Analysis of Phase-Locked Loops

using the Best Linear Approximation

Dries Peumans

Adam Cooman

Gerd Vandersteen

Nonlinear behaviour degrades the envisioned performance

2

UnwantedIdeal

Nonlinear behaviour

Analysis of Phase-Locked Loopsusing the Best Linear Approximation

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1 How to describe the PLL?

Architecture and linear models

2 How to characterise the nonlinearities?

Best Linear Approximation (BLA) and multisines

3 How to combine both?

Pitfalls and results

The PLL uses feedback to lock the phase of its oscillator to the reference

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CPPFD LF

VCO

DIV

UP

DOWN

Can we come up with an ideal model?

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CPPFD LF

VCO

DIV

UP

DOWN

PLL is best studied in the phase domain

1 Voltage and current domain

Strongly nonlinear

2 Phase domain

Linear

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𝑣 𝑡 = 𝐴 cos(𝜔𝑐𝑡 + 𝜑 𝑡 )

Voltage 𝑣 𝑡

Phase noise 𝜑 𝑡

You can linearize the behaviourin the phase domain

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PFD + CP LF VCO

DIV

The BLA combines concepts from both the linear and Volterra theory

1 Linear model

+ Easy to use / widespread

− Neglects nonlinearities

2 Volterra theory

+ Models nonlinearities

− Difficult− Weak nonlinearities

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Best Linear Approximation+ Linear+ Strong nonlinearities

The BLA extracts a linear modelfrom nonlinear systems

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Linear Distortions

Multisines make odd and even NLs distinguishable

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Multisines give more controlover the excited frequencies

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Noise Multisine

Wanted profile

Frequency Frequency

Applying multisines as time jitterallows to characterise the distortions

1 Non-ideal oscillator

2 Digital reference clock

Time jitter

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Phase domain multisine

Phase domain multisinesneed to be quantised

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Phase domain multisinesare applied as the reference signal

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A 4th-order type-II PLL is analysed using the BLA

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PFD behaves linearly in phase domain

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𝑌𝑃𝐹𝐷

Even 𝑌𝑆

Odd 𝑌𝑆

Introduce nonlinear behaviour in the CP

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1 Asymmetric delay

𝜏𝑈𝑃 ≠ 𝜏𝐷𝑁

2 Mismatch in current sources

𝐼𝑈𝑃 ≠ 𝐼𝐷𝑁

Effects of non-idealities in CP are significant

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Asymmetry of 1‰ Mismatch of 1%

𝑌𝐶𝑃 𝑌𝐶𝑃

Even 𝑌𝑆

Even 𝑌𝑆

Odd 𝑌𝑆 Odd 𝑌𝑆

Analysis of Phase-Locked Loopsusing the Best Linear Approximation

19

1 How to describe the PLL?

Architecture and linear models

2 How to characterise the nonlinearities?

Best Linear Approximation (BLA) and multisines

3 How to combine both?

Pitfalls and results