GATE ECE Electromagnetics Formula Sheet 2027, With the Traps

A GATE ECE electromagnetics formula sheet can be short, because the section is small: a mean of 7.4 marks a paper across the five EC papers the GATE ECE 2027 book counts (2010, 2019, 2020, 2025 and 2026). It has to be exact, though. Nearly every question asks for one quantity from one formula, and marks go to a missing factor of two or a missing square root.

This sheet gives the formulas those papers used, topic by topic, each with a one-line example and the trap that costs marks.

Key takeaways

• Electromagnetics carried 25 questions and 37 marks across the five counted papers, never fewer than 6 marks or more than 9 in a paper.
• The plane-wave reflection coefficient and the transmission-line reflection coefficient are the same expression, with $\eta$ in one and $Z$ in the other.
• Power formulas carry a half for peak amplitudes and none for rms values, and reflected power is $\vert\Gamma\vert^{2}$, not $\vert\Gamma\vert$.
• Every counted waveguide question was on a rectangular guide, and all of them came from one cut-off formula.
• The Smith chart is tested as algebra: one full turn is half a wavelength, and a half turn turns $z$ into $1/z$.
• Skin depth is named in the syllabus but no counted question computes it, so learn the formula without overspending time on it.

The symbols and conventions this sheet uses

The intrinsic impedance $\eta$ of a medium is the ratio of electric to magnetic field strength in a plane wave travelling through it. The characteristic impedance $Z_0$ plays the same role for a transmission line. The reflection coefficient $\Gamma$ is the ratio of reflected to incident field (or voltage), and the transmission coefficient $\tau$ is the ratio of transmitted to incident field.

The VSWR (voltage standing wave ratio) is the ratio of maximum to minimum voltage along a line. A normalised impedance is $z = Z/Z_0$. The phase constant is $\beta = 2\pi/\lambda$, in radians per metre.

Two conventions run through every formula. A peak amplitude is the coefficient in front of the cosine; an rms value is the peak divided by $\sqrt{2}$. Assume rms only when the question says so.

Maxwell's equations in integral form

The four laws relate fields round a closed path or over a closed surface to the sources inside. Counted questions: 2019 Q21, Q22 and Q46; 2025 Q50 and Q65.

The displacement current density is $\partial\mathbf{D}/\partial t$. For sinusoidal fields, conduction current divided by displacement current is $\sigma/(\omega\varepsilon)$, which is the loss tangent of the next section.

Example: a single loop of resistance $10\ \Omega$ sees its flux rise by $0.02$ Wb in $0.1$ s. The emf is $0.2$ V and the current is $20$ mA, flowing so as to oppose the rise.

Plane waves: impedance, speed and loss

In a uniform plane wave, $\mathbf{E}$, $\mathbf{H}$ and the direction of travel are mutually perpendicular, and $\vert\mathbf{E}\vert/\vert\mathbf{H}\vert = \eta$. Counted questions: 2010 Q25 and Q46; 2020 Q55; 2026 Q24 and Q28.

Example: in a non-magnetic medium with $\varepsilon_r = 4$, $\eta = 377/2 \approx 188.5\ \Omega$ and $v_p = 1.5\times10^{8}$ m/s.

On skin depth, be clear about the evidence. The syllabus names it, yet no question in the five counted papers computes it. The nearest is 2026 Q28, a loss-tangent MSQ that stops one step short of $\delta$. Learn the scaling: four times the frequency gives half the skin depth.

Remember: $\eta_0 = 120\pi \approx 377\ \Omega$, and a dielectric divides it by $\sqrt{\varepsilon_r}$, never by $\varepsilon_r$.

Power in a plane wave: the Poynting vector

The Poynting vector $\mathbf{E}\times\mathbf{H}$ gives power per unit area. Its time average for phasor fields, and its plane-wave form, are:

Here $E_0$ is a peak amplitude. With an rms value the power is $E_{\text{rms}}^{2}/\eta$.

Example: a free-space wave with $E_0 = 377$ V/m peak has $H_0 = 1$ A/m and carries $377/2 \approx 188.5\ \text{W/m}^2$.

Trap: the half belongs to peak amplitudes and is absent for rms values. Read which one the question gives before you divide.

Normal incidence: what reflects and what passes

When a wave in medium 1 meets medium 2 head-on, the reflected and transmitted fields follow from the two impedances. Every counted incidence question was at normal incidence.

Example: air to a dielectric with $\varepsilon_r = 4$ gives $\eta_2 = \eta_1/2$, so $\Gamma = -1/3$ and $\tau = 2/3$. One-ninth of the power reflects; eight-ninths passes. Going the other way, $\Gamma = +1/3$ and $\tau = 4/3$, yet the power split is unchanged.

Transmission lines: input impedance and special lengths

A load $Z_L$ seen through a lossless line of length $l$ looks like a different impedance at the input:

Counted questions: 2010 Q23 and Q47; 2020 Q31; 2025 Q64.

Example: at 1 GHz, $\lambda_0 = 30$ cm; on a line with $\varepsilon_r = 4$, $\lambda = 15$ cm, so a quarter-wave section is $3.75$ cm. With $Z_0 = 50\ \Omega$ and $Z_L = 100\ \Omega$, it presents $25\ \Omega$.

Reflection coefficient, VSWR and delivered power

The line's reflection coefficient has the same form as the plane-wave one, with impedances in place of $\eta$.

Example: $Z_L = 100\ \Omega$ on a $50\ \Omega$ line gives $\Gamma_L = 1/3$ and VSWR $= 2$. With $9$ W incident, $8$ W reaches the load.

This sheet stops at the formulas. The GATE ECE 2027 book lists all 25 counted Electromagnetics questions by year and number, and its section question bank names, in each solution, which slip produced each wrong option.

Reading the Smith chart without drawing it

The Smith chart plots $\Gamma$ on a circle, labelled in normalised impedance, with $\Gamma = (z - 1)/(z + 1)$. Both counted chart questions (2020 Q23, 2026 Q52) were pure algebra.

Example: $z = 2$ moved a quarter wavelength becomes $z = 0.5$, which is $25\ \Omega$ on a $50\ \Omega$ line, matching the quarter-wave result above.

In one line: the chart is normalised, so multiply by $Z_0$ before you quote an impedance.

S-parameters: lossless, reciprocal and matched

A two-port's S-parameters relate reflected and transmitted waves to incident ones. Counted questions: 2010 Q22; 2020 Q56; 2026 Q34.

Example: $\vert S_{11}\vert = 0.2$ and $\vert S_{21}\vert = 0.9$ give $0.04 + 0.81 = 0.85$, so $15$ per cent of the incident power is lost inside. Remember too that a three-port cannot be lossless, reciprocal and matched at all ports at once.

Rectangular waveguides: cut-off, dominant mode and guide wavelength

A rectangular guide with broad wall $a$ and narrow wall $b$ passes a mode only above its cut-off frequency:

Counted questions: 2019 Q47 and Q61; 2026 Q33.

Example: $a = 2.5$ cm gives $f_c = 6$ GHz. At $10$ GHz, $\lambda = 3$ cm and $f_c/f = 0.6$, so $\lambda_g = 3/0.8 = 3.75$ cm. Note that 2019 Q61 ran this formula backwards, from a ratio of cut-offs to the wall ratio.

Antennas: Friis and the short dipole

The Friis equation gives the fraction of transmitted power received across free space:

The radiation resistance of a short dipole of length $l$ is $R_r = 80\pi^{2}(l/\lambda)^{2}$. Counted questions: 2019 Q33; 2020 Q53; 2026 Q21.

Example: with $G_t = G_r = 10$, $\lambda = 0.1$ m and $R = 1$ km, $P_r/P_t \approx 6.3\times10^{-9}$. A short dipole of length $\lambda/50$ has $R_r \approx 0.32\ \Omega$. Make it 10 per cent longer and $R_r$ rises by $1.1^{2} = 1.21$, a 21 per cent rise.

Using the sheet in the last fortnight

Pair this sheet with practice to a number, since a large share of this section's questions are NAT with no options to eliminate. The marking scheme, which is common to every GATE paper, puts no negative marks on NAT, so attempt every one. Run each calculation on the GATE virtual calculator at least once before the exam.

For where these 7.4 marks sit against the other sections, read GATE ECE important topics, and fit the revision into the last two months strategy for GATE ECE. The book's last-minute revision sheet gives the same formula-and-trap treatment to all nine sections, as part of the GATE ECE 2027 book.

More formula sheets: all of GATE ECE · Analog Circuits · Communications · Signals and Systems

Quick revision

1. $\eta_0 = 120\pi \approx 377\ \Omega$; in a dielectric, $\eta = \eta_0/\sqrt{\varepsilon_r}$.
2. $P_{\text{avg}} = \vert E_0\vert^{2}/(2\eta)$ for peak amplitudes; drop the half for rms.
3. $\Gamma = (\eta_2 - \eta_1)/(\eta_2 + \eta_1)$ and $\tau = 1 + \Gamma$; power fractions are $\vert\Gamma\vert^{2}$ and $1 - \vert\Gamma\vert^{2}$.
4. Quarter-wave line gives $Z_0^{2}/Z_L$; half-wave line repeats $Z_L$; use the line wavelength $\lambda_0/\sqrt{\varepsilon_r}$.
5. VSWR $= (1 + \vert\Gamma\vert)/(1 - \vert\Gamma\vert)$ and delivered power is $P_{\text{inc}}(1 - \vert\Gamma\vert^{2})$.
6. Smith chart: clockwise toward the generator, one turn is $\lambda/2$, a half turn inverts $z$.
7. Lossless means unitary, reciprocal means symmetric, matched means $S_{11} = 0$.
8. TE10 cuts off at $c/(2a)$; Friis gains are ratios; $R_r = 80\pi^{2}(l/\lambda)^{2}$.