The way a transmission line behaves is fundamentally determined by the electrical impedance connected at its end, known as the load impedance (ZL). Understanding these termination conditions is crucial for designing and troubleshooting RF systems.

When a transmission line is terminated with a short circuit (a direct connection between the conductors, effectively ZL = 0 Ohms), the behavior is unique:

• Reflection Coefficient: ΓL = ZL + Z0 / ZL - Z0 = 0 + Z0 / 0 - Z0 = Z0 - Z0 = -1. In polar form, ΓL = 1∠180∘ (or 1∠-π radians). This means the entire incident voltage wave is reflected, but with a 180-degree phase shift (it flips polarity).
• Voltage and Current at the Load: At the short circuit (z=0), the voltage must be zero: V(0) = V0+ + V0− = 0. This implies V0− = -V0+, which matches ΓL = -1. The current at the short circuit will be maximum.
• Standing Waves: A perfect standing wave is formed.
• There is a voltage node (minimum voltage, ideally zero) at the short circuit.
• There is a current antinode (maximum current) at the short circuit.
• As you move away from the short circuit along the line, the voltage and current vary sinusoidally. The next voltage maximum (antinode) will occur at a distance of λ/4 from the short circuit, then a voltage node at λ/2, and so on.

When a transmission line is terminated with an open circuit (the conductors are not connected, ZL = ∞ Ohms), the behavior is also distinct:

• Reflection Coefficient: ΓL = ZL + Z0 / ZL - Z0. To evaluate this when ZL → ∞, we can divide the numerator and denominator by ZL: ΓL = 1 + Z0 / ZL / 1 - Z0 / ZL. As ZL → ∞, Z0 / ZL → 0. So, ΓL = 1 + 0 / 1 - 0 = 1. In polar form, ΓL = 1∠0∘. This means the entire incident voltage wave is reflected with no phase shift.
• Voltage and Current at the Load: At the open circuit (z=0), the current must be zero: I(0) = Z0 V0+ - Z0 V0− = 0. This implies V0+ = V0−, which matches ΓL = 1. The voltage at the open circuit will be maximum.
• Standing Waves: A perfect standing wave is formed.
• There is a current node (minimum current, ideally zero) at the open circuit.
• There is a voltage antinode (maximum voltage) at the open circuit.
• As you move away from the open circuit, the voltage and current vary sinusoidally, but shifted by λ/4 compared to the short-circuited case. The next voltage node will occur at λ/4 from the open circuit.

This is the most desirable termination condition in most RF applications. When a transmission line is terminated with a load impedance that is exactly equal to its characteristic impedance (ZL = Z0 Ohms), it is called a matched line.

• Reflection Coefficient: ΓL = Z0 + Z0 / Z0 - Z0 = 2Z0 / 0 = 0.
• No Reflection: This is the key outcome. Since ΓL = 0, there is no reflected wave. All the incident power is absorbed by the load.
• No Standing Waves: With no reflected wave, there is no interference pattern, so no standing waves are formed. The voltage and current amplitudes remain constant along the line.
• VSWR: For a matched line, VSWR=(1+|Γ|)/(1−|Γ|)=(1+0)/(1−0)=1. This is the ideal VSWR.
• Maximum Power Transfer: A matched termination ensures that the maximum possible power is transferred from the transmission line to the load, as none is reflected back. This is a fundamental principle of power transfer in RF systems.

The input impedance (Zin) of a transmission line is the impedance that you "see" looking into the line from a particular point, usually at the input terminals of the line. It's not necessarily the characteristic impedance, and it varies depending on the load impedance (ZL), the characteristic impedance (Z0), and the electrical length of the line (l).

The general formula for the input impedance of a transmission line of length l (measured from the load, so the point where you're looking in is at z=−l) is:

Zin(l) = Z0 * (Z0 + ZL) * tanh(γl) / (ZL + Z0 * tanh(γl))

Where:
- Z0 is the characteristic impedance of the line.
- ZL is the load impedance.
- γ is the propagation constant (α+jβ).
- l is the length of the transmission line from the load.
- tanh is the hyperbolic tangent function.

This general formula applies to both lossy and lossless lines.

These special cases are critical for understanding how transmission line sections can be used as circuit elements (like inductors, capacitors, or transformers) at RF.

• Short-circuited line (ZL = 0): If the line is short-circuited (ZL = 0), the formula becomes: Zin,sc(l) = jZ0 * tan(βl)
• Interpretation: The input impedance of a short-circuited lossless line is purely reactive (imaginary).
• If 0 < βl < π/2 (i.e., 0 < l < λ/4), then tan(βl) is positive, so Zin,sc is positive imaginary, meaning it behaves as an inductor.
• If π/2 < βl < π (i.e., λ/4 < l < λ/2), then tan(βl) is negative, so Zin,sc is negative imaginary, meaning it behaves as a capacitor.
• Open-circuited line (ZL = ∞): If the line is open-circuited (ZL = ∞), the input impedance behaves as:
  Zin,oc(l) = -jZ0 * cot(βl)
• Interpretation: The input impedance of an open-circuited lossless line is also purely reactive. It behaves opposite to the short-circuited line.

When the length of a lossless transmission line is exactly a quarter-wavelength (l = λ/4), then βl = (2π/λ)(λ/4) = π/2. At this length, tan(βl) approaches infinity. To evaluate the formula, we use the trick of dividing numerator and denominator by tan(βl): Zin(l) = Z0 * (Z0 / tan(βl) + jZL / tan(βl)) / (ZL / tan(βl) + jZ0)

• Interpretation: A quarter-wavelength section of transmission line acts as an impedance transformer. It can transform a purely resistive load ZL to a different purely resistive input impedance Zin. This is widely used for impedance matching.