# Nonlinear Waveform Prediction in Fast Microwave Circuits

#### A time domain, transient analysis simulator that allows the prediction of waveforms in fast, nonlinear circuits in which accurate S-parameter data can be imported and used

**Nonlinear Waveform Prediction in Fast Microwave Circuits**

**Optotek Ltd. Kanata, Ontario, Canada**

MMICAD WAVEFORM,™ a new time domain, transient analysis simulator, allows the prediction of waveforms in strongly nonlinear circuits in which accurate S-parameter data can be imported and used in the simulation. The new time domain software is intended for the design of circuits where difficulties are experienced with conventional time domain or harmonic balance simulators. Therefore, it can be advantageously used in the design of high speed communication modules, microwave packages, nonlinear transmission lines and similar components.

Beneficial applications for MMICAD WAVEFORM would include the following characteristics: complex, externally defined, voltage and/or current source driving waveforms; signals large enough to cause strongly nonlinear operation of the active devices; circuits that have inherently (and strongly) nonlinear electrical response, including assessing the stability of power MMICs; circuits consisting of a mixture of microwave passive elements (defined by S-parameter data) and nonlinear active devices; and a transient phenomenon of interest (for example, start-up of oscillation) or phase shift as a function of time in pulsed microwave circuits, such as phased-array radar systems. The simulator is also useful in predicting unintentional oscillation in amplifier circuits without the use of injected signals.

Using the simulator's capabilities, the designer can address complex, frequency-dependent effects in passive circuits such as the variation of effective dielectric constant and loss of a microstrip line vs. frequency for a broadband microwave circuit. Since the simulator can import S and Y parameters for passive networks with up to 19 ports, advanced electromagnetic simulators (or measurement data) can be used to predict the transfer response for the passive circuitry external to an MMIC, enabling the determination of packaging effects (such as the coupling between bond wires) and interactions between MMICs connected using short microstrip lines. Externally defined waveforms (of the type encountered in microwave and optical communications applications) can be used to control voltage and current sources. To ensure accurate simulation, advanced GaAs FET and high electron mobility transistor (HEMT) models, such as EEHEMT1 and TOM3, are included in addition to the SPICE diode model. For transistor parameter extraction, MMICAD WAVEFORM can be used in combination with LASIMO™ large-signal transistor modeling software.

**Example: Harmonic Generation Using a Step Recovery Diode and Filter**

This application example illustrates the use of the MMICAD WAVEFORM software to simulate the waveforms at any node in a circuit containing a mix of nonlinear and linear devices, with the electrical characteristics of some or all of the linear devices being defined through the use of measured or predicted S-parameter data. MMICAD WAVEFORM predicts the waveform from a harmonic generator circuit consisting of a silicon step recovery diode (SRD) followed by an edge-coupled microstrip filter. The circuit topology is shown in **Figure 1** . On the left side is a 50 W source impedance, 1 GHz sinewave generator sending a 3 V signal into a 10 nH inductor. The SRD is connected to the 10 nH inductor through a short, low inductance 0.35 nH bond wire. The SRD is envisaged to be in chip form to avoid package-induced performance degradation. The diode model supported in MMICAD WAVEFORM is the SPICE-compatible PN-junction diode. The SrdTestDiode parameters are device = diode, IS = 1.0E-16, RS = 5, N = 1.0, TT = 1.0E-9, CJO = 0.4P, VJ = 0.65 and M = 2.5.

The small value 0.35 pF capacitor that separates the SRD from the subsequent edge-coupled filter is used to block the 1 GHz signal from the source. In a similar manner, the 10 nH inductor between the signal source and SRD is intended to prevent high frequency components generated by the SRD from finding their way back into the signal source.

The passband filter is of the edge-coupled microstrip variety. The topology was defined using the MMICAD SYNTHESIS™ filter program. A passband center frequency of 5 GHz was selected to allow inspection of the fourth harmonic of the 1 GHz fundamental signal. The S parameters of the edge-coupled microstrip filter were characterized using the MMICAD linear simulator with the simulated response predicted in uniform frequency steps up to 50 GHz. This response is shown in **Figure 2** . The result is saved in a filter.s2p file.

Any S-parameter data that are used by MMICAD WAVEFORM need to be converted into the time domain equivalent form of the data through the IMPULSE command, as shown in **Figure 3** . This selection initiates a character mode executable that reads the data in the S-parameter file. The time domain impulse response of the filter (the time domain equivalent of the S-parameter data set) can be visualized; the H21 (equivalent of S21) response is shown in **Figure 4** . Note that the impulse response dies away with time.

To simplify the situation, the SRD-based circuit is tested with a 50 W load replacing the series combination of the bandpass filter and the 50 W load shown previously. This modification allows visualization of the extremely nonlinear waveform generated by the abrupt termination of free carrier sweep out in the silicon SRD as the diode moves into negative bias. The total simulation time period is defined from 0 to 20 ns with the software performing a prediction at 10 ps time intervals (or more frequently).

By assigning the VProbe command to the left mouse button, the user can position the cursor on any node to view the predicted waveform at that node. When the voltage across the output 50 W load is probed, the software temporarily superimposes the plot of the predicted waveform over the circuit schematic, as shown in **Figure 5** .

The response of the introduced edge-coupled filter is defined in the filter.wfi file, which was generated from the filter.s2p file using MMICAD WAVEFORM in IMPULSE mode. In this example, the filter.s2p data in the frequency domain are mapped to the filter.wfi file in the time domain. **Figure 6** shows the waveform (upper plot) at the input to the edge-coupled filter. The output from the edge-coupled filter and across the 50 W terminating load resistance is shown in **Figure 7** . A comparison of this waveform with the signal at the input to the series 10 nH inductance, shown in **Figure 8** , illustrates that the SRD circuit generates appreciable signal at the fifth harmonic. (Note that although the actual signal source in series with the 50 W source resistance is generating a sinewave with an amplitude of 3 V, the amplitude of the signal from the source, including the effect of the source impedance, is 1.5 V. The amplitude of the fourth harmonic at 5 GHz is 200 mV, as shown in the output screen image.)

To investigate domain characteristics of the waveform, the signal is sampled from 9 to 19 ns using MMICAD WAVEFORM's REFORMAT command. The predicted spectral response is displayed using the fast Fourier transform-based SPECTRUM command. **Figure 9** shows the calculation of the amplitude (50 W system) of the first three harmonics. Predicted signal levels are -7.044, -19.294 and -35.593 dBm for the first, second and third harmonics, respectively. The level of the source 1.5 V amplitude into a 50 W load is approximately 16.5 dBm.

**Conclusion**

Despite severe nonlinearities, the MMICAD WAVEFORM software is able to predict the SRD circuit waveforms. Since S-parameter data can be used to characterize the response of the linear circuit elements, the software can accurately incorporate complex effects such as the skin effect and frequency dispersion in the effective dielectric constant of a microstrip line. Using the fast Fourier transform on a sample of the waveform after the initial transients are more settled allows examination of the steady-state response of the circuit, including an indication of the level of any harmonics generated. MMICAD WAVEFORM contributes to the simulation of fast microwave and optoelectronic circuits by solving the response in the time domain. As such, MMICAD WAVEFORM complements the capabilities of harmonic balance simulations conventionally used to predict the steady-state response of RF and microwave circuits.

MMICAD WAVEFORM operates on Pentium-compatible systems under MS Windows® 95/98/NT4. The minimum recommended system is a Pentium II 350 MHz PC with 64 MB of RAM. MMICAD WAVEFORM is priced from $7500. Additional information can be obtained from the company's Web site: www.optotek.com.

*Optotek Ltd., Kanata, Ontario, Canada (613) 591-0336.*