Resonance Simulation in PI Design Xiao Dan Nokia Abstract PCB power-ground resonance theory is introduced and relationship between Z simulated parameter and resonance frequency is discussed based on the comparison of theory calculation and simulation in Sigrity PowerSI via one simple example. Additionally, considering the impact of excitation source impedance, VRM should be modeled as voltage source and drink IC should be modeled as current source to obtain accurate simulation results. Keywords PI, resonance, eigen mode, PDN self-impedance, excitation source Introduction The amout, type and position definition and optimization of decoupling capacitors which depend on the resonance hot spot position, resonance frequency and amplitude level between power and ground planes are big challenges in PI design,. In resonance analysis area, 3D full wave simulation tools are very useful assistance, on the other hand, measurements provide similar analysis as well. Benefit of simulation is eliminating test error related with test equipments, and shows good correlation with test, provided that both simulation and test are set up correctly. Sometimes wrong results will be caused by wrong setting; and still the useful information is needed to be read from results based on resonance theory,even though the simulation results are right. Commercial PI simulation tools, such as Sigrity PowerSI, Ansys SIwave,etc, all provide three kinds of analysis methods: eigen mode, S/Z parameter mode, frequency sweep mode or spatial mode. Eigenmode and parameter mode analysis base on resonance cavity and energy transmission theory. Frequency sweep mode is mostly close to the simulation of real active working PCB. One simple example is discussed as followed, separatelly simulated in Sigrity PowerSI by three modes. It's one 2-layer PCB, top layer is Power plane, bottom layer is ground plane, size: a=30 mm, b=50 mm, copper thickness=0.011 mm, ε r =4.2, and substrate layer thickness h=0.063 mm, stack-up and parameter are showed in Figure 1. Power-GND Planes Resonance Theory Figure 2 shows structure of the ideal PCB Power-GND pairs,the top and bottom surface are PEC boundary and other four surfaces are PMC boundary from non-radiation character point of view, so that this structure can be considered as rectangle resonance cavity. Figure 2 Rectangle resonance cavity formed by ideal Power- GND planes Base on eigen number of cavity,resonance frequency is derived from formula (1): c m f = π n p mnp a + π ( ) ( ) 2π ε b + π ( ) r h 2 2 2 In above formula, c= (μ 0 ε r ) -1/2, ε r is the relative dielectric constant of substrate layer, m, n, p is the eigen number, and a, b, h are the three demension size separatelly. Figure 1 Simple Power-GND structure, in Sigrity PowerSI As h value is tiny, the phase change along Z direction can be ignored by compared with the resonace frequency along X and Y direction, so that Formula (1) (1)
can be simplified as (2): 300 mπ nπ f (MHz) = ( ) + ( ) 2π ε r a b 2 2 In above formula,the units of value a and b are meter.resonance frequency depends on the size of Power and GND planes. Within the frequency range of 0 Hz~5 GHz, seven resonance (2) frequency can be calculated from formula (2). In Sigrity PowerSI tool, model is set up and to run in eigen mode solver, then we can get simulated resonance frequency and the surface voltage distribution. Comparison result of formula calculation and simulated are showed in Table 1. Simulation and formula calculation show a pretty good Table 1 Comparison, resonance of example, formula calculation Vs. simulation Eigen number m, n Formula calculation, resonance frequency /GHz Simulation, resonance frequency /GHz Surface voltage distribution 1 (0,1) 1.46 1.44 2 (1,0) 2.44 2.41 3 (1,1) 2.85 2.81 4 (0,2) 2.93 2.9 5 (1,2) 3.81 3.78 6 (0,3) 4.39 4.35 7 (2,0) 4.88 4.84 correlation. The voltage distribution shows another important VNA 2-ports measurement. Accordingly, simulation tools provide information, the resonance hot spot, which is the peak points in S/Z solver mode, by adding single port between the studied power Table 1. Eigen mode solves all the potential resonance frequency and ground planes, S 11 or Z 11 parameter of the defined port is points of power and ground planes. As long as the source is located numerically calculated. Some related concerns include: at the hot spot location, maximum resonance will be excited, (1) How to get the resonance frequency points from S or Z provided that resonance mode is not changed by the excitation itself, parameter? otherwise, resonance will be weaker or never happened. (2) How to eliminate the impact caused by port setting, to get Base on the resonance voltage distribution, if the source is right simulation results? excited at Point A (the diagonal center in Figure 1), mode 4 (f= (3) How to understand resonance from signal energy transmission 2.9 GHz) and mode 7 (f=4.84 GHz) resonance happen, meanwhile, point of view? For example, what's the relationship between other frequency resonance are not obvious or excited. If the resonance of Port A and Port B, in Figure 1? source is excited at Point B (the diagonal end in Figure 1), all the Z 11 parameter of single-end port is defined as the ratio of port resonance mode will be got, then mode 3 (f=2.81 GHz) and mode 5 voltage and current excited by ideal current source, as shown in (f=3.87 GHz) present the strongest level. In other words, the actual Figure3. resonance mode and level depend on source location. The other case is excitation source changes the resonance distribution, since eigen mode of Sigrity PowerSI is some kind of passive simulation. To introduce this impact into simulation, the source impedance should be modeled by one small value resistor, which isadded at the exact source position. Figure 3 Z 11 definition, single-end port S or Z parameter measurement for passive PCB, is usually used to confirm resonance frequency. One of the most polular methods is U1 mag ( Z11) = (3) I 1
Self-impedance of power delivery net is defined as the frequency depended impedance observed from sink IC power supply input pins to the whole PDN net, which is the Z 11 parameter got by adding port between power-gnd pairs at this input pin. According to the surface voltage distribution of eigen mode simulation, the port voltage amplitude will be maximum at one frequency as long as resonance happens at this frequency. Base on formula (3), Z 11 at this frequency point will present highest spike accordingly. So the frequency with maximum Z 11 amplitude is just the resonance frequency point. Ideal current source substitutes for sink IC. Z 11 is the inherent character of PDN, which one depends on VRM, decoupling capacitor, Power-GND pairs and sink IC package parameters, and it's not related with the port reference impedance in measurement or simulation, as showed in Figure 4. (a) S 11 for different port reference impedance Figure 4 Components of Z 11, PDN self-impedance S 11 indicates the reflection loss at defined port: Z magg( S11) = Z Z + Z 11 0 11 0 In above formula, Z 0 is defined as the port reference impedance. Under total reflection condition, S 11 =1, stationary wave is formed as long as resonance happens, the reflection at port presents minimum value, hence the frequency with minimum S 11 amplitude is just the resonance frequency point. According to formula (4), S 11 depends on the port reference impedance, different port impedance causes different S 11, in detail, the frequency points and amplitude level of minimum S 11 are both different. The frequency points of minimum S 11 amplitude level match with the frequency points of maximum Z 11 amplitude level, only happens under the precondition of Z 0 Z 11, which "o" means the port reference impedance is close to open status of ideal current source. Assuming one port is located at Point A in Figure 1, the Z 11 and S 11 under different port reference impedance (0.1 Ω, 50 Ω and 1 000 Ω) are showed in Figure 5. It's easy to see that Z 11 parameter is not impacted by different simluated port reference impedance setting. So, it's recommended to get resonance frequency from Z 11. The frequency points with maximum Z 11 amplitude are the resonance frequency. (4) (b) Z 11 for different port reference impedance Figure 5 Comparison of S 11 and Z 11, different port reference impedance, Port A Assuming the other port is excited at point B in Figure 1, between Power and GND planes, Z 11 is showed as following Figure 6. Figure 6 Z 11, 0 Hz~5 GHz, Port B In total,there are clearly 5 maximum Z 11 amplitude points in Figure 6, all the frequency points can be found in Table 1 accordingly. However,two resonance frequency in Table 1, 2.9 GHz and 4.84 GHz, are not obvious in Figure 6, since these 2 points are so close to strong resonance frequency at 2.81 GHz and 5.01 GHz, the strong resonance mode pull Z 11 floor up, so that Z 11 spike at 2.9 GHz and 4.84 GHz are covered by the increased floor. In addition, previous eigen mode simulation ever provides one conclusion: If the source is excited at point B (the diagonal end in
Figure 1), all the resonance mode will be got, meanwhile, mode 3 (f=2.81 GHz) and mode 5 (f=3.87 GHz) present the strongest level. However, in Figure 6, although Z 11 amplitude at 2.81 GHz and 3.78 GHz are higher than that ones at 1.44 GHz, 2.41 GHz and 4.35 GHz as expected, one surprise happens at 4.84 GHz, the Z 11 amplitude is obviously higher than that one at 3.78 GHz, this phenomenon is also related with the Z 11 floor increased around 4.84 GHz, which one is pulled up by near strong resonance. It follows that some resonance frequency points would be missing and the strongest resonance point would be estimated unproperly, because Z 11 floor is pulled up by strong resonance mode. There is limitation in utilizing Z 11 to estimate resonance frequency and amplitude. This matter should be highlighted. Z parameter simulation shows pretty good correlation with VNA 2-port measurement in resonance analysis.however, this kind of simulation is still passive simulation, for the case of source impact on resonance, the source impedance should be modeled by one small value resistor, which is manually added at the exact source position. Energy transmission theory provides explanation of the fixed relationship between resonance at Port A and B. Energy spreads from the excitation to outside, and reflection happens as long as the incident wave reaches at the PCB edge, because of the restrain of PMC boundary. Reflection wave stacks up with incident wave, which forms nodes as they are opposite phase and forms loop as they are in-phase. It's so called resonance. For the consideration of Port B, which one is at the end of diagonal, the simplest resonance mode is generated by the energy reflection along long edge and short edge, and the relationship between resonance wavelength and PCB size follows formula (5) 2a=λ 1 and 2b= λ 2 (5) In formula (5), λ 12, = c ε f r 12,, which is the signal wave length in PCB substrate. The resonance modes got from (5) actually are the same resonance mode 1 and 2 in formula (2), namely, m=0, n=1 and m=1, n=0. For the consideration of Port A, which is at the center of diagonal, the simplest resonance mode as followed, a=λ 1 and b=λ 2 (6) Base on formula (5) and (6), it's easy to find that if source is excited at the diagonal center, the first 2 resonance frequency points are 2.9 GHz and 4.84 GHz, if source is excited at the diagonal end, they are 1.44 GHz and 2.41 GHz. The former is exactly twice the latter. Excitation Source Impact In the real power on case, PCB is active and IC acts as excitation source. The actual excitation source is frequency depended. Simulation tools normally support frequency depended source, however, for simplified purpose, frequency un-depended ideal voltage source and current source are usually adopted as approximate simulation. Power and GND planes resonance is releated with pairs structure and the components added between pairs, so the excitation source impedance will impact the equivalent parasitic parameter (R/L/C) between Power-GND pairs, then change resonance frequency and amplitude level. Proper excitation source model is substitution for active IC in simulation, to achieve proper simulation results. Voltage regulator module and sink IC are both active components and belong to PDN net. How to properly set up excitation model for sink IC? According to the defination of PDN self-impedance Z 11, current source is substitution for sink IC. As an example, Point A in Figure 1 is assumed as sink postion, ideal voltage and current source are separately added as excitation here, run this model in spatial mode solver, the resonance frquency and voltage distribution within 1 Hz~5 GHz as shown in Table 2. Table 2 Example in Figure 1, resonance simulation, excitation at sink (Point A), ideal voltage source Vs. ideal current source Excitation Resonance mode 1 Resonance mode 2 Ideal voltage source, 1 V (resonance frequency: 3.1 GHz) Ideal current source, 1 A (resonance frequency: 2.9 GHz and 4.8 GHz) None As shown, ideal voltage source substitution changes the actual pairs resonance frequency because of the short circuit character of tiny source impedance, on the contrary, ideal current source substitution matches with actual sink IC performance as the open circuit character of huge source impedance does not change pairs resonance frequency. Things are different for VRM modeling. Voltage source is substitution for VRM because of the tiny source impedance of VRM. As an example, assuming sink IC is located at Point A in Figure 1 and modeled as ideal current source, VRM is located at Point C and modeled as voltage source with 0.01 Ω source impedance. Run
eigen mode *, Z parameter mode and spatial mode (adding voltage probe at sink location to measure the voltage-frequency curve) solver in Sigrity PowerSI separatelly. Results shown in Figure 7~Figure 9. correlation with PDN self-impedance: all the spike of impedance curve are included in the resonance frequency points got from eigen mode simulation. Figure 9 shows one clear conclusion: the resonance excited by VRM (voltage source substitution) and sink (current source substitution) shows pretty good correlation with eigen mode and PDN self-impedance curve. Accordingly, near filed scanning measurement of active poweron PCB provides the surface E or H field distribution, which one shows the resonance frequency points and hot spot excited by actual source. Summary Figure 7 Eigen mode, resonance frequency Figure 8 Z 11 parameter at point A Figure 9 Spatial mode, voltage-frequency curve at A Comparing Figure 7 with Table 1, it's easy to find that VRM source impedance not only impact IR drop performance at low frequency, but change the self-impedance character of PDN, then finally impact the resonance freuqency of Power-GND pairs. From Figure 7 and Figure 8, by taking VRM impedance impact into consideration, the resonance frequency points still show Signal energy spreadsfrom VRM to outside along Power-GND pairs, reflection wave stacks up with incident waveand resonance is formed. Resonance character depends on the pairs structure, VRM source impedance and component added on pairs. Both eigenmode and PDN self-impedance curve provide resonance frequency solver. Eigen mode calculates all the possible resonance frequency, and the actual excited resonance frequency depends on the sink position. The spikes frequency of impedance curve at sink position show the actual excited resonance frequency points, which ones are all included in the resonance frequency got from eigen mode solver. Higher spike level in self-impedance curve, higher resonance amplitude, however, this conclusion gets limitation as sometimes the strong resonance pull up Z 11 floor and makes things complicated. Eigen mode and Z parameter sovler is passive simulation. The actual powered on PCB is active, if the excitation source does impact the resonance performance, manually added small resistor at excitation position is right substitution for the real source impedance, otherwize, wrong simulation results would be got. All the resoance mode calculated by spatial solver are also included in the resoance frequency got from eigen mode, and they match with the spike ones in self-impedance curve. Voltage source is substitution for VRM, and current source is substitution for sink IC, wrong simulation results would be got if VRM and sink adopted opposite source model. [1] Eric Bogatin. Signal and Power Integrity - simplified (second edition)[m]. San Antonio: Pearson Education, Inc., 2010. * As previous discussion, the excitaion is auto ignored in eigen mode and parameter mode solver, so that one manually added small resistor is substitution for voltage source impedance, to model the impact of VRM impedance on resonance. In this example, one 0.01 Ω resistor is added between the pairs at Point C