Skip to content

COP Calculation Using Area Method

Author: YoElMiCr0 | Source: Hand-drawn notes

This page explains how to calculate the COP (Coefficient of Performance) using graphical area methods for two typical signals: - A square + exponential waveform - A sinusoidal waveform

The goal is to evaluate the energy efficiency (or signal effectiveness) by comparing the area under the waveform during its decay phase to the energy input from the initial square pulse.

Case 1: Square Pulse + Exponential Decay

Waveform

COP waveform

  • A₁ is the rectangular area of height \( E_{dc} \) and width \( T_{on} \)
  • A₂ is the area under the exponential decay from \( V_p \) to \( V_p \cdot e^{-1} \)

🔸 Area Calculations

Input Energy Area (A₁)

\[ A_1 = E_{dc} \cdot T_{on} \]

Output Area (A₂)

\[ A_2 = \int_0^{5T} V_p \cdot e^{-\frac{t}{T}} \, dt \]

Solving the integral:

\[ A_2 = V_p \cdot T \cdot (1 - e^{-5}) \]

COP Expression

\[ COP = \left| \frac{A_2}{A_1} \right| = \left| \frac{V_p \cdot (T - T \cdot e^{-5})}{E_{dc} \cdot T_{on}} \right| \]

Case 2: Sinusoidal Waveform

Waveform Area

Sine waveform

We now consider a sine wave of peak voltage \( V_p \) over one period \( T \). We only consider the positive half-cycle as contributing useful area.

Area of Half-Sine

\[ A_2 = V_p \cdot \left( \frac{2}{\pi} \right) \cdot \frac{1}{2} \cdot T = V_p \cdot \frac{T}{\pi} \]

Again, the reference energy is the same:

\[ A_1 = E_{dc} \cdot T_{on} \]

COP Expression

\[ COP = \left| \frac{A_2}{A_1} \right| = \left| \frac{V_p \cdot \left(\frac{2}{\pi}\right) \cdot \frac{1}{2} \cdot T}{E_{dc} \cdot T_{on}} \right| = \left| \frac{V_p \cdot \frac{T}{\pi}}{E_{dc} \cdot T_{on}} \right| \]

Interpretation

  • This method gives an intuitive understanding of how much energy output (area under curve) is gained per energy input (square pulse).
  • A higher COP indicates a more efficient conversion or transfer.
  • COP is dimensionless, since both \( A_1 \) and \( A_2 \) have units of V·s (or energy proxy when impedance is fixed).

Notes

  • The exponential decay simulates discharging capacitors or resonant LC circuits.
  • The sine wave analysis assumes symmetric AC behavior; only half-wave is considered for output area.
  • All units are consistent if using V, seconds, and energy in Joules (or V·s).
  • RMS or power-based analysis may be more appropriate for continuous waveforms. For pulsed or decaying waveforms, area-based COP gives better insight into waveform efficiency.

Definitions and Assumptions

  • \( V_p \): Peak voltage of the waveform
  • \( E_{dc} \): Constant DC input level
  • \( T_{on} \), \( T_{off} \): Durations of the input square pulse and decay respectively
  • \( T = T_{on} + T_{off} \): Total period of one cycle
  • \( A_1 \): Area under the DC pulse (input energy)
  • \( A_2 \): Area under the output waveform (output energy proxy)
  • COP: Ratio of output to input energy in area terms

Assumptions: - Linear, time-invariant behavior
- Negligible system losses
- Constant or known load impedance - Repeated periodic behavior over cycles

Why Use Areas to Estimate COP?

In systems such as resonant circuits or pulsed energy transfer (e.g., SMPS, flyback converters), the area under the voltage-time curve approximates energy transferred or stored when the load impedance is fixed.

We define:

\[ COP = \left| \frac{A_2}{A_1} \right| \]

This is not thermal COP. It is a dimensionless metric expressing waveform effectiveness in transferring or storing energy.

Physical Relevance

This method is physically meaningful when:

  • The circuit includes capacitive or inductive energy storage elements
  • Voltage waveforms represent energy release post excitation
  • Output is not a continuous waveform, but rather a decaying or resonant pulse

For a resistive load \( R \), energy dissipated is:

\[ E = \int_0^T \frac{V(t)^2}{R} \, dt \]

If \( R \) is constant, energy is proportional to the area under \( V(t)^2 \), and often approximated by the shape of \( V(t) \) when known.

Example Calculation

Let’s assume:

\[ E_{dc} = 5\, \text{V} \quad T_{on} = 1\, \text{ms} \quad V_p = 12\, \text{V} \quad T = 6\, \text{ms} \]

Then:

\[ A_1 = 5 \cdot 1 \, \text{ms} = 5 \, \text{mV·s} \]
\[ A_2 = 12 \cdot (6 - 6e^{-5}) \, \text{ms} \approx 11.6 \, \text{mV·s} \]
\[ COP = \frac{11.6}{5} \approx 2.32 \]

This indicates more waveform energy was present after the input than during the input — typical in reactive circuits or tuned LC systems.

References

  • Sedra & Smith, Microelectronic Circuits – Energy in time-domain waveforms
  • Erickson & Maksimović, Fundamentals of Power Electronics
  • IEEE Transactions on Power Electronics

Disclaimer: This framework is not a substitute for full electromagnetic simulation or thermal modeling. It provides a practical, intuitive approximation under well-defined electrical boundary conditions.