The Relationship of the Input Impedance of a Transmission Line to Load and Characteristic Impedance

Key Takeaways

• The input impedance of a transmission line is the impedance seen by any signal entering it. It is caused by the physical dimensions of the transmission line and its downstream circuit elements.

• If a transmission line is ideal, there is no attenuation to the signal amplitudes and the propagation constant turns out to be purely imaginary. 

• When the transmission line length is infinite, the input impedance is equal to the characteristic impedance.

Designing electrical circuits involves reading datasheets for component details and specifications

While designing a circuit, designers must read through component datasheets to see how much internal resistance is associated with each component. If this is ignored, designers might end up with a circuit drawing a current that is different from its specification. 

Resistance and impedance are inherent properties of any circuit that resists the flow of current through it. Signals face impedance while in a waveguide, integrated circuit, or transmission line. The input impedance of a transmission line is the impedance seen by any signal entering it. It is caused by the physical dimensions of the transmission line and its downstream circuit elements.  

It is important for designers to understand input impedance, which is why we’ve put together the following information—read on to learn more. 

The Input Impedance of a Transmission Line

At the entry point of a transmission line, signals encounter input impedance that limits the flow of current through it. The input impedance depends on the complete set of elements present in the circuit. In high-speed and high-frequency circuits, signals can undergo serious degradation due to input impedance. When designing a circuit, input impedance should be considered to ensure signal integrity. 

There is no way to get rid of input impedance in a circuit. Each of the elements present in the circuit provides some fraction of input impedance to the signal entering the circuit.

To determine the input impedance of a circuit with only passive elements such as resistors, capacitors, and inductors, circuit analysis concepts can be used. However, there are non-linear devices, switches, diodes, integrated circuits, cavity resonators, and waveguides in RF and microwave circuits, and, in such circuits, the input impedance is correlated to the voltage level of the signal. In structures such as waveguides, transmission lines, and cavity resonators, the geometry influences the value of input impedance. Considering all these aspects of input impedance, it is safe to say that it plays a significant role in impedance matching networks. If not properly calculated, input impedance will result in poor impedance matching.

Calculating the Input Impedance

Consider a lossless, high-frequency transmission line where the voltage and currents are given by equations 1 and 2, with the input impedance, characteristic impedance, and load impedance as Zin, Z0, and ZL, respectively. 

As the transmission line is ideal, there is no attenuation to the signal amplitudes and the propagation constant turns out to be purely imaginary. Let’s define the output terminals with axis point z=0 and input terminals z=-L. Our objective is to find the impedance of the circuit when looking from Z=-L: 

The input impedance is the ratio of input voltage to the input current and is given by equation 3. By substituting equation 5 into equation 4, we can obtain the input impedance, as given in equation 6:

From equation 6, we can conclude that the input impedance of the transmission line depends on the load impedance, characteristic impedance, length of the transmission line, and the phase constant of the signals propagating through it. 

It is already a known fact that the characteristic impedance Z0 is dependent on the distributed parameters of the transmission line, such as resistance, inductance, capacitance, and conductance (as given by equation 7), which are usually defined per unit length. Whenever any change is made in the circuit, the input impedance changes.

The relationship between the characteristic impedance and input impedance can be deduced for certain transmission lines. In the derivation of the input impedance equation, we have considered the finite length of the transmission line. When the transmission line length is infinite, then the input impedance of the transmission line is equal to the characteristic impedance. Whenever the transmission line of finite length is terminated by a load impedance that is equal to the characteristic impedance, there is no reflection of signals (according to equation 7). In this case, the input impedance equals characteristic impedance. 

The calculation of the input impedance of a transmission line over a range of frequencies is useful to understanding the signal behavior in a circuit. Luckily, Cadence’s software provides tools to calculate impedances and S-parameters at various frequencies. 

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