What is the difference between a Coplanar Waveguide (CPW) and a Microstrip line?

CPW features a structure where ground planes are placed on the same substrate surface on both sides of the signal line, offering advantages such as no need for vias, low dispersion, and ease of probe measurement. Microstrip lines have a ground plane on the bottom layer, requiring via connections.

Which simulation methods are suitable for CPW electromagnetic field analysis?

For 3D full-wave analysis, FEM (e.g., HFSS) or FDTD (e.g., CST) are appropriate. For planar structures, Method of Moments (MoM) tools like Momentum or Sonnet are efficient. The choice between frequency-domain and time-domain methods depends on the application.

What are common causes for discrepancies between CPW simulation and measurement results?

Primary causes include improper de-embedding, unaccounted frequency-dependent substrate permittivity, unmodeled surface roughness, and the launch pad effects of probes.

Electromagnetic Field Simulation of Coplanar Waveguides (CPW)

Q: How is the characteristic impedance of a CPW calculated?

It is calculated using the formula Z₀ = (30π/√εeff) × K(k')/K(k), which employs the complete elliptic integral of the first kind, K(k). Here, k is defined by the signal line width (w) and gap (s) as k = w/(w+2s).

Category: Electromagnetic Field Analysis > High Frequency | Consolidated Edition 2026-04-11

Coplanar waveguide cross-section showing signal line, ground planes, and electric field distribution on a dielectric substrate

Cross-sectional structure and electric field distribution of a Coplanar Waveguide (CPW). Ground planes are placed on both sides of the signal line (center), and electromagnetic waves propagate in a quasi-TEM mode.

Theory and Physics

Basic Structure of CPW

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What's the difference between a coplanar waveguide and a microstrip? They're both transmission lines on a substrate, right?

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The major difference is the location of the ground plane. Microstrip has a ground plane on the backside of the substrate, but CPW places the ground planes on both sides of the signal line on the same plane. Looking at the cross-section, it has a G-S-G (Ground-Signal-Ground) structure.

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What are the advantages of having the ground on the same plane?

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There are three practical benefits.

No vias required — Since the ground is on the surface, connection vias to the backside are unnecessary. This simplifies component mounting in MMICs and RFICs.
Easy probe measurement — G-S-G probes can be placed directly on it, making it the standard structure for on-wafer measurements.
Low dispersion — It propagates in a quasi-TEM mode, resulting in less frequency dispersion compared to microstrip, making it suitable for broadband design.

Since its invention by C.P. Wen in 1969, it has become the de facto standard structure in compound semiconductor circuits like GaAs or InP. Recently, it's also widely used in 5G millimeter-wave front-ends in the 28/39 GHz bands.

The parameters defining the cross-sectional structure of a CPW are as follows.

Parameter	Symbol	Description	Typical Value
Signal Line Width	$w$	Width of the center conductor	10–500 μm
Gap Width	$s$	Spacing between signal line and ground	5–200 μm
Substrate Thickness	$h$	Thickness of the dielectric substrate	100–635 μm
Conductor Thickness	$t$	Metallization thickness	0.5–5 μm
Relative Permittivity	$\varepsilon_r$	Permittivity of the substrate	2.2 (PTFE) – 12.9 (GaAs)

Characteristic Impedance and Elliptic Integrals

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Is the characteristic impedance of a CPW calculated using empirical formulas like for microstrip?

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No, for CPW there is an analytical formula using the complete elliptic integral of the first kind, $K(k)$. It's derived from the conformal mapping method.

The characteristic impedance of a CPW on an infinitely thick substrate (no backside ground) is given by the following equation.

$$ Z_0 = \frac{30\pi}{\sqrt{\varepsilon_{\text{eff}}}} \cdot \frac{K(k')}{K(k)} $$

Here, each variable is defined as follows.

$$ k = \frac{w}{w + 2s}, \qquad k' = \sqrt{1 - k^2} $$

$w$: Signal line width, $s$: Gap width
$K(k)$: Complete elliptic integral of the first kind $\displaystyle K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - k^2 \sin^2\theta}}$
$k'$: Complementary parameter of $k$

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I've never seen elliptic integrals before... Do you actually calculate this by hand for every design?

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No, we don't calculate by hand. There are convenient approximation formulas for the ratio $K(k)/K(k')$, which are often used in practice.

Approximation for $K(k)/K(k')$ (Hilberg approximation):

$$ \frac{K(k)}{K(k')} \approx \begin{cases} \displaystyle \frac{1}{\pi} \ln\!\left(2\,\frac{1+\sqrt{k}}{1-\sqrt{k}}\right) & (0.707 \leq k \leq 1) \\[10pt] \displaystyle \frac{\pi}{\displaystyle \ln\!\left(2\,\frac{1+\sqrt{k'}}{1-\sqrt{k'}}\right)} & (0 \leq k \leq 0.707) \end{cases} $$

The error of this approximation is below 0.01% across the entire range, providing sufficient accuracy even with calculator-level computation.

Derivation Background: Conformal Mapping Method

The CPW characteristic impedance formula is derived using a conformal mapping called the Schwarz-Christoffel transformation. By mapping the electrode arrangement of the CPW cross-section (z-plane) to a parallel plate capacitor (w-plane), the capacitance per unit length $C$ can be determined analytically. The characteristic impedance is obtained from the relation $Z_0 = 1/(v_p \cdot C)$. Conformal mapping is applicable because CPW is a quasi-TEM structure, allowing the potential distribution to be described by the 2D Laplace equation.

Effective Permittivity

Since the electric field of a CPW is distributed both within the substrate and in the air, the effective permittivity is a weighted average of the substrate's relative permittivity $\varepsilon_r$ and air ($\varepsilon_r = 1$).

$$ \varepsilon_{\text{eff}} = 1 + \frac{\varepsilon_r - 1}{2} \cdot \frac{K(k')}{K(k)} \cdot \frac{K(k_1)}{K(k_1')} $$

Here, $k_1$ is a modified parameter that accounts for the substrate thickness $h$.

$$ k_1 = \frac{\sinh\!\left(\dfrac{\pi w}{4h}\right)}{\sinh\!\left(\dfrac{\pi(w+2s)}{4h}\right)}, \qquad k_1' = \sqrt{1 - k_1^2} $$

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Does the effective permittivity change if the substrate is thin?

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Yes, it changes. When the substrate becomes thinner, more of the electric field leaks into the air side, so $\varepsilon_{\text{eff}}$ moves away from $\varepsilon_r$ and approaches 1. Conversely, if the substrate is sufficiently thick ($h \gg w + 2s$), then $k_1 \to k$, converging to the infinite-thickness substrate approximation $\varepsilon_{\text{eff}} \approx (\varepsilon_r + 1)/2$. For example, on an alumina substrate ($\varepsilon_r = 9.8$) with $h = 635\,\mu\text{m}$, $w = 50\,\mu\text{m}$, $\varepsilon_{\text{eff}}$ becomes about 5.4.

Loss Mechanisms

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What factors determine CPW loss? It seems significant in millimeter-wave, right?

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CPW loss can be broken down into three factors. All become more significant as frequency increases, so none can be ignored in millimeter-wave.

The total loss $\alpha_{\text{total}}$ is expressed as the sum of the following three components.

$$ \alpha_{\text{total}} = \alpha_c + \alpha_d + \alpha_r \quad [\text{dB/mm}] $$

1. Conductor Loss $\alpha_c$

Due to the skin effect, high-frequency currents concentrate on the conductor surface, increasing the effective resistance. The skin depth $\delta_s$ is given by:

$$ \delta_s = \sqrt{\frac{2}{\omega \mu_0 \sigma}} = \sqrt{\frac{1}{\pi f \mu_0 \sigma}} $$

For gold (Au), with $\sigma = 4.1 \times 10^7\,\text{S/m}$, $\delta_s \approx 0.79\,\mu\text{m}$ at 10 GHz, and becomes as thin as $\delta_s \approx 0.28\,\mu\text{m}$ at 77 GHz. The conductor thickness $t$ needs to be at least $3\delta_s$.

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Does surface roughness also have an effect?

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It has a significant effect. When the conductor surface roughness (RMS value $R_q$) is comparable to the skin depth, the current path lengthens, increasing resistance. The Hammerstad-Jensen model applies a correction factor $K_{sr} = 1 + \frac{2}{\pi}\arctan\!\left(1.4\left(\frac{R_q}{\delta_s}\right)^2\right)$ to the conductor loss. In the mmWave band, a difference of 0.1μm in $R_q$ can change insertion loss by 0.5 dB/cm.

2. Dielectric Loss $\alpha_d$

$$ \alpha_d = \frac{\pi}{\lambda_0} \cdot \frac{\varepsilon_r}{\sqrt{\varepsilon_{\text{eff}}}} \cdot \frac{\varepsilon_{\text{eff}} - 1}{\varepsilon_r - 1} \cdot \tan\delta $$

$\tan\delta$ is the substrate's loss tangent. Selection of low-loss substrates is crucial; representative values are shown below.

Substrate Material	$\varepsilon_r$	$\tan\delta$ (@10 GHz)	Application
Fused Silica	3.78	0.0001	High-precision filters
Alumina (Al₂O₃)	9.8	0.0003	MMIC substrate
GaAs	12.9	0.0006	RFIC
Rogers RO4003C	3.55	0.0027	PCB high-frequency circuits
FR-4	4.4	0.02	Low frequency only (unsuitable for GHz band)

3. Radiation Loss $\alpha_r$

Electromagnetic waves radiate from CPW discontinuities (bends, T-junctions, gap mismatches) into the substrate. Especially on substrates with high $\varepsilon_r$, surface wave modes are easily excited, becoming a cause of crosstalk to adjacent circuits. Radiation loss increases proportionally to the square of frequency and can become dominant in the mmWave band.

CPW Variant Structures

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I heard there are various types of CPW. How do you choose which one to use?

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There are three main variant structures. They are chosen based on the design objective.

Structure	Features	Benefits	Applications
CBCPW (Conductor-Backed CPW)	Ground also on substrate backside	Improved heat dissipation, mechanical strength	Power amplifiers, package internal wiring
FGCPW (Finite-Ground CPW)	Ground width limited to finite size	Miniaturization, isolation from adjacent circuits	MMIC, high-density integrated circuits
CPW + Air Bridge	Grounds connected by bridges	Suppression of slot-line mode	T-junctions, bends in MMICs

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Particularly, CBCPW (with backside ground) requires caution in simulation. A parallel-plate mode can be excited between the backside ground and the surface ground, coupling with the intended CPW mode. To prevent this...