---
title: "Maximum power transfer theorem in antennas"
subject: "Propagation and Antenna"
chapter: 1
key: "semester_6/propagation_and_antenna:maximum-power-transfer-theorem"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/maximum-power-transfer-theorem
site: BCT Engineering
---

# Maximum power transfer theorem in antennas

Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals

- Asked in: 2080 Chaitra, 3 marks; 2080 Asoj, 3 marks; 2077 Chaitra, 2 marks; 2076 Bhadra, 3 marks (asked 4 times)
- Video: [MAXIMUM POWER TRANSFER THEOREM](https://www.youtube.com/watch?v=1foMNuZE2Io) — RF Microwave & Antenna
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/maximum-power-transfer-theorem
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1

## Question

State and explain the maximum power transfer theorem used in antenna, with necessary diagrams.

## Short answer

Maximum power transfer occurs when load impedance is the complex conjugate of the source impedance: Z_L=Z_A*, so R_L=R_A and X_L=−X_A. For a receiving antenna, P_max=|V_oc|²/(8R_A), with half the power lost or re-radiated in R_A. For transmitting, match the antenna to avoid reflection and mismatch loss.

## Answer

**Maximum power transfer theorem:** A source delivers maximum power to a load when the load impedance is the **complex conjugate** of the source impedance: Z_L = Z_S*, i.e. R_L = R_S and X_L = −X_S.

#### Receiving antenna
By Thevenin's theorem, the receiving antenna is a voltage source V_oc in series with its impedance Z_A = R_A + jX_A, where R_A = R_r + R_loss (radiation resistance plus loss resistance).

```text
     Z_A = R_A + jX_A
   +----/\/\/----+
   |             |
 (~) V_oc       Z_L = R_L + jX_L   (receiver)
   |             |
   +-------------+
```

Maximum power reaches the receiver when Z_L = Z_A*, i.e. R_L = R_A and X_L = −X_A:

```text
P_max = |V_oc|² / (8 R_A)      (V_oc peak value)
```

Under this condition an equal amount of power is dissipated or **re-radiated** in R_A, so at most half of the captured power reaches the receiver.

#### Transmitting antenna
The antenna is the load of the transmitter (through a transmission line). For maximum radiated power, the antenna input impedance must be matched to the line/transmitter: Z_A = Z_S*. If they are not matched, part of the power is reflected, giving VSWR > 1 and **mismatch loss**:

```text
Mismatch efficiency = 1 − |Γ|²,  Γ = (Z_A − Z0)/(Z_A + Z0)
```

In practice, matching networks, baluns and stubs are used to meet this condition.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the condition for maximum power transfer from a source of impedance Z_S?
   A. Z_L = Z_S
   B. Z_L = -Z_S
   C. Z_L = 1/Z_S
   D. Z_L = Z_S* (complex conjugate)
   Answer: D. Maximum power is delivered when R_L = R_S and X_L = -X_S, i.e. the load is the conjugate of the source impedance. (Key point: Matching condition)

2. How is a receiving antenna modelled to apply the theorem?
   A. A current source in parallel with a capacitor
   B. A Thevenin voltage source V_oc in series with its impedance Z_A
   C. A pure resistor
   D. An ideal transformer only
   Answer: B. By Thevenin's theorem the antenna is V_oc in series with Z_A = (R_r + R_L) + jX_A. (Key point: Receiving antenna model)

3. Under conjugate matching of a receiving antenna, what is the maximum power to the receiver (V_oc peak)?
   A. |V_oc|^2 / R_A
   B. |V_oc|^2 / (2 R_A)
   C. |V_oc|^2 / (8 R_A)
   D. |V_oc|^2 / (4 R_A)
   Answer: C. P_max = |V_oc|^2/(8 R_A) for peak V_oc; an equal amount is dissipated or re-radiated in R_A. (Key point: Power at match)

4. What happens in a transmitting antenna if Z_A is not matched to the line?
   A. Part of the power reflects, giving VSWR > 1 and mismatch loss
   B. Radiation efficiency becomes 100%
   C. VSWR becomes exactly 1
   D. Gain doubles
   Answer: A. Mismatch reflects power; mismatch efficiency is 1 - |Γ|^2, so matching networks, baluns and stubs are used. (Key point: Transmitting mismatch)

## Related in Propagation and Antenna

- [Superposition theorem in antennas](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Asuperposition-theorem) — Propagation and Antenna, chapter 1, asked 3×
- [Reciprocity theorem for antennas](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areciprocity-theorem) — Propagation and Antenna, chapter 1, asked 7×
- [Compensation theorem for antennas](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Acompensation-theorem) — Propagation and Antenna, chapter 1, asked 6×
- [Thevenin theorem for antennas](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Athevenin-theorem) — Propagation and Antenna, chapter 1
- [Aperture antennas and rectangular horn radiation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aaperture-antenna-rectangular-horn) — Propagation and Antenna, chapter 3
- [Reflector antenna features and feed types](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Areflector-features-feeds) — Propagation and Antenna, chapter 3

## Related in other subjects

- [Maximum power transfer theorem proof](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Amax-power-transfer-theorem-prove) — Basic Electrical Engineering, chapter 3, asked 2×
- [Maximum power transfer, find R](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Amax-power-transfer-6v-8v-2a) — Basic Electrical Engineering, chapter 3
- [Superposition theorem and maximum power to RL](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Asuperposition-max-power-transfer) — Fundamental of Electrical and Electronics Engineering, chapter 1
- [Maximum power transfer, find R](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Amax-power-r-5-10-10v-20v) — Basic Electrical Engineering, chapter 3