# Digital PLL's -- Part 2

In Part 1, we found the time response of a 2^{nd} order PLL with a proportional + integral (lead-lag) loop filter. Now let’s look at this PLL in the Z-domain [1, 2]. We will find that the response is characterized by a loop natural frequency ω_{n} and damping coefficient ζ.

Having a Z-domain model of the DPLL will allow us to do three things:

- Compute the values of loop filter proportional gain K
_{L}and integrator gain K_{I}that give the desired loop natural frequency and damping. Deriving these formulae is somewhat involved, but the good news is we only have to do it once. - Compute the linear-system time response to a step in the reference phase.
- Compute the frequency response.

**Figure 1. DPLL Time Domain Model Block Diagram**

Figure 1 is the time-domain DPLL model we derived in part 1. To convert this to a useful Z-domain model, we replace the accumulators in the Loop Filter and NCO with the transfer function z^{-1}/(1 – z^{-1}), who’s numerator and denominator we can multiply by z to get 1/(z – 1). This gives us the model in Figure 2, where we have also indicated the phase detector gain K_{p}.

**Figure 2. DPLL in the Z-domain**

The open-loop response of this system is the product of the transfer functions of the three blocks:

$$ G_1(z) = \frac{K_pK_LK_{nco}}{z-1} + \frac{K_pK_IK_{nco}}{(z-1)^2} \tag{1} $$

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*2*^{nd} Order System in s and z

*2*

^{nd}Order System in s and zA 2^{nd} order continuous-time system with a Lead-lag filter is shown in Figure 3. [3, 4]. See Appendix B for a derivation of the closed-loop response. If we convert this to the Z-domain, we’ll see the response has the same form as that of our DPLL. This will allow us to relate K_{L} and K_{I }of the DPLL to ω_{n} and ζ.

**Figure 3. 2 ^{nd} order system in s with a zero in the closed-loop response**

To convert this system to the z-domain equivalent, make the following replacement:

$$ s \rightarrow \frac{z-1}{T_s} $$

This approximation is valid as long as the loop natural frequency is much less than the sample frequency (see Appendix C). The Z-domain block diagram is shown in Figure 4, where we have allowed for the possibility that the loop filter could have a sample rate T_{s_filt }different from the NCO sample rate T_{s_nco}.

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**Figure 4. 2 ^{nd} order system in the z-domain**

The open-loop response G_{2}(z) is:

$$ G_2(z) = \frac{2\zeta\omega_nT_{s\_nco}}{z-1} + \frac{T_{s\_filt}T_{s\_nco}\omega_n^2}{(z-1)^2} \tag{2} $$

By equating open-loop response G_{1 }of our DPLL to G_{2}, we can find K_{L} and K_{I} in terms of ω_{n} and ζ.

Equating G_{1} and G_{2}:

$$ \frac{K_pK_LK_{nco}}{z-1} + \frac{K_pK_IK_{nco}}{(z-1)^2} = \frac{2\zeta\omega_nT_{s\_nco}}{z-1} + \frac{T_{s\_filt}T_{s\_nco}\omega_n^2}{(z-1)^2} $$

Solve for K_{L }and K_{I}:

$ K_pK_LK_{nco} = 2\zeta\omega_nT_{s\_nco} $ → $ K_L = 2\zeta\omega_n/K_p\; * \; T_{s\_nco}/K_{nco} $ (3)

$ K_pK_IK_{nco} = T_{s\_filt}T_{s\_nco}\omega_n^2 $ → $ K_I = T_{s\_filt}\omega_n^2/K_p \; * \; T_{s\_nco}/K_{nco} $ (4)

A given DPLL design has defined values for T_{s_filt},T_{s_nco}, K_{p}, and K_{nco}. Given those values, K_{L} and K_{I} are uniquely determined by the choice of ω_{n} and ζ. Note that the units of K_{p} are cycle^{-1}. See Appendix D for an alternate form of the equations for K_{L} and K_{I}.

*Computing the Closed-loop response*

*Computing the Closed-loop response*

The closed loop phase response of the DPLL in Figure 2 is given by:

$$ CL(z) = \frac{Z[u]}{Z[ref\_phase]} = \frac{G_1}{1 + G_1} $$

Thus from equation 1,

$$ \frac{K_pK_LK_{nco}} {z-1} + \frac{K_pK_IK_{nco}} {(z-1)^2} $$

$ CL(z) = $ __________________________________

$$ 1 + \frac{K_pK_LK_{nco}} {z-1} + \frac{K_pK_IK_{nco}} {(z-1)^2} $$

$$ CL(z) = \frac {b_0 + b_1z^{-1}} {1 + a_1z^{-1} + a_2z^{-2}} \qquad (5) $$

Where

b_{0} = K_{p}K_{L}K_{nco}b_{1 }= K_{p}K_{I}K_{nco} - K_{p}K_{L}K_{nco}a_{1} = K_{p}K_{L}K_{nco} – 2

a_{2} = 1 + K_{p}K_{I}K_{nco} - K_{p}K_{L}K_{nco}

Equation 5 is in the form of an IIR filter transfer function, which allows for straightforward computation of the time and frequency responses in Matlab.

*Example*

This example uses the same parameters as Example 2 in Part 1. We will compute K_{L} and K_{I}, then we will find the time and frequency response using CL(z). For this example, f_{s_nco }= f_{s_filt }= f_{s}. The Matlab script is listed in Appendix A. The DPLL parameters are as follows. As you can see, not all of the parameters from the time domain model apply to the Z-domain model.

f_{s} 25 MHz

Reference frequency NA

Initial reference phase NA

NCO initial frequency error NA

K_{nco} 1/4096

K_{p} 2 cycle^{-1}f_{n} 400 Hz loop natural frequency. f_{n} = ω_{n}/(2π)

ζ 1.0 loop damping coefficient

1. Find K_{L} and K_{I}

KL= 2*zeta*wn*Ts/(KP*Knco) % loop filter proportional gain KI= wn^2*Ts^2/(KP*Knco) % loop filter integral gain KL = 0.4118 KI = 2.0698e-005

2. Compute the time response to a step in the reference phase. Since CL(z) is in the form of a digital filter transfer function, we can find the time response using the Matlab "filter" function.

b0= KP*KL*Knco; b1= KP*Knco*(KI - KL); a1= KP*KL*Knco - 2; a2= 1 + KP*Knco*(KI - KL); b= [b0 b1]; % numerator coeffs a= [1 a1 a2]; % denominator coeffs x= ones(1,N); % step function y= filter(b,a,x); % step response pe= y-1; % phase error response

The phase error response is shown in Figure 5. Because this model is linear, the non-linear acquisition behavior we saw in the time-domain model of Part 1 (Figure 3.4) is missing. Thus we see that the Z-domain model is less capable than the time-domain model for computing the time response. Finally, one detail worth mentioning: the response has some overshoot. This is caused by the zero in CL(z). (An all-pole system would not have overshoot for ζ = 1).

3. Compute the frequency response CL(z).

u = 0:.1:.9; f= 10* 10 .^u; % log-scale frequencies f = [f 10*f 100*f 1000*f]; z = exp(j*2*pi*f/fs); % complex frequency z CL= (b0 + b1*z.^-1)./(1 + a1*z.^-1 + a2*z.^-2); % closed-loop response CL_dB= 20*log10(abs(CL)); semilogx(f,CL_dB),grid

The closed-loop frequency response is shown in Figure 6. Comparing this response to that of the equivalent continuous-time system in Figure B.2, we see that they match. Note the peak in the response occurs near the loop natural frequency of 400 Hz. The slope of the response in the stopband is -20 dB/decade.

**Figure 5. Phase Error for unit-step change in reference phase.** f_{n} = 400 Hz, ζ = 1.0.

**Figure 6. Closed-Loop Frequency Response.** f_{n} = 400 Hz, ζ = 1.0.

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4. Plot the step response and the frequency response for different values of damping. Figure 7 shows the step response and Figure 8 shows the frequency response.

**Figure 7. Step Response.** f_{n} = 400 Hz; ζ = 0.5 (blue), 1.0 (green), 2.0 (red)

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**Figure 8. Closed-Loop Frequency Response.** f_{n} = 400 Hz; ζ = 0.5 (blue), 1.0 (green), 2.0 (red)

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**Appendix A. Z-Domain model of DPLL with f**_{n} = 400 Hz

_{n}= 400 Hz

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%pll_response_z2.m nr 5/24/16 % Digital 2nd order type 2 PLL % step response and closed loop frequency response Knco= 1/4096; % NCO gain KP= 2; % 1/cycles phase detector gain wn = 2*pi*400; % rad/s loop natural frequency fs = 25e6; % Hz sample rate zeta = 1; % damping factor Ts= 1/fs; % s sample time KL= 2*zeta*wn*Ts/(KP*Knco) % loop filter proportional gain KI= wn^2*Ts^2/(KP*Knco) % loop filter integral gain % Find coeffs of closed-loop transfer function of u/ref_phase % CL(z) = (b0 + b1z^-1)/(a2Z^-2 + a1z^-1 + 1) b0= KP*KL*Knco; b1= KP*Knco*(KI - KL); a1= KP*KL*Knco - 2; a2= 1 + KP*Knco*(KI - KL); b= [b0 b1]; % numerator coeffs a= [1 a1 a2]; % denominator coeffs % step response N= 100000; n= 1:N; t= n*Ts; x= ones(1,N); % step function y= filter(b,a,x); % step response pe = y – 1; % phase error response plot(t*1e3,pe),grid xlabel('ms'),ylabel('Phase Error = u/ref-phase -1'),figure %plot phase error % Closed-loop frequency response u = 0:.1:.9; f= 10* 10 .^u; % log-scale frequencies f = [f 10*f 100*f 1000*f]; z = exp(j*2*pi*f/fs); % complex frequency z CL= (b0 + b1*z.^-1)./(1 + a1*z.^-1 + a2*z.^-2); % closed-loop response CL_dB= 20*log10(abs(CL)); semilogx(f,CL_dB),grid xlabel('Hz'),ylabel('CL(z) dB')

## Appendix B. 2^{nd} order continuous-time system closed loop response in s

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**Figure B.1.** **2 ^{nd} order system in s with a zero in the closed-loop response**

$ A(s) = 2\zeta\omega_n + \frac {\omega_n^2} {s} $ lead-lag filter

$ G(s) = \frac {1} {s} A(s) = \frac {2\zeta\omega_n} {s} + \frac {\omega_n^2} {s^2}$

$CL(s) = G/(1+G) $

$ CL(s) = \frac {2\zeta\omega_n + \omega_n^2} {s^2 + 2\zeta\omega_n + \omega_n^2} $

where

ω_{n}= 2πf_{n} = loop natural frequency

ζ= damping factor

**Figure B.2. Closed-Loop Frequency Response.** f_{n} = 400 Hz, ζ = 1.0.

**Appendix C. Converting H(s) to H(z)**

This is a way to approximate H(s) when a system’s passband frequency range is much less than the sample frequency. We choose this method because it results in a block diagram and transfer function that have the same form as that of our DPLL in Figure 2.

The definition of z is:

z = exp(sT_{s})

where s is complex frequency and T_{s} is the sample time. If we approximate z by the first two terms in the Taylor series for e^{x}, we have

z ~= 1 + sT_{s} (1)

Here we are assuming 2πfT_{s } << 1, or f/f_{s} << 1/2π. For our examples, we have been using f_{n} = 5 kHz or less and fs = 25 MHz. So f_{n}/f_{s} = .0002 << 1/2π.

Rearranging equation 1, we get

s ~= (z – 1)/T_{s}

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To convert H(s) to H(z) we replace each occurrence of the variable s by (z – 1)/T_{s}.

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**Appendix D. Alternative formulae for loop filter coefficients**

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The gain block in front of the NCO has gain Knco. The output frequency of the NCO due just to Vtune is

f = Vtune*K_{nco}*f_{s_nco}

If we define K_{v} = K_{nco}*f_{s_nco} Hz,

then K_{nco} can be replaced by K_{v}*T_{s_nco} in the formulae for K_{L} and K_{I}. So equations 3 and 4 become:

$ K_L = 2\zeta\omega_n/(K_pK_v) $

$ K_I = \omega_n^2T_{s\_filt}/(K_pK_v) $

Here, the units of K_{p }are cycle^{-1}, which is consistent with K_{v} in Hz. Alternative units for K_{p} and K_{v} are radian^{-1} and rad/s, respectively.

**References**

1. Gardner, Floyd M., __Phaselock Techniques, 3 ^{rd} Ed.__, Wiley-Interscience, 2005, Chapter 4.

2. Rice, Michael, __Digital Communications, a Discrete-Time Approach__, Pearson Prentice Hall, 2009, Appendix C.

4. Rice, C.1.3

6/9/2016 Neil Robertson

**Previous post by Neil Robertson:**

Digital PLL's -- Part 1

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