Investigation of a meander stripline
Abstract Generated abstract
This paper examines the miniaturization of strip transmission delay lines using a meander, or zigzag, central conductor while retaining an approximately linear phase-frequency characteristic. It analyzes phase delay and phase distortion in coupled meander sections, relating them to the ratio of self and mutual capacitances, and then derives the per-unit-length capacitances for even and odd excitation by conformal mapping to a plane-capacitor region. The resulting formulas connect electrical parameters, characteristic impedances, geometry, and dielectric properties, and calculated curves are presented for representative ratios of strip width, spacing, and substrate dimension to support optimization of meander stripline delay lines.
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UDC 621.372.828
INVESTIGATION OF A MEANDER STRIPLINE
N. D. MALYUTIN, P. P. GALINSKII
(Presented by the scientific seminar of the TOE Department)
This article considers ways of miniaturizing strip transmission lines while preserving the linearity of their phase-frequency characteristics, and calculates the per-unit-length capacitances and characteristic impedances of a meander line.
A reduction in the dimensions of strip delay lines can be achieved in two ways: by using substrate materials with large values of relative dielectric and magnetic permeability, \(\varepsilon_r\) and \(\mu_r\), and by employing a zigzag-shaped layout of the central current-carrying strip and multilayer printed assembly. However, increasing \(\varepsilon_r\) and \(\mu_r\) requires reducing the width of the central strip and moving from printed to film technology, which has higher resolving power; and reducing the dimensions by more densely arranging the “turns” of the conductor leads to nonproportionality between the phase delay in the line and the geometric length of the current-carrying strip, as well as to a mismatch of the line in characteristic impedance, as determined for a straight line.
Fig. 1. Construction of a meander stripline: 1 — meander current-carrying strip; 2 — dielectric substrates; 3 — grounded bases
Fig. 2. Cross section of a meander line
The construction of a zigzag-shaped line is shown in Fig. 1, and its cross section in Fig. 2.
If the number of conductors is large, the outer conductors will introduce small errors into the phase shift [1], and the phase delay in the line can be determined as follows:
\[ \varphi = N_1 \vartheta + N_2 \beta l_2, \tag{1} \]
where
\(\psi\) is the difference in the voltage phases at the beginning and end of the bend of one strip;
\(N_1\) is the number of coupled strips;
\(N_2\) is the number of strips connecting the coupled conductors;
\(\beta\) is the phase constant of the TEM wave in an uncoupled line having the same dimensions as the zigzag line;
\(l_2\) is the length of the connecting strips.
The angle \(\psi\) can be found [1] from the equation
\[ \cos \psi = \frac{\sqrt{4m^2(m^2-1)\cos^2 \beta l_1+1}-1} {2(m^2-1)\cos \beta l_1} \tag{2} \]
\(l_1\) is the length of the coupled strips;
where
\[ m^2=\frac{C_0+2C_1}{C_0} \]
depends on the ratio of the self-capacitance \(C_0\) to the mutual capacitance \(C_1\) (Fig. 2), and the phase constant is related to the angle \(\psi\) by the expression
\[ \cos \beta l_1 = \frac{\cos \psi}{\cos^2 \psi + m^2 \sin^2 \psi}. \tag{3} \]
From equations (2) and (3) it is seen that the zigzag stripline structure introduces no phase distortions for values of \(m^2\) close to unity. However, this case corresponds to weak coupling between the strips of the meander and to large overall dimensions of the line. Using (1) and (3), it is easy to establish the relation between \(m^2\) and the permissible deviation \(\Delta \varphi\) of the phase shift of the bent line from the phase shift in a straightened line of the same length,
\[ m^2 = \frac{\cos \psi - \cos^2 \psi \cos(\psi+\Delta \varphi/N_1)} {\sin^2 \psi \cos(\psi+\Delta \varphi/N_1)}. \tag{4} \]
At a low frequency, for which \(\beta l_1\) is less than \(10\text{--}15^\circ\), the angle \(\psi\) can be approximated by the equation
\[ \psi = \beta l_1 \frac{1}{\sqrt{2m^2-1}}. \tag{5} \]
The actual phase shift in a meander line varies nonlinearly with frequency, since
\[ \frac{d\psi}{d\beta l_1} \]
increases as the frequency increases [1]. The magnitude of the relative phase distortion, defined as the deviation of the phase characteristic from a linear one, is written as
\[ \delta \varphi = \frac{ N_1\psi - N_1\beta l_1 \dfrac{1}{\sqrt{2m^2-1}} }{ N_1\beta l_1 \dfrac{1}{\sqrt{2m^2-1}} + N_2\beta l_2 }. \tag{6} \]
To optimize the structure according to any criterion, equations (1)—(6) must be supplemented with expressions establishing the relation between the primary electrical parameters \(C_0\), \(C_1\) and the geometrical dimensions of the structure and the properties of the substrate materials.
The principal parameters of lines with a TEM wave usually include [3] the capacitances under even and odd excitation \(C_{0e}\) and \(C_{00}\), as well as the characteristic impedances under even and odd excitation \(Z_{0e}\) and \(Z_{00}\).
Between the chosen primary parameters there is the following relation:
\[ C_0=\frac{C_{00}+C_{0e}}{2};\quad 2C_1=\frac{C_{00}-C_{0e}}{2}; \]
\[ C_{00}=C_0+2C_1;\quad C_{0e}=C_0-2C_1; \tag{7} \]
\[ Z_{00}=\frac{\sqrt{\varepsilon_r}}{3\cdot 10^8 C_{00}};\quad Z_{0e}=\frac{\sqrt{\varepsilon_r}}{3\cdot 10^8 C_{0e}}. \]
These parameters completely characterize the meander line. From them the phase delay, the characteristic impedance, and the input resistance can be found [1].
Fig. 3. Odd (a) and even (b) excitation of the strips of a meander line
Fig. 4. Sequence of conformal mappings of the cell of even excitation (a) onto the canonical region of a plane capacitor (d)
Even and odd excitation of the strips are shown in Fig. 3, b and a, respectively. The capacitance of the strips under even excitation is equal to \(N_1 C_{0e}\), where \(C_{0e}\)—the capacitance of one strip—can be found for the elementary cell of even excitation (Fig. 4, a). The cell is bounded by part of the shields and by magnetic walls (dashed lines) separating the field region of two neighboring cells. The capacitance of the strips under odd excitation is likewise equal to \(N_1 C_{00}\), where \(C_{00}\) is the capacitance of one strip; and, unlike the cell of even excitation, the cell for the odd type of excitation includes vertical metallic walls.
To find the per-unit-length capacitances, the method of conformal mappings [2] was used, which made it possible to solve the problem of finding the primary parameters for specified dimensions. To map the inner region of the cells onto the canonical region of a plane capacitor and to find the function, the following sequence of operations was carried out (Fig. 4).
The Jacobi functions
\[ z_2 = sn\,(z_1,k_1) \]
map the rectangle in the \(z_1\) plane, representing the cell of the even type of excitation, onto the \(z_2\) plane with the correspondence of points shown in Fig. 4, a, b. In this case the coordinate of point \(A_1\) is expressed in terms of the strip width as
\[ z_1(A_1)=w/2=a, \]
and the coordinates of point \(A_1\) in the \(z_2\) plane are
\[ z_2=sn\,(A_1,k_1)=d. \]
The quantity \(k_1\) entering into the expression for \(d\) is called the modulus of the elliptic function \(sn(z_1,k_1)\) and can be found from tables [2] for the ratio of the complete elliptic integrals of the first kind \(K_1\) and \(K_1'\), equal to
\[ \frac{K_1}{K_1'}=\frac{w+s}{b}. \]
Next we use the linear transformation
\[ z_2'=\frac{z_2}{d}. \]
It maps the \(z_2\) plane onto itself. In this case the point \(A_2\) moves to \(+1\), and the point \(C_2\) will have the coordinate (Fig. 3c)
\[ c=\frac{1}{k_1 d}. \]
We now map the plane \(z_2'\) into the region of a plane capacitor in the \(z_3\) plane by means of an elliptic integral of the first kind
\[ z_3=F(z_2'; k_2). \]
From the correspondence of points under the mapping (Fig. 4c, d) we establish that
\[ \frac{1}{c}=\frac{1}{k_2}, \]
whence
\[ k_2=k_1 d. \]
Since the coordinates of the points \(A_1\) and \(C_1\) in the \(z_3\) plane have, respectively, the values \(K_2\) and \(K_2+iK_2'\), the capacitance per unit length in the even-excitation mode is found from the formula for a plane capacitor
\[ C_{0e}=\varepsilon_0\varepsilon_r\frac{4K_2'}{K_2}, \tag{8} \]
where \(K_2\) and \(K_2'\) are complete elliptic integrals of the first kind, depending on the moduli
\[ k_2 \quad \text{and} \quad k_2'=\sqrt{1-k_2^2}. \]
The sequence of mappings for the odd-type cell differs from that given above by the absence of the intermediate linear transformation. Proceeding from this,
\[ C_{00}=\varepsilon_0\varepsilon_r\frac{4K_3}{K_3'}, \tag{9} \]
where the complete elliptic integrals of the first kind \(K_3\) and \(K_3'\) depend on the moduli \(k_3\) and \(k_3'=\sqrt{1-k_3^2}\), and \(k_3=d\).
Thus, the calculation of the capacitances by formulas (8) and (9) reduces to the calculation of the moduli \(k_1\), \(k_2\), \(k_3\) and the values of the elliptic integrals. Using expressions (8), (9), and (7), the capacitances per unit length \(C_{00}\), \(C_{0e}\), \(C_0\), and \(C_1\) were calculated for a meander line whose gap between strips \(s\) is equal to the width of the current-carrying strip \(w\). The calculated curves are shown in Fig. 5. Figure 6 gives the dependence of the wave impedances in the even \(Z_{0e}\) and odd \(Z_{00}\) excitation modes on \(w/b\) at \(s/b=0.5\).
Thus, formulas (7–9), establishing additional relationships between the electrical and structural parameters, make it possible to obtain optimal characteristics of strip meander delay lines according to the following criteria: linearity of the phase-frequency characteristic, dimensions, and material properties.
Fig. 5. Dependence of the per-unit-length capacitances of a meander line on the ratio \(w/b\) at \(w/b=s/b\) (dielectric—air). The transverse dimensions of the structure are shown at the top.
Fig. 6. Dependence of the characteristic impedances of the even \(Z_{oe}\) and odd \(Z_{oo}\) modes of excitation of a meander line on \(w/b\) at \(s/b=0.5\) (dielectric—air).
In conclusion, the authors express their deep gratitude to P. L. Vorob'ev for useful advice and assistance in the work.
REFERENCES
- R. Pregla. Mäanderleitung als Laufzeitausgleichsschaltung. AEÜ, 1967, 21, No. 5, 219–227.
- V. I. Lavrik, V. N. Savenkov. Handbook of Conformal Mappings. Kiev, “Naukova dumka,” 1970.
- A. L. Fel'dshtein, L. R. Yavich. Synthesis of Four-Terminal Networks and Eight-Terminal Networks at UHF. Moscow, “Svyaz’,” 1971.