### Introduction

As described in the references, a standard procedure can be used to determine the values of R_{0}, C_{0}, and C_{P} for a second-order loop filter in a phase-locked loop (PLL). It uses open-loop bandwidth (ω_{0}) and phase margin (ϕ_{M}) as design parameters, and can be extended to third-order loop filters to determine R_{2} and C_{2} (Figure 1). The procedure solves for C_{P} directly and subsequently derives the remaining values.

In some cases, C_{P}, R_{2}, and C_{2} may be fixed-value components integrated within the PLL, leaving only R_{0} and C_{0} available for controlling the loop response. This nullifies the aforementioned procedure because C_{P} cannot be adjusted. This article proposes an alternative procedure that can be used when the value of C_{P} is fixed, and addresses limitations imposed by the inability to control the value of C_{P}.

### Assumptions

This loop filter design method relies on two assumptions that are typically used in third-order passive filter designs that extend a second-order loop filter design to third-order by compensating for the presence of R_{2} and C_{2} through adjustment of R_{0} and C_{0}.

- The pole frequency resulting from R
_{2}and C_{2}should be at least an order of magnitude greater than ω0 (the desired open-loop unity-gain bandwidth); specifically f_{0}≤ 0.1/(2πR_{2}C_{2}), where f_{0}= ω_{0}/(2π). - The load of the series combination of R
_{2}and C_{2}on the R_{0}-C_{0}-C_{P}network should be negligible.

### Transfer Function of a Second-Order Loop Filter

A second-order loop filter has two time constants (T_{1} and T_{2}) associated with its components:

The loop filter’s transfer function, in terms of T_{1}, T_{2}, and C_{P}, is important because it plays a significant role in the overall response of the PLL:

### PLL System Function

The small signal model shown in Figure 2 provides the means for formulating the PLL response and a template for analyzing phase variation at the output resulting from a phase disturbance at the input. Note that the voltage-controlled oscillator (VCO), being a frequency source, behaves like an ideal phase integrator, so its gain (K_{V}) has a 1/s factor (the Laplace transform equivalent of integration). Hence, the small signal model of a PLL has frequency dependence (s = σ + jω).

The closed-loop transfer function (H_{CL}) for a PLL is defined as θ_{OUT}/θ_{IN}. The open-loop transfer function (H_{OL}), defined as θ_{FB}/θ_{IN}, is related to the closed-loop transfer function. It is instructive to express H_{CL} in terms of H_{OL} because the open-loop transfer function contains clues about closed-loop stability:

K represents the combined gains of the phase-frequency detector (PFD), charge pump, and VCO—that is, K = K_{D}K_{V}, where K_{D} is the charge pump current in amperes and K_{V} is the VCO gain in Hz/V. H_{OL}, H_{CL}, and H_{LF} are all functions of s. The negative sign in Equation 4 shows the phase inversion implied by the negative feedback to the summing node in Figure 2. Defining H_{OL} as in Equation 4 leads to subtraction in the denominator of Figure 5, which provides an intuitive explanation of closed-loop stability.

Inspection of Equation 5 reveals a potential loop stability problem. Given that H_{OL} is a function of complex frequency (s = σ + jω), it necessarily has frequency dependent magnitude and phase components. Therefore, if H_{OL} simultaneously exhibits unity gain and zero phase shift (or any integer multiple of 2π radians) for any particular value of s, the denominator of H_{CL} becomes zero, the closed-loop gain becomes undefined, and the system becomes completely unstable. This implies that stability is governed by the frequency-dependent magnitude and phase characteristics of H_{OL}. In fact, at the frequency for which the magnitude of H_{OL} is unity, the phase of H_{OL} must stay far enough from zero (or any integer multiple of 2π) to avoid a zero denominator in Equation 5.

The frequency, ω_{0}, at which the magnitude of H_{OL} is unity, holds great importance. The phase of H_{OL} at ω_{0} defines the phase margin of the system ϕ_{M}. Both ω_{0} and ϕ_{M} can be derived from H_{OL}.

### Defining R_{0} and C_{0} in Terms of ω_{0} and ϕ_{M}

Using the design parameters ω_{0} and ϕ_{M} to determine the values of R_{0} and C_{0} requires expressions containing those four variables and other constant terms. Start with Equation 4, because it defines H_{OL}. This includes H_{LF}, which includes R_{0} and C_{0} via T_{1} and T_{2}. Since H_{OL} has magnitude and phase, it stands to reason that ω_{0} and ϕ_{M} can be incorporated as well.

Substituting Equation 3 into Equation 4 and rearranging terms yields Equation 6, which presents H_{OL} in terms of T_{1} and T_{2} along with constants K, N, and C_{P}:

Evaluation at s = *j*ω, yields the frequency response of H_{OL}:

The (*j*ω)^{2} term in the denominator simplifies to –ω^{2}:

The magnitude and phase of H_{OL} are:

Keep in mind that T_{1} and T_{2} are shorthand expressions for algebraic combinations of R_{0}, C_{0}, and C_{P}. Evaluating Equation 9 at ω = ω_{0} and setting |H_{OL}| = 1 defines the unity gain frequency, ω_{0}, as the frequency at which the magnitude of H_{OL} is unity.

Similarly, evaluating Equation 10 at ω = ω_{0} and setting ∠H_{OL} = ϕ_{M} defines the phase margin, ϕ_{M}, as the phase of H_{OL} at frequency ω_{0} (the unity gain frequency).

It is a trivial matter to expand Equation 11 and Equation 12 by substituting Equation 1 for T_{2} and Equation 2 for T_{1}, which brings R_{0} and C_{0} into the equations. Hence, we have succeeded in relating ω_{0} and ϕ_{M} to the variables R_{0} and C_{0} along with constants K, N, and C_{P}.

Simultaneously solving the resulting equations for R_{0} and C_{0} is no trivial task. The symbolic processor available in MathCad^{®} can solve the two simultaneous equations, but arccos must be substituted for arctan. This transformation enables the symbolic processor to solve for R_{0} and C_{0}, yielding the following solution sets (R_{0A}, C_{0A}; R_{0B}, C_{0B}; R_{0C}, C_{0C}; and R_{0D}, C_{0D}). See the Appendix for details on transforming Equation 12 to use the arccos function.

This result is problematic because the goal was to solve for R_{0} and C_{0} given ω_{0} and ϕ_{M}, but this indicates four possible R_{0}, C_{0} pairs instead of a unique R_{0}, C_{0} pair. However, closer inspection of the four results leads to a single solution set as follows.

Note that in the context of modeling a PLL, all of the variables in the above equations possess positive values, including cos(ϕ_{M}) because ϕ_{M} is constrained to values between 0 and π/2. As a result, C_{0A} and R_{0B} are clearly negative quantities. Therefore, solution sets R_{0A}, C_{0A} and R_{0B}, C_{0B} are immediately ruled out because negative component values are not acceptable. The R_{0C}, C_{0C} and R_{0D}, C_{0D} results require further analysis, however.

Note that the four equations involving R_{0C}, C_{0C} and R_{0D}, C_{0D} possess the common factor:

Closer inspection reveals that Expression 13 has the form a^{2} – (2ac)cos(β) + c^{2}. Equating this with the arbitrary quantity, b^{2}, yields:

Equation 14, the Law of Cosines, relates a, b, and c as the lengths of the three sides of a triangle with β being the interior angle of the vertex opposite side b. Since b^{2} is the square of the length of one side of a triangle, it must be a positive quantity, which implies the right side of Equation 14 must also be positive. Thus, Expression 13 must be a positive quantity, which means the denominator of R_{0D} is positive. The numerator of R_{0D} is also positive, therefore R_{0D} must be negative, which rules out the R_{0D}, C_{0D} solution set. This leaves only the R_{0C}, C_{0C} pair as a contender for the simultaneous solution of Equation 11 and Equation 12.

### Constraints on R_{0} and C_{0}

Although Equation 15 and Equation 16 are contenders for the simultaneous solution of Equation 11 and Equation 12, they are only valid if they result in positive values for both R_{0} and C_{0}. Close inspection of R_{0} shows it to be positive—its numerator is positive, because the range of cos^{2}(x) is 0 to 1—and its denominator is the same as Expression 13, which was previously shown to be positive. The numerator of C_{0} is also the same as Expression 13, so C_{0} is positive as long as its denominator satisfies the following condition:

This is depicted graphically in Figure 3, in which the left and right sides of Equation 17 are each equated to y (blue and green curves) with the horizontal axis sharing ω_{0} and ϕ_{M}. The intersection of the two curves marks the boundary condition for ω_{0} and ϕ_{M}. The condition under which Equation 17 is true appears as the red arc. The portion of the horizontal axis beneath the red arc defines the range of ϕ_{M} and ω_{0} that ensures C_{0} is positive. Note the point on the horizontal axis directly below the intersection of the blue and green curves establishes ϕ_{M_MAX}, the maximum value of ϕ_{M} to ensure C_{0} is positive.

Equation 18 requires that C_{P}Nω_{0}^{2} be less than K in order to satisfy the constraints of the arccos function for ϕ_{M_MAX} between 0 and π/2. This establishes ω_{0_MAX}, the upper limit on ω_{0} to ensure C_{0} is positive.

### Compensating for R_{2} and C_{2} (Third-Order Loop Filter)

In the case of a third-order loop filter, components R_{2} and C_{2} introduce additional phase shift, Δϕ, relative to the second-order loop filter:

To deal with this excess phase shift, subtract it from the desired value of ϕ_{M}:

Applying ϕ_{M_NEW} to Equation 15 and Equation 16 results in different values for R_{0} and C_{0} than for the second-order solution, with the new values compensating for the excess phase shift introduced by R_{2} and C_{2}. The presence of R_{2} and C_{2} also affects ϕ_{M_MAX}, the maximum allowable value of ϕ_{M}. The new maximum value of ϕ_{M} (ϕ_{M_MAX_NEW}) is

### Conclusion

This article demonstrates using open-loop unity-gain bandwidth (ω_{0}) and phase margin (ϕ_{M}) as design parameters for second-order or third-order loop filters when only components R_{0} and C_{0} are adjustable. Simulation of a PLL with a second-order loop filter using R_{0} and C_{0} yields an exact match to the theoretical frequency response of H_{OL} and the resulting phase margin, thereby validating the equations. The parameters ω_{0} and ϕ_{M} have upper bounds for a second-order loop filter per Equation 19 and Equation 18, respectively.

The procedure for determining R_{0} and C_{0} assumed a second-order loop filter, but is extendible to third-order loop filter designs by adjusting the desired phase margin (ϕ_{M}) to a new value (ϕ_{M_NEW}) per Equation 21, yielding a new upper bound (ϕ_{M_MAX_NEW}) per Equation 22.

Although simulations using a second-order loop filter validated Equation 15 and Equation 16, validating the equations that extend the design procedure to third-order loop filter designs requires a redefinition of the loop filter response, H_{LF}(s), to include R_{2} and C_{2} as follows:

The incorporation of this form of H_{LF} into the H_{OL} and H_{CL} equations enables simulations of third-order loop filter designs using R_{0} and C_{0}. Such simulations reveal the calculated values of R_{0} and C_{0} deviate from the theoretical frequency response and phase margin associated with H_{OL} for a PLL when using a third-order loop filter. This is predominantly due to the effect of R_{2} and C_{2} on H_{OL} in a third-order loop filter.

Recall that the formulas for R_{0} and C_{0} assume a second-order loop filter, but R_{2} and C_{2} do not exist in a second-order filter, so including them as part of the loop filter constitutes a source of error in spite of adjusting R_{0} and C_{0} to compensate for the phase shift introduced by R_{2} and C_{2}. Even in the presence of this error, however, simulation indicates that using the adjusted values of R_{0} and C_{0}, but limiting the choice of ω_{0} to a maximum of ¼ of the value dictated by Equation 19 yields acceptable results. In fact, the simulated open-loop bandwidth and phase margin results deviate only slightly from the design parameters (ω_{0} and ϕ_{M}) for a PLL using a third-order loop filter.

### Simulation Results

The following is the result of running four simulations of a PLL with a third-order loop filter. The simulations all have the following fixed-loop filter components and PLL parameters:

C_{P} = 1.5 nF

R_{2} = 165 kΩ

C_{2} = 337 pF

K_{D} = 30 µA

K_{V} = 3072 (25 ppm/V at 122.88 MHz)

N = 100

Simulation 1 and Simulation 2 use ω_{0} = 100 Hz, which is near the calculated upper limit of 124.8 Hz (ω_{0_MAX}). As such, Simulation 1 and Simulation 2 deviate from the design parameter values (ω_{0} and ϕ_{M}) by nearly 10%. On the other hand, Simulation 3 and Simulation 4 use ω_{0} = 35 Hz, which is approximately ¼ the upper limit. As expected, Simulation 3 and Simulation 4 hold much closer to the design parameters (ω_{0} and ϕ_{M}), yielding an error of only about 1%.

Table 1 summarizes the simulation results and also includes the calculated values of R_{0}, C_{0}, ω_{0_MAX}, and ϕ_{M_MAX} for the given design parameters, ω_{0} and ϕ_{M}. Note that for the purpose of comparison it would be preferable for both Simulation 1 and Simulation 3 to use ϕ_{M} = 80°, but Simulation 1 must satisfy the constraint imposed by Equation 22 of ϕ_{M} < 48° (hence the choice of 42°).

**
Table 1: Simulation Summary**

Simulation 1 | Simulation 2 | Simulation 3 | Simulation 4 | |||||

Parameter | ω_{0} |
ϕ_{M} |
ω_{0} |
ϕ_{M} |
ω_{0} |
ϕ_{M} |
ω_{0} |
ϕ_{M} |

Design | 100 Hz | 42° | 100 Hz | 30° | 35 Hz | 80° | 35 Hz | 30° |

Simulation | 93.1 Hz | 38.7° | 92.5 Hz | 27.1° | 34.9 Hz | 79.0° | 34.7 Hz | 29.3° |

R_{0} |
969.6 kΩ | 1118 kΩ | 240.1 kΩ | 139.9 kΩ | ||||

C_{0} |
14.85 nF | 3.670 nF | 225.5 nF | 21.24 nF | ||||

ω_{0_MAX} |
124.8 Hz | 124.8 Hz | 124.8 Hz | 124.8 Hz | ||||

ϕ_{M_MAX} |
48.0° | 48.0° | 84.8° | 84.8° |

Figure 4 and Figure 5 show the open- and closed-loop response for each simulation.

### Appendix—Converting the Discontinuous Arctan Function to the Continuous Arccos Function

Equation 10 demonstrates that the angle ϕ is the difference between angle θ_{2} and angle θ_{1}, where θ_{2} = arctan(ωT_{2}) and θ1 = arctan(ωT_{1}). Furthermore, ωT_{2} is expressible as x/1 and ωT_{1} as y/1:

This implies the geometric relationship shown in Figure 6, with θ_{1} and θ_{2} defined by the triangles of Figure 6 (b) and (a), respectively. Figure 6 (c) combines these two triangles to show ϕ as the difference between θ_{1} and θ_{2}.

The Law of Cosines relates an interior angle (θ) of a triangle to the lengths of the three sides of the triangle (a, b, and c) as follows:

Applying the Law of Cosines to angle ϕ in Figure 6 (c) yields:

Solving for ϕ:

But, x/1 = ωT_{2} and y/1 = ωT_{1}, allowing ϕ to be expressed in terms of T_{1} and T_{2}.

### Ссылки

Brennan, Paul V. *Phase-Locked Loops: Principles and Practice*. McGraw-Hill, 1996.

Keese, William O. AN-1001, National Semiconductor Application Note, *An Analysis and Performance Evaluation of a Passive Filter Design Technique for Charge Pump Phase-Locked Loops*. May 1996.