VSWR

On a mismatched transmission line, the forward and reflected waves superimpose to create a standing wave pattern of voltage and current along the line. The Voltage Standing Wave Ratio or VSWR is the ratio of the maximum voltage amplitude to the minimum voltage amplitude in this standing wave pattern. It is a practical measurement tool widely used in antenna and RF system testing, and it appears frequently in GATE electromagnetics.

Core Concept Explanation

On a lossless transmission line with a mismatched load, the incident wave V+ and the reflected wave V- exist simultaneously. These two waves travel in opposite directions and superimpose. At certain points along the line, they add constructively (in phase) creating voltage maxima. At other points, they add destructively (out of phase) creating voltage minima. This spatial variation of voltage amplitude is the standing wave, and it repeats with a period of lambda/2 along the line.

The VSWR is defined as the ratio of the maximum to minimum voltage envelope: VSWR = Vmax/Vmin. This ratio is always greater than or equal to 1. A VSWR of 1 means Vmax equals Vmin, which means no reflected wave, which means perfect matching. A VSWR approaching infinity means the minimum voltage approaches zero, which happens only when |Gamma| = 1, that is for short circuit, open circuit, or purely reactive loads.

VSWR is related to the reflection coefficient by: VSWR = (1 + |Gamma|) / (1 - |Gamma|). Inverting this, |Gamma| = (VSWR - 1) / (VSWR + 1). These two relations are the most important VSWR formulas for GATE. Notice that VSWR depends only on the magnitude of Gamma, not its phase, because the standing wave pattern shifts spatially but its envelope shape depends only on |Gamma|.

The practical significance of VSWR is that it is easy to measure directly with a slotted line measurement setup, even when the reflection coefficient phase is unknown. By physically moving a probe along the transmission line and measuring voltage, the VSWR can be found without needing to know the electrical length or phase of the load reflection.

Mathematical Expression

The total voltage on a lossless line at position z from the load is V(z) = V+ * [exp(-j*beta*z) + Gamma * exp(j*beta*z)]. The magnitude |V(z)| varies between V+ * (1 + |Gamma|) and V+ * (1 - |Gamma|). From these extremes, the VSWR is derived.

The locations of voltage maxima occur where the incident and reflected waves are in phase. The first voltage maximum from the load is at a distance d_max from the load given by d_max = -angle(Gamma) / (2*beta), where angle(Gamma) is the phase of the load reflection coefficient. The first voltage minimum is lambda/4 away from the nearest maximum.

The current standing wave ratio ISWR equals VSWR numerically, because current maxima occur at voltage minima and vice versa. The product Vmax * Imin = Vmin * Imax = |V+|^2 / Z0 = incident power. These properties are important for understanding impedance measurement on a slotted line.

Practical Understanding

In antenna systems, VSWR is the standard metric for how well the antenna is matched to the feedline. A VSWR of 2:1 corresponds to |Gamma| = 1/3, which means about 11% of the incident power is reflected. A VSWR of 1.5:1 corresponds to |Gamma| = 0.2, meaning only 4% power is reflected. Most commercial transmitters specify maximum allowable VSWR on their antenna port (typically 2:1) to prevent damage to the final amplifier stage.

High VSWR causes several practical problems beyond power loss. The standing wave creates voltage and current hot spots along the line. At current maxima, the line conductor carries higher than expected current, increasing ohmic loss and local heating. At voltage maxima, the dielectric between conductors experiences higher electric field stress, risking dielectric breakdown at high power levels.

Numerical Example

A 50 ohm lossless transmission line is terminated in a load ZL = 150 ohm (purely resistive). Calculate the VSWR. This is a direct GATE-type problem where ZL is real and greater than Z0.

Given:
Z0 = 50 ohm
ZL = 150 ohm  (real, greater than Z0)

Why this formula applies:
First find Gamma, then use VSWR = (1 + |Gamma|)/(1 - |Gamma|)

Formula:
Gamma = (ZL - Z0)/(ZL + Z0)
VSWR = (1 + |Gamma|)/(1 - |Gamma|)

Substitution:
Gamma = (150 - 50)/(150 + 50) = 100/200 = 0.5
|Gamma| = 0.5  (real positive, so phase = 0 deg)

Calculation:
VSWR = (1 + 0.5)/(1 - 0.5)
     = 1.5 / 0.5
     = 3

Verification: |Gamma| = (VSWR-1)/(VSWR+1) = (3-1)/(3+1) = 2/4 = 0.5 (correct)

Reflected power = |Gamma|^2 = 0.25 = 25% of incident power

Final Answer:
VSWR = 3  (written as 3:1)
|Gamma| = 0.5
Return Loss = -20 log10(0.5) = 6 dB
25% of incident power is reflected back toward source

Exam Tip: For a real ZL greater than Z0, VSWR = ZL/Z0. For a real ZL less than Z0, VSWR = Z0/ZL. Both cases give a real, positive Gamma with no phase. This shortcut works only when ZL is purely resistive. GATE often sets up exactly this case. Also remember: VSWR = 1 means perfect match. VSWR = infinity means short or open circuit.

Mechanism: How Standing Waves Form

• The incident wave V+ travels in the +z direction and the reflected wave V- = Gamma*V+ travels in the -z direction. Their superposition creates a spatially varying amplitude envelope.
• At voltage maxima (anti-nodes), the current is minimum. At voltage minima (nodes), the current is maximum. Voltage and current standing waves are spatially shifted by lambda/4.
• The separation between consecutive voltage maxima (or minima) is lambda/2. This is used in slotted line measurements to determine the wavelength and hence the frequency.
• For a short-circuit termination, the voltage at the load end is zero (voltage node at load). For an open-circuit termination, the voltage at the load end is maximum (voltage anti-node at load).
• VSWR depends only on |Gamma|. Two loads with different phases but the same |Gamma| produce the same VSWR, but with the standing wave pattern shifted spatially.
• In a lossy line, the standing wave is not perfectly periodic. The Vmax envelope decreases and Vmin envelope increases as you move away from the load toward the source, reducing the apparent VSWR measured at the source.

Quick Revision

• VSWR = Vmax/Vmin = (1 + |Gamma|)/(1 - |Gamma|). Range: 1 to infinity.
• Inverse: |Gamma| = (VSWR - 1)/(VSWR + 1). Useful when VSWR is given and Gamma is needed.
• For real ZL greater than Z0: VSWR = ZL/Z0. For real ZL less than Z0: VSWR = Z0/ZL. No complex arithmetic needed.
• VSWR = 1 means perfect match (|Gamma|=0). VSWR = infinity means full reflection (|Gamma|=1).
• Voltage nodes are lambda/2 apart. Voltage node to nearest current node is lambda/4.
• Reflected power = |Gamma|^2. At VSWR=2, |Gamma|=1/3, reflected power = 11%.
• GATE trap: VSWR depends only on |Gamma|, not on phase. Different complex ZL values can give the same VSWR.