << 1, we have: cos φ( t) 1 and sin φ( t) φ( t), giving: x( t) = A cos ωt Aφ ( t)sinωt. max A φ(t) A The above phasor diagram illustrates that the signal x(t) is the projection on the real axis o the sum o two phasors rotating at the same speed ω and making an angle o 9 between them. We see that this phase shit produces the phase modulation. Furthermore, the Fourier transorm o the expression o x(t) can be evaluated. So, i [ φ t ] Φ ( ) =F ( ), then A A X ( ) = [ δ ( ) + δ ( + ) ] [ Φ( ) Φ ( + ) ]. j I the signal is PM, then φ( t) = ( φ) s( t) giving Φ ( ) = ( φ) S( ). So, i the signal is bandlimited to W, then the PM signal will be limited to a bandwidth B = W. t I the signal is FM, then φ( t) = π ( ) s( λ) dλ giving ( ) Φ ( ) = S( ). Here also the bandwidth o the FM signal is W. j 36

7 Sinusoidal modulation The other case that has a simple analytic expression is when the modulating signal is sinusoidal. When a signal is sinusoidal, its derivative is also sinusoidal. So, we can use the same analysis or both requency and phase modulation. The modulated signal in both cases = +. φ( t) = ( φ) s( t) or PM and is x( t) A cos ( ω t φ( t) ) t φ( t) = π ( ) s( λ) dλ or FM. For FM modulation, we assume that s( t) = cosω m t. This gives: π ( ) φ( t) = sinω m t. ω m ( ω β ω ) x t = A t + t. ( ) cos sin m β = is called the modulation index. So, m For PM modulation, the modulating signal is s( t) = sinω m t. The instantaneous phase deviation becomes: φ( t) = ( φ)sinω m t. In this case, the modulation index is β = ( φ) and we obtain the same expression. So, or both cases, the expression o the modulated signal is: ( ω β ω ) ( β ω ) ( ω ) x( t) = A cos t + sin mt = A Re exp j sin mt exp j t exp β sinω m is periodic with In the above expression, the unction ( j t) a period T m development is: π =. It can be developed in Fourier series. The ω m = + exp β sin ω ( β )exp ω ( j t) J ( jn t) m n m n= 37

8 The Fourier coeicients J ( β ) are the Bessel unctions o the irst n kind o order n and argument β. These unctions are tabulated and can be easily computed. They appear as solutions o dierential equations. For positive order, we can use the ollowing Mc Lauren series: J n n k β ( 1) β ( β ) = k= k!( n + k)! and when n is negative, we use the ollowing relation: n J n( β ) = ( 1) J n( β ) The ollowing igure shows the behavior o the irst 6 Bessel unctions. k They look like damped sinewaves. From the Mc Lauren series we can deduce their properties or β around. For very small values o β, we have the ollowing approximations: J n ( β ) J ( β ) 1 n 1 β n! or n > This means that the only unctions that we should consider around zero are J and J 1. So, or β <.1, J ( β ) 1 and ( ) β J1 β. 38

9 The ollowing table gives the value o the irst Bessel unctions. The exponentially modulated signal is: + n n= ( ) x( t) = A J ( β )cos ω + nω m t We see that the signal contains a large number o components around the carrier requency. For example, let us assume β = 1. Then only J, J 1 and J are signiicantly dierent rom zero. We can then write: n n= [ ω ω ] x( t) A J (1)cos ( + n ) t The modulated signal is the sum o ive sinewaves. The spectrum is displayed below. The values o the dierent amplitudes are read rom the above table. m 39

10 m m + m + m Single sided spectrum o the signal The above spectrum is approximately bandlimited. For the exponentially modulated signal, we can use as transmission bandwidth the band o requency that contains most o the power o the signal. The total power o the transmitted waveorm can be computed as ollows. From the series development o x(t), we obtain: + P =< x ( t) >=< A Jn( β )cos ( ω + nω m ) t > n= I the dierent sinewave are independent, the total power is the sum o the individual powers. We obtain: A A P = J β = + n ( ) n= We have used the ollowing property o the Bessel unctions: + n= J ( β ) = 1 n 4

11 I we keep n components on each side o the carrier, we obtain: n, = J k ( β ) k= n ( β ) P n The ratio o this power to the total power is: A n (, β ) P n P = k= n J k ( β ) This ratio is very close to 1 (.95) or n = β + 1. So, the band o requency components containing the sidebands rom nm + n can be used as an approximation or the bandwidth o the m exponentially modulated waveorm or sinusoidal modulating waveorm. So the transmission bandwidth is approximately: For FM signal, m B = n = ( β + 1) m β =, we obtain: B = ( ) + m The above rule is called the "Carson's Rule". It has been ound empirically that this rule can be applied even i the modulating signal is not sinusoidal. In this case, we can deine the "Deviation Ratio"; δ = where W is the bandwidth o the baseband modulating signal. W Carson's rule generalizes to B = ( δ + 1) = ( ) + W. For example, broadcast FM uses a value o = 75 khz or transmitting a baseband signal having a bandwidth W = 15 khz. This gives a transmission bandwidth B = = 18 khz. The normalized m m to 41

12 bandwidth is khz. Carson's rule underestimates it, but the error is small. A general remark about FM and PM is that the bandwidth o the resulting signal is in general much larger than W. This is due to the act that these modulations are highly non-linear. Another remark about FM is that it is very resistant to perturbations induced by noise and intererence. So, we can say that FM protects the inormation o the signal at the expense o a bandwidth increase. When β is very small, we can use J ( β ) 1 and ( ) β J1 β to express the modulated signal: x( t) A cos t A β β = ω + cos ( ω + ωm ) t A cos ( ω ωm) t) giving x( t) = A [ cosω t β sinω t sinω t] = A [ cos ω t β s( t)sinω t] m which is the narrow band approximation. Filtering the FM signal Being a non linear modulation, the usual method o iltering the complex envelop o the FM signal by the equivalent lowpass ilter does not work or general ilter shapes. In some speciic cases, this technique can be used. In order to use it, this FM signal must be bandpass. In this case, the complex envelop is easily extracted. ( ω φ ) x( t) = A cos t + ( t) giving a complex envelop ( φ ) m ( t) = A exp j ( t). One case where this technique can be used is x the case o a ilter with an amplitude response o the type: 4

13 ( θ ) H ( ) = M ( )exp j ( ) where M ( ) = M + k( ) or > and θ ( ) = πτ g ( ) + θ. The equivalent lowpass ilter is: [ ] ( πτ g ) H ( ) = M + k exp j exp( jθ ). In the requency domain, lp the complex envelop o the output is: [ ] ( πτ ) M ( ) = M ( ) H ( ) = M + k exp j exp( jθ ) M ( ) So: y x lp g x ( θ ) ( πτ ) ( θ ) x ( πτ g ) M ( ) = M exp j M ( )exp j y x g + k exp j M ( )exp j dx( t) Given that F = π jx ( ) dt, the multiplication by in the requency domain becomes a dierentiation in the time domain. So, k dmx ( t τ g ) my ( t) = M mx ( t τ g ) + exp j π j dt ( θ ) From the expression o the input complex envelop, we obtain: dm ( t τ ) dφ( t τ ) x g = ja g exp ( jφ ( t ) ) and inally: dt dt k dφ( t τ g ) my ( t) = A M + exp j ( t ) exp j π dt The output signal is then: ( φ τ g ) ( θ ) k dφ( t τ g ) y( t) = A M + cos t + ( t ) + π dt ( ω φ τ g θ ) dφ( t) I the signal is FM, = π ( ) s( t). In this particular case: dt 43

14 ( t τ ) y( t) = A M + k( ) s( t τ ) cos ( ) g g ω t + π s( λ) dλ + θ We remark that the output signal has two dierent modulations: FM and AM. The inormation signal is contained in the envelop o the output signal. So, an envelop detector will demodulate the signal. I the ilter has a dierent transer unction, we can use the concept o "quasi-static" approximation. I the carrier requency is much higher than the baseband modulating requency, then we can saely assume that the requency is constant over a quite long time. The FM signal will behave almost like a constant requency sinewave. We know that i the input o a ilter with transer unction H() is a pure sinewave ( t) A cos π, the output is also a sinewave at the same requency: [ ] y( t) = A H ( ) cos π t + Arg H ( ). In the quasi- static approximation, we replace by the instantaneous requency ( t ). So, given ( t) = + ( ) s( t), we obtain: [ ( )] t y( t) = A H ( + ( ) s( t) cos π t + π ( ) s( λ) dλ + Arg H + ( ) s( t) Example: Using H ( ) M ( )exp( jθ ( )) = with M ( ) = M + k( ) or > along with θ ( ) = πτ g ( ) + θ, we obtain: H ( ( t)) M k( ) s( t) = + and [ ] g Arg H ( ( t)) = πτ ( ) s( t) + θ. We see that we obtain the same amplitude variation as in the previous case (except or a time delay). 44

15 FM through nonlinear system Consider the ollowing memoriless nonlinear system. [.] x( t ) T y( t ) I the input is x( t) = A cos θ ( t), the output will be = [ θ ]. The output signal T [ A t ] y( t) T A cos ( t) cos ( ) θ is not periodic as a unction o the variable t, it is periodic, with a period π i we consider it unction o θ. We can thus develop the output y in Fourier series. a y t a n t + ( ) = + n cos θ ( ) n= 1 The Fourier series is a cosine series because the input is A cos θ. It is even. So, T [ A ] cos θ is also even. The coeicients are given by: 1 π an = T [ cosθ ] cosnθ dθ π π Replacing the argument θ by its value leads to: a y t a n t n t + ( ) = + n cos + ( ) n= 1 [ ω φ ] The above signal is a superposition o an ininite number o exponentially modulated waveorms. For the FM case, we can write: ( t ) ( t ) ω π λ λ ω π λ λ a y( t) = + a1 cos t + ( ) s( ) d + a cos t + ( ) s( ) d + We can remark that the output is a sum o FM signals at carrier 45

16 requencies n each one with a maximum deviation n. I the dierent spectra do not intersect, we can select one o them using a bandpass ilter tuned at n and obtain at the output: FM Generation Direct Method ( ω π λ λ ) t z( t) = an cos n t + ( n ) s( ) d The FM signal can be generated directly using a Voltage Controlled Oscillator (VCO) like the one used in the lab generators (GW-Instek GFG855A). The output signal is a sinewave with an instantaneous requency given by = v + kmvin. The requency v is called the ree running requency and the constant k m is called the VCO gain (It is measured in Hz/Volts). In general, these VCOs can use a variable reactance in a parallel RLC circuit used to tune an oscillator. We can use "varactors" or example. The output requency o this type o oscillator is the resonant requency o the RLC circuit. 1 = where C = C cvin. The input voltage is proportional to LC s(t). v = V s( t), so: max in in 1 1 c ( t) = = 1 vin ( t) LC LC C 1. I c v ( ) 1 in t C max <<, we can use the approximation [ ε ] ε. The instantaneous requency is given by: c v 1 + v C in. The ree 46

17 running requency is v 1 = and the VCO gain is LC k m cv =. The C maximum requency deviation is = V k in max m. There exist a large number o VCO circuits. The most common ones (the ones that are ound in integrated circuits and in signal generators) generate triangular waves using integrators or capacitors charged by controlled current sources. A good reerence is "K. K. Clarke & D. T. Hess, Communication Circuits: Analysis and Design, nd ed. Krieger Pub Co, 1994" which is the textbook or the communication circuit course. (The part between the two is or reading only) Frequency mixing In this part, we introduce an important technique used in receiver and transmitter design: Frequency mixing. The mixer is a device capable o changing a carrier requency or any type o modulation. It is based on the requency translation theorem o Fourier theory. We start irst with real signal mixing. Consider a general bandpass signal xr ( t) = r( t)cos ωr t + φ( t). I we multiply this signal by a sinewave x ( t) = Bcosω t, the result is: z( t) = xr ( t) xlo ( t) = Br( t)cos ωrt + φ( t) cosωlot Using trigonometric identities, we see that this signal is the sum o two bandpass signals: B B z( t) = r( t)cos ( ωr ωlo ) t φ( t) r( t)cos ( ωr ωlo ) t φ( t) lo lo 47

18 Using a bandpass ilter tuned at either the sum or the dierence requency, we obtain a bandpass signal having the same complex envelop (i.e. the same inormation) but a dierent carrier requency. The new requency is usually called "intermediate requency" i. I i = r + lo, we say that we are doing "up mixing". On the other hand, i i = r lo or i = lo r, we say that we are perorming down mixing. The "mixer" is an important electronic subsystem in any communication receiver or transmitter. It is the basic building block o the "superheterodyne" receiver. This concept o receiver was introduced in order to solve the very complex problem o ampliying and selecting one radio station among a large number o stations transmitting at dierent requencies. The irst solution that comes to mind is to use a "tunable" bandpass ilter. However, the construction o a very selective tunable bandpass ilter is very complex. Furthermore, due to component aging, such system is prone to random changes and mistuning ater a certain time. It is much easier to build a ixed requency very selective ilter. So, instead o translating the center requency o a tunable ilter beore the dierent signals, it is much easier to translate the requency o the signals beore the center requency o a ixed bandpass ilter. This is the concept o the super heterodyne receiver. The superheterodyne receiver is composed o a tunable local oscillator ganged with a wide band tunable RF ampliier, a mixer and a ixed requency IF ampliier. It is built using the block diagram shown below. 48

19 Let us assume we are using down mixing. The r ampliier pre-selects a band o requencies containing a small number o stations around the station at requency c. The bandwidth B RF is large compared to the bandwidth B required by the modulation used (FM, AM, any linear one) but smaller than IF, the intermediate requency. Using down mixing, we must have: IF = c 1 giving c = 1 + IF or IF = 1 cgiving c 1 =. From the above two relations, we see that i the r ilter does not exist, then we can receive two dierent stations i we simply use the local oscillator or tuning. These two stations are separated by IF. These two requencies are called "image requencies". The job o the tunable r ampliier is to eliminate one o them so that it will not interere with the station that we want to receive. Intermediate requency or broadcast receivers has been standardized to the values o 455 khz or AM and 1.7 MHz or FM. IF 49

20 Another technique is used to avoid this problem o image requencies. It is based on requency translation using complex phasors. Consider the high requency signal: xr ( t) = r( t)cos ωr t + φ( t). The associated analytic signal is: xr ( t) xr ( t) jxˆ r ( t) r( t)exp ( jφ( t) ) exp( jωrt ) + = + =. I we multiply this signal by the phasor: exp( jω t) signal: z( t) r( t)exp ( jφ( t) ) exp( j( ωr ωlo ) t) lo, we obtain the =. The real part is: [ ] ( ) xi ( t) = Re z( t) = r( t)cos ωr ωlo t + φ( t). This is the correct translated signal. So, the process o perorming the above operation is: ( ˆ )( ω ω ) xi ( t) = Re xr ( t) + jx( t) cos lot jsin lot giving: x ( t) = x ( t)cos ω t + xˆ ( t)sinω t i r lo r lo This leads to the ollowing block diagram: cosω lo t Imageless Mixer The above circuit can be used without any image rejection ilter beore. π π 5

21 Indirect FM generation This technique o FM generation is the one that is commonly used in FM transmitters. This is due to the act that the carrier requency and the maximum requency deviation can be set with high precision. It is based on a Narrow Band Frequency modulator cascaded with nonlinear ampliiers that are used as requency multipliers. Mixers are also used to translate the carrier because requency multiplication leads usual to impractically high carrier requencies. A general block diagram o such system is: s( t ) t (.) du NBPM N1 N cosπ t 1 cosπ t The requency multipliers are implemented using a nonlinear ampliier (Class C) ollowed by a narrow bandpass ilter tuned at the proper harmonic. I we consider the above block diagram, the carrier requency is given by = N ( N1 ) or N ( N1 ) 1 =. I the requency 1 deviation at the output o the NBFM is 1, the inal deviation is 1 = N1 N. In general the requency multiplication is achieved by a cascade o requency doublers and triplers. It is impossible to achieve an eicient ampliier i the multiplication actor is larger than three. 51

22 FM Demodulation FM demodulation by dierentiation (FM to AM conversion): I we compute the derivative o an FM signal, we obtain: d dφ A cos t ( t) A sin t ( t) dt dt ( ω + φ ) = ω + ( ω + φ ) dφ Since the signal is an FM one, = π ( ) s( t), we get: dt d A cos t ( t ) A ( ) s ( t ) sin t ( ) s ( ) d dt We can see that the output o a dierentiation circuit will produce a ( ) ( ) ( t ) ω + φ = ω + π ω + π λ λ modulation o the amplitude. This modulation can be detected using any AM demodulator. The ollowing circuit is a dierentiator built using an OP-Amp simulated using Multisim. XSC1 G T R1 1kΩ A B 3 C1 1 U1 V1 1nF FM 5 V 1kHz 1 Hz OPAMP_3T_VIRTUAL 5

23 The above picture displays the output signal. We clearly see the two modulations: AM and FM. The signal produced by the FM source is a sinewave modulated FM signal with β = 5, carrier requency = 1 khz and modulating requency = 1 Hz. Dierentiator using an op-amp i1 R 3 C 1 Vin i1 Op-Amp Vout The Operational Ampliier is an ampliier that possesses a very high gain, dierential input and very high input impedance. I the voltage 53

24 at the inverting input is v, the input at the non-inverting input is v + and the gain o the op-amp is G, its output is given by G( v v ) + is assumed very large. This means that a very small dierence will produce a measurable output. We can saely assume that this dierence is zero. At that time, the inverting input is practically at ground in the above schematic. This implies that the capacitor is in dvin parallel with Vin. So, i1 = C. The input impedance o the op-amp dt is very large. The same current will low though the resistance R. So, Vout = Ri1. Ater the elimination o i1, we obtain: dvin Vout = RC dt There exist a large number o other FM demodulators. The interested student should consult the previous reerence (Clarke & Hess). The Phase Locked Loop The phase locked loop (PLL) is a eedback system composed principally o a voltage controlled oscillator (VCO), a phase detector (PD) and a lowpass ilter (LPF). The phase detector is usually modeled as a multiplier.. G x ( ) c t LPF K a y( t ) v( t ) VCO 54

25 The input signal is x ( t) = A cos θ ( t) and the output o the VCO is c c c v( t) = A cos θ ( t). The output o the multiplier is v v Ac Av z( t) = cos c ( t) v ( t) + cos c ( t) + v ( t) ( θ θ ) ( θ θ ). The lowpass ilter eliminates the sum term and ilters the dierence term. So, we can consider that the output o the lowpass ilter (ater ampliication) is Ac Av y( t) = Kah( t) cos ( θc ( t) θv ( t) ). π Let us introduce a variable ε such that: θv ( t) = θc ( t) ε ( t) +. The output o the lowpass ilter becomes: Ac Av y( t) = Kah( t) [ sin ε ( t) ] We have a one to one relationship between y(t) and ε(t) i π π ε ( t). The relation becomes linear i ε << 1. To simpliy the analysis o the PLL and eliminate the eect o the amplitudes, let us make A c = and A v = 1. The input signal has a carrier requency. This makes θ ( t ) = ω t + φ c ( t ). The VCO ree running requency v is shited rom by an oset : v =. The instantaneous phase π π o the VCO is: θv ( t) = π vt + φv ( t) +. The constant is added in order to introduce the variable ε(t) in the ollowing expressions. The instantaneous phase deviation φ ( t ) is produced by the output signal v 55

26 y(t): φ v ( t) = π Kv y( t) 4. The constant K v is the VCO gain expressed in Hz/Volts. Using the deinition o ε(t), we can write: Replacing each phase, we obtain: Ater dierentiation: π ε ( t) = θc ( t) θv ( t) + ε ( t) = π t + φ( t) φ ( t) ε ( t) = π + φ( t) π K y( t) However, y(t) is the output o the lowpass ilter ampliied by K a. So: v v ( ) y( t) = K h( τ )sin ε ( t τ ) dτ a + The PLL is thus governed by the ollowing integro-dierential equation: + ε ( t) + π K h( τ )sin ε ( t τ ) dτ = φ( t) ( ) K = K a K v is called the "Loop Gain". The above equation is a nonlinear equation that is quite complex to solve. In our course, we are going to consider two dierent cases: The irst order loop (with no iltering) to analyze in a simple manner the "locking mechanism" and the linear approximation when sinε ε. Frequency Acquisition Consider a ilter with a transer unction H ( ) = 1 over the requency band o interest. The impulse response will be a Dirac impulse: h( t) = δ ( t) and we will have: 4 A dot above a unction means that the unction is dierentiated. 56

27 + ( ) = ( ) h( τ )sin ε ( t τ ) dτ sin ε ( t) We obtain the ollowing irst order dierential equation: ε + π K sin ε = φ( t) + π Let us assume that we apply the signal x t ( ω t φ ) ( ) c = cos + at the time t =. In this case, φ( t) = φ and its derivative is zero. We also assume that the ree running requency o the VCO is dierent rom the carrier we want to lock on ( ). The dierential equation becomes ε + π K sinε = π or t. This equation can be written as: ε π K = sinε + This equation relates the derivative o ε with ε. The set o points K ε, ε is called the phase plane. In our case, it is better to use ε, ε. This describes a single curve with t as a parameter. As t π K varies, the point is going to move along the curve shown below. The points on the curve where ε = are called "equilibrium points". We distinguish two dierent types o equilibrium points: A point is called stable i ater a small perturbation around the point, the trajectory in the phase plane will go back to the equilibrium point. The point is 57

28 called unstable i ater a small perturbation, the trajectory will go away rom the equilibrium point. The ollowing mechanical analogy will show the dierence between the points. Stable Equilibrium Unstable Equilibrium I a trajectory starts at an equilibrium point, it will remain there. ε π K K ε Stable Equilibrium Point Unstable Equilibrium Point Phase Plan Plot ε The above curve shows the locus o the points, ε as t goes π K rom zero to ininity. The trajectory starts at an initial point ε and it will move. It is easy to see that or any starting point, the trajectory will move toward a stable equilibrium point. However, we can have 58

29 equilibrium points i and only i the curve intersects the ε axis. This is possible i 1 K. So, i this condition is satisied, then lim ε ( t ) = t and the steady state value o ε will be: ε st = sin The steady state value o y(t) will be: y 1 K = K sinε = st a st π and since θv ( t) = θc ( t) ε ( t) +, the steady state output o the VCO will be: π vst ( t) = cos ωt + φ ε st + I the loop gain is very large, then ε st will be very small. The VCO output will have exactly the same requency as the input signal with a π phase shit that is practically. I > 1, it is impossible to have a lock. The trajectory will not K intersect the ε axis and there is no equilibrium point and no steady state solution. The VCO requency will keep changing. In order to have a lock, the VCO ree running requency must be in the ollowing range: [ K, K ] K v +. This range is called the "lock range". The value ound is valid or a irst order loop only. It will be dierent or another ilter. 59

30 FM Demodulation In order to analyze the PLL when the input is requency modulated, we assume that the VCO ree running requency is the same as the input carrier requency. This means that =. We also assume that ε remains small all the time. We can replace sinε by ε in our analysis. With these hypotheses, we have: and ( ) ( ω φ ) x t t t ( ) c = cos ( ) + π v( t) = cos ωt + φv ( t) + y( t) = K h( τ )sin ε ( t τ ) dτ K h( τ ) ε ( t τ ) dτ a + + a ε ( t) = φ( t) φ ( t) Since y is linearly related to ε and φ v is also linearly related to y, it is more interesting to use the phase deviations as primary variables instead o using x c and v. the dierent relationships are better described by the ollowing block diagram. v φ ( t) + ε ( t) LPF K a y( t ) φ ( t ) v VCO The above block diagram is completely linear. The VCO transer unction is given by the ollowing relation: 6

31 t φv ( t) = π Kv y( u) du This gives the ollowing relation in the requency domain: Kv Φ v ( ) = Y ( ) j The closed loop transer unction (in the requency domain) is: =F. where H ( ) [ h( t) ] 1 jh ( ) Y ( ) = ( ) K Φ v H ( ) + j K I the input is an FM wave, φ( t) = π s( u) du, then The transer unction becomes: Φ ( ) = S( ) j H ( ) Y ( ) = S( ) K v H ( ) + j K We see that the output signal is the baseband inormation signal iltered. proportional to s(t). I the loop gain is very high, the output signal will be y( t) s( t) K So, i we can make sure that the error signal is small at all times, the PLL can be used with advantage as a requency demodulator. Another important application o the PLL is the implementation o requency synthesizers. v t 61

32 Frequency Synthesis When a PLL is locked, the requencies o the signals arriving at the two inputs o the phase detector (multiplier) are equal. 1 N 1 LPF K a y( t ) N VCO The blocks labeled N k are requency dividers (usually implemented by presettable logic counters). At the inputs o the phase detector (assuming that the PLL is locked), we can write: N = N 1 1 N So, the VCO will produce: = 1. This means that we can N produce a signal with a requency that can be set using digital hardware and that can be very stable i the reerence oscillator producing 1 is very stable. This technique o requency generation is commonly used in modern receivers. The PLL is usually built in a microcontroller. 1 6