Reflections and power losses in RF circuits are due to mismatched impedance in transmission lines.
Multi-section and tapered transformers are constructed using transmission lines. Once constructed, they are known as transmission line transformers.
Circuit designers use multi-section and tapered transformers in RF and microwave circuits to match impedance.
The impedance mismatch in transmission lines is the root cause of most reflections and power losses in RF circuits
The reflection of electromagnetic waves can result in overheating, power loss, and RF and microwave circuit malfunction. The impedance mismatch in transmission lines is the root cause of most reflections and power losses. There are several approaches in RF and microwave engineering for impedance matching, including stub matching, LC network matching, and controlled impedance routing. However, RF circuits use multi-section and tapered transformers to match impedance.
Both multi-section and tapered transformers are constructed using transmission lines, and are known as multi-section and tapered transmission line transformers. Let’s take a closer look at how these transformers affect performance in RF and microwave circuits.
Multi-Section and Tapered Transformers
Impedance matching is an essential process in RF and microwave circuits that reduce reflections and power loss. Transmission lines can be introduced between the input and output to match the impedance. These sections of transmission lines are collectively called transmission line transformers. Multi-section and tapered transformers connect between the input and output ports to match the impedance. The required impedance and passband properties are attained by varying the number of sections or length of transmission line transformers.
Multi-section transformers use many transmission line sections to match the load impedance. Tapered transformers are the alternative to the infinite number of transmission line sections present in multi-section transformers.
Let’s explore both types of transformers more closely.
Multi-section transformers are formed by connecting ‘n’ transmission lines in a series. To achieve matched impedance, multi-section transformers are connected between the feeder transmission line of characteristic impedance Z0 and the load impedance ZL.
A quarter long wavelength transmission line—also called a quarter-wave transformer—connected to the load is used for real load impedance matching. A quarter-wave transformer is a basic example of a multi-section transmission line transformer. Bandwidth is limited in the quarter-wave transformer, however, the introduction of multi-sections in the quarter-wave transformer can increase bandwidth.
The figure below shows a multi-section transformer with overall reflection coefficient , and each section’s reflection coefficient, shown as Γ1, Γ2, Γ3...Γn, respectively. Because all sections are the equal length of ‘l’, the individual reflection coefficients share the same sign.
A quarter-wave transformer is a basic example of a multi-section transmission line transformer
The overall reflection coefficient of a multi-section transformer can be given by the following equation, where :
The equation shows that any frequency response can be obtained by including multi-sections with proper reflection coefficients. By choosing the proper characteristic impedance of the transmission line sections, the required reflection coefficient can be achieved as a function of frequency.
Although circuit designers usually use tapered transformers in circulators, filters, and couplers, they are also frequently used in impedance-matching circuits. Tapering causes physical dimensions, such as diameter, width, and thickness, to be diminished to match the impedance. Transmission lines are either continuously or intermittently tapered to match impedance. The figure below shows a continuously tapered transformer used to match impedance.
The tapered transmission line transformer is frequently used in impedance matching circuits
A tapered transformer's reflection and transmission characteristics are expressed using the nonlinear Riccati equation, where β is the phase constant and ẑ is the normalized characteristic impedance:
The impedance profile of tapered transformers is slowly varying. Tapered transformers are essentially non-uniform transmission lines described by their length and taper function. The tapered profile of the transmission line governs the passband characteristics of the tapered transformer. The transformer's profile can be:
The tapered profile will change passband characteristics. In practical tapered transformers, the physical dimensions of the transmission line are often tapered linearly or exponentially. The Klopfenstein taper offers the shortest matching section, with minimum reflection coefficient at the expense of equal ripples in the passband.
The performance of multi-section and tapered transformers in microwave and RF circuits can be quantified by understanding how well they reduce signal reflections and power losses. Cadence’s software provides tools to implement impedance matching in your RF and microwave circuits.