Designing effective low-pass and high-pass filters is a fundamental task in radio frequency (RF) and microwave engineering. These filters selectively pass signals below or above a specified cutoff frequency while attenuating others, making them essential for signal conditioning, frequency selection, and impedance matching. Among the various design approaches, the Smith chart stands out as a graphical tool that simplifies the complex impedance calculations inherent in filter synthesis. By visualizing impedance and reflection coefficient transformations, engineers can design filters with precision and intuition. This article explores the theory and practice of designing low-pass and high-pass filters using Smith chart techniques, providing a detailed workflow from initial specifications to component selection and verification.

The Smith Chart: A Foundation for Filter Design

The Smith chart is a polar plot of the complex reflection coefficient Γ, overlaid with constant resistance and constant reactance circles. Originally developed by Phillip H. Smith in the 1930s, it remains a cornerstone of RF design because it transforms complicated impedance equations into simple geometric operations. A normalized impedance z = Z / Z₀ (where Z₀ is the system characteristic impedance, typically 50 Ω) is mapped onto the chart as a point. The chart’s real axis represents pure resistance, while the arcs above and below correspond to inductive (positive) and capacitive (negative) reactance, respectively. Its dual—the admittance chart—uses the same circles but reinterprets them as conductance and susceptance, which is equally valuable when dealing with parallel components.

For filter design, the Smith chart allows engineers to determine the required reactive elements (capacitors and inductors) that transform an input impedance to an output impedance while achieving the desired frequency response. By plotting the normalized impedance at the cutoff frequency and then following constant resistance or constant conductance circles, one can deduce series or shunt element values. This graphical approach eliminates the need for solving transcendental equations and provides immediate insight into bandwidth and matching trade-offs. Modern simulation software often embeds the Smith chart as a visualization tool, but understanding the manual technique fosters deeper intuition for parametric adjustments and parasitic compensation.

Key properties for filter design:

  • Constant resistance circles – horizontal lines? Actually, constant resistance appears as circles tangential to the rightmost point. These are useful when adding series reactance.
  • Constant reactance arcs – these intersect the real axis at specific points and indicate the impedance trajectory as frequency changes.
  • Admittance transformation – rotating a point by 180° around the center converts impedance to admittance, which is helpful when analyzing shunt elements.

To leverage these properties, designers must become fluent in reading the chart: moving along a constant resistance circle corresponds to adding series inductance (clockwise) or capacitance (counterclockwise), while moving along a constant conductance circle corresponds to adding shunt inductance or capacitance. These movements form the basis for synthesizing low-pass and high-pass filter networks.

Designing Low-Pass Filters with the Smith Chart

A low-pass filter (LPF) passes frequencies from DC up to a cutoff frequency fc and attenuates signals above it. The ideal response has zero insertion loss in the passband and infinite rejection in the stopband, but practical filters achieve a gradual roll-off determined by the number of reactive elements (order). Common prototype configurations include Butterworth (maximally flat passband), Chebyshev (sharper cutoff with ripple), and Bessel (linear phase). The Smith chart aids in translating these prototypes into exact component values for a given impedance level.

Step-by-Step LPF Design Procedure

Step 1: Define specifications. Choose the cutoff frequency fc, system impedance Z₀ (e.g., 50 Ω), filter order (e.g., third-order), and prototype type (e.g., 0.5 dB ripple Chebyshev). For a low-pass filter, the normalized element values (g-parameters) for the chosen prototype are available from standard tables. For example, a third-order Butterworth LPF with Z₀ = 50 Ω uses g₁ = 1 H, g₂ = 2 F, g₃ = 1 H (scaled).

Step 2: Normalize and plot the impedance at cutoff. At the cutoff frequency, the filter’s input impedance is purely real and equal to Z₀. Therefore, the normalized impedance z = 1 + j0 is the starting point at the center of the Smith chart.

Step 3: Synthesize the components via chart trajectories. For a series-first, shunt-second topology (typical for LPF), the design proceeds as follows:

  • First element (series inductor): Adding a series inductor moves the impedance point along a constant resistance circle clockwise (increasing inductive reactance). The required normalized reactance xL = g₁ (normalized inductance) = 1 for Butterworth. On the chart, locate the constant resistance circle for r=1, then move along it from the center (z=1) to the point where the normalized reactance equals 1. This is the intersection of the r=1 circle with the x=1 arc. The new impedance is z = 1 + j1.
  • Second element (shunt capacitor): It is easier to work in admittance when adding shunt elements. Rotate the impedance point 180° around the center to obtain the corresponding admittance y = 0.5 - j0.5. Adding a shunt capacitor increases the susceptance upward (positive imaginary part). The required normalized susceptance bC = g₂ = 2. On the admittance Smith chart (or using the same chart with admittance interpretation), move from the admittance point upward along a constant conductance circle (g=0.5) until the total susceptance becomes y = 0.5 + j1.5. This corresponds to the net shunt capacitive susceptance of 2.
  • Third element (series inductor): Rotate back to impedance: the new admittance point (0.5 + j1.5) gives an impedance of z = 0.2 - j0.6. Adding the final series inductor moves along the constant resistance circle r=0.2 clockwise. The required normalized reactance is g₃ = 1. The movement ends at z = 1 + j0 (center), confirming that the network is perfectly matched at the cutoff frequency.

Step 4: Denormalize components. Convert the normalized reactance and susceptance to physical values using the cutoff frequency:

  • Inductance: L = (xL * Z₀) / (2π fc)
  • Capacitance: C = bC / (2π fc * Z₀)

For fc = 1 GHz and Z₀ = 50 Ω, the first inductor becomes 7.96 nH, the shunt capacitor 6.37 pF, and the second inductor also 7.96 nH.

Verification and Tuning

After constructing the filter, plot the impedance at several frequencies across the passband and stopband on the Smith chart. At low frequencies (well below fc), the impedance should remain near 50 Ω (center). At high frequencies, the impedance moves toward the edge of the chart (high reflection), indicating effective rejection. If the passband return loss is insufficient, slight adjustments to component values can be guided by the chart—for example, adding a small series inductance to compensate for parasitic capacitance. Practical RF components also have self-resonant frequencies that must be considered; the Smith chart can help visualize these parasitics when measured data is overlayed.

Alternative Topologies and Prototypes

The same procedure applies to Chebyshev and elliptic prototypes, but the normalized reactance and susceptance values are different. For a Chebyshev LPF with 0.1 dB ripple, g₁ = 0.843, g₂ = 0.622, g₃ = 1.355. The Smith chart trajectories will end at the same center only if the order is odd; for even orders, the filter may require an impedance transformer at one port. The chart can also be used to design π- and T-sections by swapping series and shunt element sequences. In all cases, the graphical nature of the Smith chart reveals how the impedance spiral converges to the matched condition.

Designing High-Pass Filters with the Smith Chart

High-pass filters (HPF) pass frequencies above the cutoff and attenuate low frequencies. They are duals of low-pass filters: series capacitors replace series inductors, and shunt inductors replace shunt capacitors. The design procedure is analogous, but the impedance movements on the Smith chart are in the opposite direction because capacitive reactance decreases with frequency (negative imaginary part moving counterclockwise for series elements) and inductive susceptance decreases with frequency (negative imaginary part moving downward on the admittance chart for shunt elements).

Step-by-Step HPF Design Procedure

Step 1: Obtain normalized low-pass prototype. Use the same g-values from a low-pass prototype (e.g., third-order Butterworth: g₁=1, g₂=2, g₃=1). For a high-pass filter, the normalized values are converted via the frequency transformation s → 1/s. The equivalent high-pass element values become: series capacitance with normalized susceptance = 1/g₁, shunt inductance with normalized reactance = 1/g₂, and series capacitance again with 1/g₃.

Step 2: Plot the impedance at cutoff. At cutoff, the high-pass filter also presents a matched impedance of Z₀, so the starting point is again the center of the Smith chart with z=1.

Step 3: Synthesize components via chart trajectories (moving counterclockwise for series capacitance, downward for shunt inductance).

  • First element (series capacitor): A series capacitor adds capacitive (negative) reactance. From the center, move along the constant resistance circle r=1 counterclockwise. The required normalized reactance xC = -1/g₁ = -1. The new impedance is z = 1 - j1.
  • Second element (shunt inductor): Rotate to admittance: y = 0.5 + j0.5. Adding a shunt inductor increases inductive susceptance, which is negative on the admittance chart. The required normalized susceptance bL = -1/g₂ = -0.5. Move downward along the constant conductance circle g=0.5 until the total susceptance becomes y = 0.5 - j0.0? Actually, starting from y=0.5+j0.5, we need to add b=-0.5, so new y = 0.5 + j0.0. This is a pure conductance of 0.5.
  • Third element (series capacitor): Rotate back to impedance: the admittance 0.5 + j0 gives impedance z = 2 + j0. Adding the final series capacitor requires moving counterclockwise along constant resistance r=2. The required normalized reactance xC = -1/g₃ = -1. The movement from z=2 to z=2 - j1. For the impedance to return to the center (z=1), the third element must also involve an impedance transformation? Wait, careful: In a high-pass filter, the third element is also a series capacitor, but after two elements the impedance is 2 - j1. Adding another series capacitor of x=-1 moves to z=2 - j2. That does not reach the center. The issue is that the topology and sequence matter: a standard T-network high-pass filter uses series capacitors at the input and output and a shunt inductor in the middle. However, the impedance at the center after the final element should be 1+j0 if the filter is symmetric. In this example, the denormalized values may need scaling or the prototype may require a different order. Actually, for a third-order high-pass, the network is not directly obtained by substituting capacitors for inductors in the low-pass ladder; the element values must be recalculated using the frequency transformation. Using the transformation s → 1/s yields a network where the shunt inductor is in parallel with the load, and the series capacitors are at input and output. The Smith chart approach must account for the fact that after the shunt inductor, the impedance may not be 1+j0; the final series capacitor brings it back. Let's correct the trajectory:

Corrected HPF synthesis (third-order Butterworth): Normalized high-pass values: C₁ = 1 F, L₂ = 0.5 H, C₃ = 1 F. At cutoff, the filter is matched: impedance at all nodes? Actually, the standard table for high-pass gives element values directly. For a 50 Ω, 1 GHz high-pass: C₁ = 1/(2π×1e9×50×g₁) = 3.18 pF, L₂ = (50×g₂)/(2π×1e9) = 7.96 nH (since g₂=1 for the transformed value? Wait, confusion arises. Let's use the standard low-pass to high-pass transformation: For a low-pass series inductor L, in high-pass it becomes a series capacitor C = 1/(ω_c² L). More practically, designers often use the normalized low-pass g-values and then apply the transformation: series branch elements become capacitors with normalized impedance 1/g, shunt branch elements become inductors with normalized admittance 1/g. So for g₁=1, series capacitor has normalized reactance -1/1 = -1. For g₂=2, shunt inductor has normalized reactance 1/2 = 0.5 (but admittance perspective). The Smith chart procedure works if we carefully track impedance and admittance conversions.

Simplified approach: Use the admittance Smith chart for shunt elements. Starting at center impedance z=1 (admittance y=1). For a high-pass T-network:

  1. Series capacitor: move along constant resistance r=1 counterclockwise by x=-1. Arrive at z=1-j1 (y=0.5+j0.5).
  2. Shunt inductor: rotate to admittance space (already y=0.5+j0.5). To add shunt inductor, we need to add positive inductive susceptance? Inductive susceptance is negative (since Y = jωC for capacitor, but for inductor Y = 1/jωL = -j/(ωL)). So adding an inductor in shunt reduces the total susceptance (makes it more negative). The required normalized susceptance for the shunt inductor is b_L = -1/g₂ = -0.5 (since g₂=2). Starting from y=0.5+j0.5, subtract 0.5 from the imaginary part: new y = 0.5 + j0.0. This is a point on the zero reactance line.
  3. Rotate back to impedance: y=0.5+j0.0 gives z=2+j0. Now add the final series capacitor: move counterclockwise along r=2 by x=-1. Arrive at z=2 - j1 (y=0.4+j0.2). This does not match to the center. The reason is that the high-pass T-network is not symmetric in the sense of impedance after each element; the final series capacitor must be chosen such that the overall impedance at the input is Z₀. In practice, the element values are pre-calculated from prototype tables, not derived stepwise on the chart. The chart is more useful for impedance matching after the filter is designed, or for verifying the response at various frequencies. Therefore, a more practical use of the Smith chart for high-pass filters is to plot the frequency sweep and adjust components to achieve the desired cutoff and return loss. Alternatively, one can design the high-pass filter as the impedance complement of a low-pass filter using the Z-to-admittance transformation, but that is beyond this scope.

Practical design tip: Instead of stepwise synthesis, designers often start with a known low-pass prototype and convert using the chart: plot the low-pass impedance trajectory at the cutoff frequency, then invert it with respect to the center (rotate 180°) to obtain the high-pass response. This graphical dual transformation is less common but illustrates the symmetry. For most practical work, engineers use software tools that implement these transformations, but understanding the underlying chart movements helps in debugging prototype measurements.

Verification and Component Selection

For a high-pass filter built with surface-mount components, the Smith chart can display the measured S-parameters. At frequencies well above cutoff, the impedance should cluster near the center (matched). As frequency decreases toward cutoff, the impedance spirals outward toward the open or short circuit, depending on the topology. At DC, a high-pass filter blocks all signals, so the input impedance becomes an open circuit (infinite VSWR) reflected as the rightmost point on the Smith chart. By overlaying measured data on the chart, engineers can identify parasitic resonances or insufficient attenuation and then add compensating elements.

Advanced Techniques and Practical Considerations

Beyond basic synthesis, the Smith chart facilitates several advanced filter design tasks. For instance, when designing multi-stage filters (e.g., fifth-order), the chart helps visualize the impedance transformation across each section. Each stage brings the impedance closer to the center at the cutoff frequency, but away from it at other frequencies. By plotting the loci for several frequencies, the designer can verify that the combined response meets the rejection requirements.

Matching networks integrated with filters: Often a filter must also match the input impedance to a non-50 Ω source or load. The Smith chart can incorporate a matching section (e.g., an L-network or stub) directly into the filter by adjusting the first or last element values. For example, a low-pass filter with a 75 Ω input can be designed by starting the chart with a normalized impedance of z=1.5 (for 75 Ω reference) and then following the same trajectories but scaling components accordingly. The final element restores the impedance to 75 Ω instead of 50 Ω.

Dealing with parasitic effects: Real inductors have self-resonant frequencies and equivalent series resistance (ESR), while capacitors have equivalent series inductance (ESL). These parasitics add spurious reactance that shifts the impedance on the Smith chart. A common technique is to measure the component’s impedance at several frequencies and plot it on the chart, then adjust the nominal values to compensate. For instance, if a capacitor intended for a high-pass filter shows inductive behavior above self-resonance, the filter may become low-pass at high frequencies. The chart immediately reveals such frequency-dependent anomalies.

Comparison with other design methods: While Smith chart design is graphical and intuitive, it is limited to moderate complexity (up to about fifth-order) due to manual effort. For higher orders, software tools like ADS, HFSS, or MATLAB provide optimization. However, the chart remains especially useful for quick prototyping, educational purposes, and when working with legacy equipment or hand calculations. It also serves as a powerful diagnostic tool when analyzing measured filter data.

For further reading on Smith chart fundamentals, refer to the Microwaves101 guide to Smith chart basics. A detailed tutorial on filter design using the chart is available from All About Circuits, which provides step-by-step examples with actual component values. For those looking for a comprehensive reference on RF filter synthesis, the classic text by Matthaei, Young, and Jones is invaluable, but a practical online resource is the RF Cafe Smith Chart page, which includes downloadable chart templates and calculator tools.

Conclusion

The Smith chart remains an enduring tool for designing low-pass and high-pass filters in RF engineering. Its graphical approach transforms tedious impedance calculations into straightforward geometric constructions, enabling engineers to quickly synthesize and verify filter networks. By mastering the technique of moving along constant resistance and constant conductance circles, one can design Butterworth, Chebyshev, and other prototype filters with confidence. Moreover, the Smith chart serves as a powerful verification and troubleshooting aid when working with real components that include parasitics. While modern simulation software has automated many of these steps, a solid grasp of Smith chart techniques provides deeper insight into the underlying principles of impedance and frequency response. Whether used for initial design or final tweaking, the Smith chart remains a cornerstone of practical RF filter engineering.