#### 1. Introduction

It is time consuming to measure and plot the overall load characteristics (Rieke diagram), but it has been used to express the characteristics of microwave oscillators, since the load characteristics can be represented within a circle of the finite extent. Previously a method was reported to determine the mathematical representation of the microwave oscillator admittance by using Rieke diagram and numerical calculation [1]-[6]. When analyzing the load characteristics (Rieke diagram) by using the oscillator admittance formula (1) that the voltage-dependent susceptance component (\(B_v\)) and frequency dependent components (\(G_{\omega}\), \(Y_{\omega2}\)) are added to the van der Pol oscillator model [6]-[9], the analysis results meet well with the experimental results [8]. The coefficients \(G_{\omega}\), \(G_{\omega2}\) make the constant power contour more practical than the van der Pol oscillator model on Rieke diagram, and \(B_v\) curves the constant frequency contours on Rieke diagram [6]. As for the mutual synchronization phenomena, \(G_{\omega2}\) makes unstable domain of anti-phase mode wider [10]. \(B_v\) well explains the injection-locking characteristics [8]. \(B_v\) also makes the anti-phase locking frequency a little bit different from in-phase locking frequency and makes anti-phase locking range asymmetric in shape in mutual synchronization characteristics [11]. The formula (2) does not have the frequency dependent components (\(G_{\omega}\), \(Y_{\omega2}\)) but it has \(B_v\). So, when \(|B_v|\) becomes larger, the formula (2) reproduces the synchronization characteristics better than the van der Pol oscillator model [8], [11]-[16]. This paper describes a method to conveniently determine the mathematical representation not using numerical calculation but using manual calculation.

#### 2. Mathematical Expression of Oscillator Admittance

The mathematical representation of microwave oscillator admittance is shown as (1) and (2) [6], [8].

\[\begin{align} Y(\mkern1.5mu j\omega, |V|^2) & = -G_0 + j B_0 + (G_{\omega} + j B_{\omega}) \cdot \mathit{\Delta} \omega \nonumber \\ & \hphantom{{}=} + (G_v + j B_v) |V|^2 + Y_{\omega2} \cdot \mathit{\Delta} \omega^2 \tag{1} \end{align}\] |

Where, \(\mathit{\Delta}\omega = \omega - \omega_0\), \(\omega_0/2\pi\) is the center frequency that is the oscillation frequency when the matched load (\(G_L = Y_0\), \(B_L = 0\)) is connected to the line. \(Y_0\) is the characteristic admittance of the line, and the coefficient \(Y_{\omega2} = G_{\omega2} + j B_{\omega2}\) is complex number.

\[\begin{align} Y(\mkern1.5mu j\omega, |V|^2) & = -G_0 + j B_0 + j B_{\omega} \cdot \mathit{\Delta} \omega \nonumber \\ & \hphantom{{}=} + (G_v + j B_v) |V|^2 \tag{2} \end{align}\] |

Now, let the load admittance be \(Y_L = G_L + j B_L\), and connect \(Y_L\) to the oscillator admittance (2).

\[\begin{equation*} Y(\mkern1.5mu j\omega, |V|^2) + Y_L = 0 \tag{3} \end{equation*}\] |

The real part of (3) and the output power of oscillator are

\[\begin{align} & -G_0 + G_v |V|^2 + G_L = 0 \tag{4} \\ & P = G_L |V|^2 \nonumber \end{align}\] |

From (4), the oscillator generates its maximum output power \(P_m\), when \(G_L = G_0/2\) (Fig. 1).

\[\begin{equation*} P_m = G_0^2 / 4G_v \tag{5} \end{equation*}\] |

Substitute (4) into the imaginary part of (3) to eliminate \(|V|^2\). The constant frequency contour is

\[\begin{equation*} B_0 + B_L = (G_L - G_0) B_v/G_v - B_{\omega} \cdot \mathit{\Delta} \omega \tag{6} \end{equation*}\] |

The coefficients ratio \(B_v/G_v\) is the slope (K) of the constant frequency contour (\(\mathit{\Delta} \omega / 2\pi = 0\)) (Fig. 2). When connecting the matched load (\(G_L = Y_0\), \(B_L = 0\)) to the oscillator, the coefficient \(B_0\) is

\[\begin{equation*} B_0 = (Y_0 - G_0) B_v/G_v \tag{7} \end{equation*}\] |

As is shown in Fig. 2, putting \(G_L = Y_0\), the coefficient \(B_{\omega}\) can be determined by measuring the changes in frequency and load susceptance (Appendix).

The coefficients \(G_0\), \(G_v\) and \(B_{\omega}\) are usually positive, so the stability criteria \(B_{\omega} G_v > 0\) are satisfied [6].

##### 2.1 Normalized Formula of Oscillator Admittance

Let (2) be normalized by using (9) [8].

\[\begin{align} & \hat{Y}(\mkern1.5mu jx, |\hat{V}|^2) = -\hat{G}_0 + j \hat{B}_0 + jx + (\hat{G}_0/2 + j \hat{B}_v) |\hat{V}|^2 \tag{8} \\ & \! \left. \begin{array}{@{}l@{\,}} \delta = (\omega - \omega_0)/\omega_0,\ Q = \omega_0 B_{\omega}/2Y_0,\ x = 2\delta Q \\ |\hat{V}|^2 = |V|^2\!/|V_m|^2,\ |V_m|^2 = G_0/2G_v, \\ -\hat{G}_0 + j \hat{B}_0 = ( - G_0 + j B_0)/Y_0 \\ \hat{B}_{\omega} = B_{\omega} \omega_0/ 2QY_0 = 1 \\ \hat{G}_v + j \hat{B}_v = (G_v + j B_v) |V_m|^2\! / Y_0 = \hat{G}_0/2 + j \hat{B}_v \\ \hat{P} = P/P_m,\ \hat{Y}_{L} = Y_L/Y_0 = \hat{G}_L + j \hat{B}_L \end{array} \right\} \tag{9} \end{align}\] |

Next, (7) is normalized as below.

\[\begin{equation*} \hat{B}_0 = (1 - \hat{G}_0) \hat{B}_v/\hat{G}_v = (1 - \hat{G}_0) 2\hat{B}_v/\hat{G}_0 \tag{10} \end{equation*}\] |

Now, the normalized formula (8) can be determined.

##### 2.2 Coefficients Determination Procedure

- To determine \(Y_0\): the characteristic admittance of the line.
- To change \(\hat{G}_L\) to find the maximum power \(P_m\), when \(\hat{B}_L = 0\). Let \(\hat{G}_L\) at the power \(P_m\) be \(\hat{G}_{Lpmax}\).
\[\begin{align} & \hat{G}_0 = 2\hat{G}_{Lpmax}, \\ & \hat{G}_v = \hat{G}_0/2 \end{align}\] - To measure the slope (K) of the constant frequency contour (\(\mathit{\Delta} \omega / 2\pi = 0\)) that passes through the matched load (\(\hat{G}_L = 1\), \(\hat{B}_L = 0\)) (Fig. 2).
\[\begin{gathered} \hat{B}_v = (\text{slope K}) \times \hat{G}_v = (\text{slope K}) \times \hat{G}_0/2 \\ \hat{B}_0 = (1 - \hat{G}_0) \hat{B}_v/\hat{G}_v = (1 - \hat{G}_0) 2\hat{B}_v/\hat{G}_0 \end{gathered}\]

##### 2.3 Examples

A High Frequency (HF) Clapp oscillator is used to plot Rieke diagram using the circuit simulation, since the HF oscillator can easily change its circuit constant. The oscillator has a Field Effect Transistor (2SK241) and an inductor (9.4 \(\mu H\)). The connection capacitor 0.1\(\mu F\) between the oscillator and the load \(Y_L\) is selected to get the standardized Rieke diagram, which means that in the case of van der Pol’s oscillator the center frequency line (\(\mathit{\Delta} \omega / 2\pi = 0\), \(x = 0\)) is equal to the zero susceptance and non-oscillation region exists in the heavy-load-admittance region [6] (Fig. 3).

- The characteristics of the line \(Y_0 = 0.002\) [S].
- \(\text{Pm} = 16.532\) [
*mW*] and \(\hat{G}_{Lpmax} = 1.388\), therefore\[\hat{G}_0 = 2\hat{G}_{Lpmax} = 2.776,\quad \hat{G}_v = \hat{G}_0/2 = 1.388\] - The slope (K) of the constant frequency contour (\(\mathit{\Delta} \omega / 2\pi = 0\)) passing through the matched load (\(\hat{G}_L = 1\), \(\hat{B}_L = 0\)). The slope (K) \(= +0.193\), accordingly
\[\begin{align}& \hat{B}_v = (\text{slope K}) \times \hat{G}_v = 0.268 \\ &\hat{B}_0 = (1 - \hat{G}_0) \hat{B}_v/\hat{G_v} = -0.343 \end{align}\]

As a results, the normalized formula of oscillator admittance is

\[\begin{align} \hat{Y}(\mkern1.5mu jx, |\hat{V}|^2) & = -2.776 - j\,0.343 + jx \nonumber \\ & \hphantom{{}=} + (1.388 + j\,0.268) |V|^2 \tag{11} \end{align}\] |

Since \(B_v\) is positive, the constant frequency contour (\(\mathit{\Delta} \omega / 2\pi = 0\)) curves to the left toward the periphery (Fig. 5).

#### 3. Conclusion

By limiting the measurement items to the slope (K) of the constant frequency contour (\(\mathit{\Delta} \omega / 2\pi = 0\)) and the load conductance \(G_L\) which generates maximum output power \(P_m\), the mathematical representation of microwave oscillator admittance can be determined by manual. The validity of the method was confirmed by comparison with the circuit simulation results (Fig. 4).

#### Acknowledgments

The author would like to thank Dr. Masamitsu Nakajima for his guidance over the years after graduating from the university. This paper is a development of his guidance.

#### References

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#### Appendix: Method to Find \(B_v/G_v\) from \(B_{\omega}\)

When the change in frequency and load susceptance are measured as 400 [*KHz*] and \(2.19Y_0\) [*S*] respectively (Fig. 2), the coefficient \(B_{\omega}\) is given by

\[B_{\omega} = 2.19 \times 0.002/(400 \times 2\pi)\] |

\(B_{\omega} = 0.1743 \times 10^{-5}\) [*S*/*KHz*]. Eliminating \(B_0\) from (6) and (7). Considering \(B_L = 0\),

\[\begin{equation*} (Y_0 - G_0) B_v/G_v = (G_L - G_0) B_v/G_v - B_{\omega} \mathit{\Delta} \omega \tag{A$\cdot $1} \end{equation*}\] |

Taking Fig. A\(\cdot\)1 into consideration, \(B_v/G_v\) is given by

\[\begin{equation*} B_v/G_v = -B_{\omega} \mathit{\Delta} \omega_1 / (Y_0 - G_L) \tag{A$\cdot $2} \end{equation*}\] |

Substituting \(G_L= 0.5Y_0\) [*S*], \(\mathit{\Delta} \omega_1 / 2\pi = -17\)[*KHz*] and \(Y_0 = 0.002\) [*S*] into (A\(\cdot\)2). Then \(B_v/G_v = 0.186\).