### A.1 Oscillation Amplitude

The steady-state base-to-emitter voltage V

_{BE} for a Colpitts oscillator is given by [83]

$$ V_{BE} = \frac{{2I_{CC} }}{{g_{mc} }}, $$

(A.1)

where I

_{CC} is the collector bias current and g

_{mc} is the minimum required transconductance for oscillation. For the conventional Colpitts oscillator, (

A.1) can be written as

$$ V_{BE} = V_{O} - n_{COLP} V_{O} = \frac{{2I_{CC} }}{{\omega_{COLP}^{2} C^{\prime}_{1} C_{2} R_{S} }}, $$

(A.2)

where R

_{S} is the series tank resistance. Replacing R

_{S} by its parallel equivalent resistor R

_{T}, we get

$$ V_{BE} = (1 - n_{COLP} )V_{O} = \frac{{2I_{CC} R_{T} }}{{\omega_{COLP}^{4} L_{B}^{2} C^{\prime}_{1} C_{2} }}, $$

(A.3)

resulting in a peak oscillation amplitude of

$$ V_{O} = 2I_{CC} R_{T} n_{COLP} . $$

(A.4)

For the LC emitter-degenerated oscillator,

$$ V_{BE} = V_{O,LC} - n_{LC} V_{O,LC} = \frac{{2I_{CC} R_{T} }}{{\omega_{LC}^{4} L_{B}^{2} C^{\prime}_{1} C_{2} \left( {1 - k} \right)}} = 2I_{CC} R_{T} \cdot \frac{{k^{2} }}{1 - k} \cdot \frac{{L_{DEG}^{2} C_{2} }}{{L_{B}^{2} C^{\prime}_{1} }} $$

(A.5)

from which we obtain

$$ V_{O,\;LC} = 2I_{CC} R_{T} \cdot \frac{{k^{2} }}{{n_{LC} \left( {1 - k} \right)^{2} }} \cdot \frac{{L_{DEG}^{2} }}{{L_{B}^{2} }}. $$

(A.6)

It is clear from (

A.4) and (

A.6) that for a given bias current, LC emitter degeneration provides more flexibility in setting the oscillation amplitude compared to the conventional Colpitts topology. Assuming the same R

_{S} for the two topologies

^{3} and assuming that the tank capacitors are kept constant for the purpose of comparison (which implies that the inductors in the tank are varied to obtain the same oscillation frequency for the two topologies), we can obtain the ratio of the two oscillation amplitudes as

$$ \frac{{V_{O,\;LC} }}{{V_{O} }} = \frac{{\left( {1 - n_{COLP} } \right)}}{{\left( {1 - n_{LC} } \right)}} \cdot \frac{1}{{\left( {1 - k} \right)}} $$

(A.7)

for the same oscillation frequencies. Since k < 1, n

_{LC} > n

_{COLP} and therefore, V

_{O;LC} is always greater than V

_{O}.

### A.2 Phase Noise Analysis Using Leeson’s Model

From [71], the 1/f

^{2} phase noise of the Colpitts oscillator can be expressed as

$$ S_{\Updelta \upphi out,\;COLP} = 2\frac{{\left\langle {\overline{{I_{n}^{2} }} } \right\rangle }}{{V_{O}^{2} }} \cdot \left| {\frac{{A_{0} }}{{C^{\prime}_{I} }}} \right|^{2} \cdot \frac{1}{{\Updelta \omega^{2} }}, $$

(A.8)

where A

_{o} is the element A of the ABCD matrix of the feedback network (i.e., the tank),

\( C^{\prime}_{I} = dC_{I} /d\omega \left| {_{{\omega_{0} }} } \right. \), and C

_{I} is the imaginary part of the element C of the matrix. For a Colpitts oscillator,

\( A_{0} = - C_{2} /C^{\prime}_{1} \) and

$$ C_{I} = (C^{\prime}_{1} + C_{2} )\omega - LC^{\prime}_{1} C_{2} \omega^{3} $$

(A.9)

Replacing C

_{2} by C

_{2;eff}, differentiating C

_{I} with respect to ω, and then setting

\( \omega = \omega_{0} = \omega_{LC} \), we obtain, for an LC emitter-degenerated oscillator,

$$ C^{\prime}_{I} = - 2\left( {C^{\prime}_{1} + \left. {C_{2,\;eff} } \right|_{{\omega_{LC} }} } \right) - \omega_{LC} C^{\prime}_{1} \left| {\frac{1}{{C_{2,eff} }}\frac{{dC_{2,\;eff} }}{d\omega }} \right|_{{\omega_{LC} }} . $$

(A.10)

From (

5.3),

$$ \left. {C_{2,\;eff} } \right|_{{\omega_{LC} }} = C{}_{2}\left( {1 - k} \right) ,\;{\text{and}} $$

(A.11)

$$ \frac{1}{{C_{2,\;eff} }} \cdot \left. {\frac{{dC_{2,\;eff} }}{d\omega }} \right|_{{\omega_{LC} }} = \frac{2k}{{\omega_{LC} }}, $$

(A.12)

where k is given by (

5.7). Substituting (

A.11) and (

A.12) in (

A.10), we obtain

$$ C^{\prime}_{I} = - 2\left[ {(1 + k)C^{\prime}_{1} + (1 - k)C_{2} } \right]. $$

(A.13)

Substituting the values of A

_{0} and

\( C^{\prime}_{I} \) in (

A.8) and re-arranging the result, we obtain the expression for the close-in phase noise of an emitter-degenerated Colpitts oscillator as

$$ S_{\Updelta \upphi out,\;LC}
= \frac{1}{2}\frac{{\left\langle {\overline{{I_{n}^{2} }} }
\right\rangle }}{{V_{O,\;LC}^{2} }} \cdot \frac{{C_{2}^{2}
}}{{C^{{\prime}^{2}}_{1} \left[ {\left( {\frac{1 + k}{1 - k}}
\right)C^{\prime}_{1} + C_{2} } \right]^{2} }} \cdot
\frac{1}{{\Updelta \omega^{2} }}. $$

(A.14)

The phase noise of the conventional Colpitts topology is readily obtained from (

A.14) by replacing k with 0,

$$ S_{\Updelta \upphi out,\;COLP}
= \frac{1}{2}\frac{{\left\langle {\overline{{I_{n}^{2} }} }
\right\rangle }}{{V_{O}^{2} }} \cdot \frac{{C_{2}^{2}
}}{{C^{{\prime}^{2}}_{1} (C^{\prime}_{1} + C_{2} )^{2} }} \cdot
\frac{1}{{\Updelta \omega^{2} }}. $$

(A.15)

### A.3 Phase Noise Analysis Using Linear Time-Variant Model

Following the analysis in [74] for a Colpitts oscillator, the phase noise due to collector current noise can be expressed as

$$ S_{{\Updelta \upphi out,\;i_{c} ,\;COLP}} = \frac{{k_{B} T}}{{4V_{O}^{2} }} \cdot \frac{1}{{R_{T} C_{2}^{2} }} \cdot \frac{1}{{n_{COLP} \left( {1 - n_{COLP} } \right)}} \cdot \frac{1}{{\Updelta \omega^{2} }}, $$

(A.16)

and that due to R

_{T} as

$$ S_{{\Updelta \upphi out,\;R_{T} ,\;COLP}} = \frac{{k_{B} T}}{{2V_{O}^{2} }} \cdot \frac{1}{{R_{T} C_{2}^{2} }} \cdot \frac{1}{{n_{COLP}^{2} }} \cdot \frac{1}{{\Updelta \omega^{2} }} $$

(A.17)

The overall phase noise for the conventional Colpitts topology is then given by

$$ S_{\Updelta \upphi out,\;COLP} = \frac{{k_{B} T}}{{4V_{O}^{2} }} \cdot \frac{1}{{R_{T} C_{2}^{2} }} \cdot \frac{{2 - n_{COLP} }}{{n_{COLP}^{2} \left( {1 - n_{COLP} } \right)}} \cdot \frac{1}{{\Updelta \omega^{2} }} $$

(A.18)

As discussed earlier, the LC degeneration impedance appears capacitive at the oscillation frequency of the emitter-degenerated oscillator. Therefore, the current and voltage waveforms of the emitter-degenerated oscillator are similar to those of the conventional Colpitts oscillator. This in turn implies that the device noise currents are injected into the tank at the voltage peaks, thereby reducing the amount of device noise conversion to phase noise [72]. Thus, the phase noise analysis carried out for the Colpitts topology is also valid for the LC emitter-degenerated oscillator topology.

By substituting C

_{2} with

\( \left. {C_{2,eff} } \right|_{{\omega_{LC} }} = (1 - k)C_{2} \) and n

_{COLP} with n

_{LC} in (

A.18), the phase noise of an LC emitter-degenerated Colpitts oscillator is readily expressed as

$$ S_{\Updelta \upphi out,\;LC} = \frac{{k_{B} T}}{{4V_{O,\;LC}^{2} }} \cdot \frac{1}{{R_{T} C_{2}^{2} }} \cdot \frac{{2 - n_{LC} }}{{n_{LC}^{2} \left( {1 - n_{LC} } \right)}} \cdot \frac{1}{{\left( {1 - k} \right)^{2} }} \cdot \frac{1}{{\Updelta \omega^{2} }} $$

(A.19)

The ratio of the phase noises of the two oscillator topologies is

$$ \frac{{S_{\Updelta \upphi out,\;LC} }}{{S_{\Updelta \upphi out,\;COLP} }} = \frac{{V_{O}^{2} }}{{V_{O,\;LC}^{2} }} \cdot \frac{{\left( {2 - n_{LC} } \right)}}{{\left( {2 - n_{COLP} } \right)}} \cdot \frac{{\left( {1 - n_{COLP} } \right)}}{{\left( {1 - n_{LC} } \right)}} \cdot \frac{{n_{COLP}^{2} }}{{n_{LC}^{2} }} \cdot \frac{1}{{\left( {1 - k} \right)^{2} }} $$

(A.20)

Using the result of (A.7), and with the same assumptions, (A.20) is simplified to

$$ \frac{{S_{\Updelta \upphi out,\;LC} }}{{S_{\Updelta \upphi out,\;COLP} }} = \frac{{\left( {2 - n_{LC} } \right)}}{{\left( {2 - n_{COLP} } \right)}} \cdot \frac{{\left( {1 - n_{LC} } \right)}}{{\left( {1 - n_{COLP} } \right)}} \cdot \frac{{n_{COLP}^{2} }}{{n_{LC}^{2} }} $$

(A.21)

Since *n* _{ LC } > *n* _{ COLP }, the ratio in (A.21) is always less than 1, indicating that the LC emitter-degenerated topology exhibits better phase noise than the conventional Colpitts oscillator.

From the foregoing analysis, the importance of using a linear time-variant model is clearly seen. In (A.8) and (A.14), all device noise sources are converted to phase noise by the same transfer function, whereas (A.16–A.19) indicate different transfer functions for different noise sources. The key concept that enables higher accuracy in the LTV model is the impulse sensitivity function (ISF), which is different for different noise sources and different circuit topologies [72]. Furthermore, the ISF takes into account the cyclo-stationary nature of device noise sources, whereas the Leeson’s model treats all noise sources as stationary processes.