#### What is in this article?:

- How to Use Lumped Constants in Microwave Circuit Design
- Transistor parameters
- Design example (continued)

**March, 1968**

The latest microelectronic techniques now make possible the design of lumped-constant circuits at microwave frequencies. Whereas a few hundred megacycles appeared to be the limit for lumped-constant “conventional” circuits in the past, new methods extend this to over 1 Gc.

The usual criteria for the use of lumped components is that the largest linear dimensions of such components should not exceed 1/100 wavelength. So-called “conventional components” usually fail this test at less than 300 Mc.

At 1.0 Gc, the free-space wavelength is 30 cm. For a medium with a dielectric constant of four, the wavelength reduces to 15 cm. In this medium, the maximum linear dimension of a component (based on 1/100 λ) is 1.5 mm, or about 0.060 in.

Fortunately, the new thin-film components meet the spatial requirement in the lower microwave region. Although tin-film inductors, for instance, are only practical into the nanohenry range, needed inductance above 300 Mc is usually low enough to make such inductance useful.

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**Criteria for thin-film components**

By imposing the spatial restrictions on thin-film components, attainable values at microwave frequencies are as follows:

**Resistors:** The value of a thin-film resistor is

where

R_{S} = specific resistance of a film in ohms per square,

L = resistor length, and

W = resistor width.

The specific resistances generally used are in the order of a few hundred ohms per square (e.g., 200), and usable line widths can be as small as 0.001 in. Thus, thin-film resistors for use at microwaves can be fabricated in the range from 4 to 400,000 ohms.

**Capacitors:** The value of simple two-plate thin-film capacitor is

C = C_{S}A,

where

C_{S} = specific capacitance in picofarads per square mil, and

A = capacitor area.

The specific capacitance is limited by the breakdown voltage desired for the capacitor. For small-signal semiconductor circuits, 50 V is usually adequate; for this breakdown voltage, a specific capacitance of 0.05 pF per square mil is not difficult to achieve. The smallest practicable capacitor area is 1 square mil. Thus, thin-film capacitors for use at microwave frequencies range from 0.05 to 125 pF.

**Inductors:** For flat spiral thin-film inductors, the inductance is approximately

where

L = inductance in microhenries,

N = number of turns,

g = mean coil diameter in inches, and

c = the radial coil depth in inches, assuming the conductor width equals the conductor spacing.

In a 0.050-in. dia circle, the maximum inductance obtainable is 125 nH. The specific resistivity of thin-film conductors is normally about 0.05 ohms per square, from which it follows that the inductors will have Qs of about 20. The minimum inductance for a 0.050-in. conductor pad is on the order of 0.2 nH.

The advantage of using lumped constants at the lower microwave frequencies is essentially one of size. If a distributed technique is used (such as microstrip on alumina) each distributed element will be about a quarter-wavelength on a side. Alumina has a dielectric constant of nine; thus, a quarter wavelength at 1.0 Gc is about 1.0 in. A single-stage microstrip amplifier will consequently occupy an area of about 2 in.^{2}. If thin-film lumped components on a glass substrate are used, a two-stage amplifier occupies an area of only about 0.08 in.^{2}. Thus the reduction in size is appreciable.