Varactor diodes

1. Where the capacitance comes from (first principles)

WHAT do we have physically? Two conductive regions (the p-side and n-side neutral bulk, full of carriers) separated by an insulating gap (the depletion region). That is literally the geometry of a parallel-plate capacitor:

$C = \frac{\varepsilon A}{W}$

where $\varepsilon$ is the semiconductor permittivity, $A$ the junction area, and $W$ the depletion width.

2. Deriving the junction capacitance formula

HOW does $W$ depend on voltage? Start from the depletion approximation of an abrupt junction. Solving Poisson's equation across the junction gives the depletion width:

$W = \sqrt{\frac{2\varepsilon (V_{bi}+V_R)}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)}$

$V_{bi}$ = built-in potential (the junction's own barrier, ~0.7 V for Si).
$V_R$ = external reverse voltage.
$q$ = electron charge, $N_A,N_D$ = doping.

Why this step? Reverse bias adds to the built-in barrier, so the total voltage across the depletion region is $V_{bi}+V_R$ . More total voltage ⇒ wider depletion.

Now substitute into $C=\varepsilon A/W$ :

$C_j = \varepsilon A \cdot \left[\frac{2\varepsilon (V_{bi}+V_R)}{q}\left(\tfrac{1}{N_A}+\tfrac{1}{N_D}\right)\right]^{-1/2} \propto (V_{bi}+V_R)^{-1/2}$

So for an abrupt junction $C_j \propto (V_{bi}+V_R)^{-1/2}$ .

Why this step? We rewrote $(V_{bi}+V_R)^{-1/2}$ as $C_0(1+V_R/V_{bi})^{-1/2}$ because at $V_R=0$ the bracket $=1$ and $C_j=C_0$ — matching the datasheet's specified zero-bias value.

3. Using it: tuning an LC circuit

WHY is a variable $C$ useful? The resonant (tank) frequency of an LC circuit is:

$f_0 = \frac{1}{2\pi\sqrt{L\,C}}$

Why this formula? At resonance inductive and capacitive reactances cancel: $X_L=X_C \Rightarrow 2\pi f L = \tfrac{1}{2\pi f C}$ , solve for $f$ .

If $C$ is the varactor's $C_j(V_R)$ , then changing the DC tuning voltage changes $f_0$ — electronic tuning. Because $C_j\propto V_R^{-n}$ and $f_0\propto C^{-1/2}$ :

$f_0 \propto C_j^{-1/2} \propto (V_{bi}+V_R)^{n/2}$

Raising $V_R$ raises $f_0$ (less capacitance, higher frequency). A hyperabrupt device ( $n\approx2$ ) gives $f_0\propto (V_{bi}+V_R)$ — nearly linear tuning, which circuit designers love.

Worked example Worked example 2 — tuning range of a radio

The varactor above tunes an $L=2.2\ \mu$ H coil. Find $f_0$ at $V_R=0$ and $V_R=4$ V. At $V_R=0$ : $C=80$ pF ⇒ $f_0=\frac{1}{2\pi\sqrt{2.2\times10^{-6}\cdot 80\times10^{-12}}}=\frac{1}{2\pi\sqrt{1.76\times10^{-16}}}\approx 12.0\text{ MHz}$ At $V_R=4$ V: $C=30.9$ pF ⇒ $f_0=\frac{1}{2\pi\sqrt{2.2\times10^{-6}\cdot 30.9\times10^{-12}}}\approx 19.3\text{ MHz}$ Why this step? Just recompute $f_0$ with the new $C$ . A single 0→4 V sweep moved the station from 12 to 19.3 MHz — that's the tuning range.

4. Circuit rules (HOW to bias one correctly)

Isolate DC from AC: a coupling capacitor blocks DC; an RF choke (large L) feeds DC bias while blocking RF.
Two varactors are often placed back-to-back so the RF signal never forward-biases either one, reducing distortion.
Model: reverse-biased varactor ≈ $C_j$ in series with a small series resistance $R_s$ → defines its Q: $Q=\dfrac{1}{2\pi f R_s C_j}$ . High Q = low loss = sharp tuning.

5. Common mistakes (Steel-manned)

6. Active recall

Recall Test yourself (hide and answer)

Why does a varactor's capacitance decrease with reverse voltage?
What determines the grading coefficient $n$ , and what value gives near-linear tuning?
Why must a varactor never be forward biased in operation?
Derive $f_0 \propto (V_{bi}+V_R)^{n/2}$ .

What is a varactor diode used as?

A voltage-controlled capacitor (variable capacitance set by reverse-bias voltage).

Which bias region does a varactor operate in?

Reverse bias (depletion/junction capacitance region), never forward.

Physical origin of a varactor's capacitance?

The insulating depletion region separating the conductive p and n bulk, i.e.

C=\varepsilon A/W

As reverse voltage increases, capacitance...?

Decreases, because the depletion width

W

increases.

General varactor capacitance law?

C_j = C_0 /(1+V_R/V_{bi})^{n}

What is the grading coefficient

n

for an abrupt junction?

n=1/2

What

n

gives a hyperabrupt varactor (wide, near-linear tuning)?

n\approx 1

2

Resonant frequency of an LC tank?

f_0 = 1/(2\pi\sqrt{LC})

How does resonant frequency scale with reverse voltage?

f_0 \propto (V_{bi}+V_R)^{n/2}

— increasing

V_R

raises

f_0

Why avoid forward biasing a varactor?

It collapses the depletion region and conducts, replacing low-loss junction C with lossy diffusion C.

What limits a varactor's Q factor?

Series resistance

R_s

Q = 1/(2\pi f R_s C_j)

Two other names for a varactor?

Varicap / tuning (varicap) diode.

Recall Feynman: explain to a 12-year-old

Imagine two crowds of people (electrons and holes) standing on either side of an empty corridor. That empty corridor is like the space inside a battery-store called a capacitor — it can hold charge. If you pull the crowds further apart (by turning a voltage knob up), the corridor gets wider, and a wider gap stores less charge. So turning the knob changes how much the diode stores — and because a radio "hums" at a pitch set by how much charge its parts hold, turning the knob changes the radio station. No moving parts — just a voltage.

Connections

PN Junction Diode — the depletion region physics the varactor exploits.
Reverse Bias & Depletion Region — source of the voltage-dependent width $W$ .
Junction vs Diffusion Capacitance — why we use reverse, not forward, bias.
LC Resonant Circuits — where varactors set $f_0$ .
Voltage Controlled Oscillator (VCO) & Phase Locked Loop (PLL) — modern applications.
Zener Diode — contrast: another special-purpose reverse-bias diode.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Varactor diode ek aisa diode hai jise hum current chalane ke liye nahi, balki ek voltage se control hone wala capacitor ki tarah use karte hain. Har reverse-biased PN junction ke andar ek depletion region banta hai — yeh ek khaali (insulator jaisa) gap hota hai jo p-side aur n-side ke beech me aata hai. Do conducting side aur beech me insulator gap — yeh bilkul ek capacitor ki tarah hai, jiska formula $C=\varepsilon A/W$ hota hai.

Ab magic yeh hai: jab aap reverse voltage badhate ho, carriers aur door chale jaate hain, depletion gap $W$ chaura ho jaata hai, aur chaura gap matlab kam capacitance. Yaani voltage knob ghumao aur capacitance change karo — bina koi mechanical part hilaye. Formula yaad rakho: $C_j = C_0/(1+V_R/V_{bi})^n$ , jahan $n$ doping profile batata hai (abrupt me $1/2$ , hyperabrupt me $\approx 2$ ).

Iska sabse bada use hai tuning. LC circuit ki frequency $f_0 = 1/(2\pi\sqrt{LC})$ hoti hai. Agar $C$ varactor ka $C_j$ ho, to sirf DC voltage badalke aap radio ya TV ki station (frequency) change kar sakte ho. Isi liye purane radios, TV tuners, aur aaj ke VCO/PLL circuits me varactor lagta hai.

Ek important baat: varactor ko kabhi forward bias mat karo. Forward me depletion region gayab ho jaata hai, diode conduct karne lagta hai, aur lossy diffusion capacitance aa jaati hai — tuning ke kaam ka nahi. Hamesha reverse bias me rakho, DC se operating point set karo, aur RF signal usके upar chhota sa ride karta hai.

1. Where the capacitance comes from (first principles)

2. Deriving the junction capacitance formula

3. Using it: tuning an LC circuit

4. Circuit rules (HOW to bias one correctly)

5. Common mistakes (Steel-manned)

6. Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Diodes & Applications

Connections