A Low-Pass and Band-Pass Filter Using Lumped Elements

Passive devices can be designed using lumped element features if both the operating frequency of the device and the insertion loss of lumped elements are low. This example simulates two types of lumped element filters. First, a 5-element maximally flat low-pass filter is built to compute frequency responses that show the cutoff at the intended frequency. Second, a band-pass filter transformed from the low-pass filter design is simulated in the same frequency range. Both filter models present S-parameters and electric field distribution by default.

Figure 1: 0402 surface-mount device (SMD) inductors and capacitors are modeled using lumped element features on 2D boundaries. The figure describes only a low-pass filter model.

Model Definition

This example shows a filter design with five SMD devices soldered to a microstrip line that sits on a 60 mil substrate with a ground plane (Figure 1). The performance of each SMD device is assumed to be ideal in the simulated frequency range. All metal parts − the microstrip line, the thin copper ground plane and the metalized via holes − are modeled as perfect electric conductors (PECs). Each end of the microstrip line is configured with a lumped port. One of the lumped ports is the source for the input signal.

The geometry of each SMD device is simplified as a 2D boundary and the electrical performance is modeled using a lumped element. Lumped elements are similar to lumped ports except that they are strictly passive and there are predefined choices for inductances and capacitance.

The dielectric constant of the substrate is εr = 3.38 and the remaining volume in the model is air. The device is assumed to sit in open space, and thus the air domain is truncated using the scattering boundary condition to absorb radiation from the circuit.

Low-Pass Filter

The filter in this example consists of inductors connected in series and capacitors connected in parallel. The capacitors are connected to a small section of microstrip line that is, in turn, connected to the metal vias shorted to the ground plane. The element values − either L or C − of the desired filter are defined by

The calculated values are shown in Table 1.

Table 1: Maximally flat low-pass filter element values, N = 5.
g1	g2	g3	g4	g5	g6
0.618	1.618	2	1.618	0.618	1

These values are determined assuming that the reference impedance is 1 Ω and the cutoff frequency of the filter is 1 Hz. However, for the particular filter in the model, the reference impedance is 50 Ω and the cutoff frequency is 1 GHz. The filter element values, L and C, are scaled using the equations below, where R0 is reference impedance and fc is cutoff frequency of the filter.

The resulting scaled element values are L1 = L3 = 4.9179 nH, L2 = 15.915 nH and C1 = C2 = 5.1503 pF.

Band-Pass Filter

The low-pass filter can be converted to a band-pass filter by transforming series inductors (L) and parallel capacitors (C) to a series LC and a parallel LC circuit, respectively. The value of transformed lumped elements are scaled by impedance and frequency based on the element values for the low pass filter prototype (Table 1).

In a series LC circuit,

In a parallel LC circuit,

where fc is used for the center frequency of the filter and Δ is the fractional bandwidth defined as bandwidth/fc.

Results and Discussion

The norm of the electric field underneath the microstrip lines at 2 GHz is plotted in Figure 2. This frequency is outside the passband of the filter and the plot illustrates that the signal is blocked. In Figure 3, the norm of the field is plotted at 0.3 GHz, which is within the passband of the filter. The plot shows that the wave propagates through the filter.

Figure 2: The norm of the electric field at 2 GHz, which is outside the passband of the filter, shows that the energy from the input port cannot reach the output port.

Figure 3: The norm of the electric field is plotted at 0.3 GHz, which is within the passband of the filter. The SMD filter lets the signal pass from the input port to the output port.

The calculated S-parameters, plotted in Figure 4, show a low-pass filter frequency response with 1 GHz cutoff frequency. A small frequency shift is observed due to the parasitic reactance caused by the metal pads and via holes connecting the SMD devices to the ground plane.

Figure 4: The S-parameters are plotted as a function of frequency to illustrate the filter response. The cutoff frequency of the low-pass filter is observed near 1 GHz.

Figure 5: The norm of the electric field at 2 GHz, which is outside the passband of the filter, shows that the energy from the input port cannot reach the output port.

Figure 6: The norm of the electric field is plotted at 1 GHz, which is within the passband of the filter. The SMD filter lets the signal pass from the input port to the output port.

Figure 7: The S-parameter plot illustrates the frequency response of the band-pass filter. The center frequency is observed at 1 GHz.

The S-parameters of the band-pass filter are plotted in Figure 7. Due to the parasitic reactance from the metal pads and via holes, the frequency responses outside the passband are not ideal as those obtained from a circuit theory.

Reference

1. D.M. Pozar, Microwave Engineering, John Wiley & Sons, 1998.

2. R.E. Collin, Foundation of Microwave Engineering, McGraw-Hill, 1992.

RF_Module/Filters/lumped_element_filter

Modeling Instructions

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Study 1

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
thickness	60[mil]	0.001524 m	Substrate thickness
fc	1[GHz]	1E9 Hz	Cutoff frequency
z0	50[ohm]	50 Ω	Reference impedance
l1	z0*0.618/2/pi/fc	4.9179E-9 H	Inductance l1
c1	1.618/z0/2/pi/fc	5.1503E-12 F	Capacitance c1
l2	z0*2/2/pi/fc	1.5915E-8 H	Inductance l2

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Layer name	Thickness (mm)
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Set up the physics. Use lumped ports for the S-parameter analysis and lumped elements to model the SMD devices.

Electromagnetic Waves, Frequency Domain (emw)

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Perfect Electric Conductor 2

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Lumped Element 1

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Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	3.38	1	Basic
Relative permeability	mur_iso ; murii = mur_iso, murij = 0	1	1	Basic
Electrical conductivity	sigma_iso ; sigmaii = sigma_iso, sigmaij = 0	0	S/m	Basic

Mesh 1

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Definitions

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Study 1

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Multislice

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Electric Field (emw)

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S-parameter (emw)

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Smith Plot (emw)

Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
del	500[MHz]/fc	0.5	Fractional bandwidth
l1_inductor	0.618*z0/2/pi/fc/del	9.8358E-9 H	Scaled inductance from l1
l1_capacitor	del/2/pi/fc/0.618/z0	2.5753E-12 F	Scaled capacitance from l1
l2_inductor	2*z0/2/pi/fc/del	3.1831E-8 H	Scaled inductance from l2
l2_capacitor	del/2/pi/fc/2/z0	7.9577E-13 F	Scaled capacitance from l2
c1_inductor	del*z0/2/pi/fc/1.618	2.4591E-9 H	Scaled inductance from c1
c1_capacitor	1.618/2/pi/fc/del/z0	1.0301E-11 F	Scaled capacitance from c1

Electromagnetic Waves, Frequency Domain (emw)

Lumped Element 1

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Lumped Element 2

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Lumped Element 3

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Lumped Element 4

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Lumped Element 5

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Add Study

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Study 2

Step 1: Frequency Domain

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Results

Multislice

Modify the settings to show the plot only on the xy-plane.

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type 0.

Find the subsection. From the list, choose .

In the text field, type -4+thickness/2.

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This reproduces Figure 5 showing the electric field distribution outside the passband.

Electric Field (emw) 1

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Compare the reproduced plot with Figure 6 showing the electric field distribution inside the passband.

S-parameter (emw) 1

The frequency response (S-parameters) is shown in Figure 7.

Smith Plot (emw) 1

Analyze the same bandpass filter model with a much finer frequency resolution using based on asymptotic waveform evaluation (AWE). When a device presents a slowly varying frequency response, the AWE method provides a faster solution time when running the simulation on many frequency points. The following example with the Adaptive Frequency Sweep can be computed about twenty times faster than regular Frequency Domain sweeps with a same finer frequency resolution.

Electromagnetic Waves, Frequency Domain (emw)

Lumped Port 1

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Click

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Lumped Port 2

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Study 3

Step 1: Adaptive Frequency Sweep

In the window for , locate the section.

In the text field, type range(0.2[GHz],0.001[GHz],2[GHz]).

Use a 50 times finer frequency resolution.

A slowly varying scalar value curve works well for AWE expressions. When is set to in the study settings, abs(comp1.emw.S21) is used automatically for two-port devices.

Because such a fine frequency step generates a memory-intensive solution, the model file size will increase tremendously when it is saved. When only the frequency response of port related variables are of interest, it is not necessary to store all of the field solutions. By selecting the check box in the section, we can control the part of the model on which the computed solution is saved. We only add the selection containing these boundaries where the port variables are calculated. The lumped port size is typically very small compared to the entire modeling domain, and the saved file size with the fine frequency step is more or less that of the regular discrete frequency sweep model when only the solutions on the port boundaries are stored.