How is the RC circuit simulated?

The RC circuit uses Euler forward integration of the first-order ODE dVc/dt = (Vin - Vc)/(RC) with a time step chosen as t/200 for stability. A square wave input drives the charging and discharging cycles, showing the exponential approach V(t) = V_s(1 - e^(-t/t)) on the oscilloscope display.

What is the LC resonance frequency?

The natural (resonant) frequency of an LC circuit is f0 = 1/(2pv(LC)). At this frequency, inductive and capacitive reactances are equal (XL = XC), and the circuit draws maximum current from the source. The quality factor Q = (1/R)v(L/C) determines the sharpness of the resonance peak � higher Q means narrower bandwidth.

What does the Filter Design preset show?

The Filter Design preset shows an RC lowpass filter with cutoff frequency f_c = 1/(2pRC). Below f_c, Vout � Vin (the capacitor charges quickly and tracks the input). Above f_c, the capacitor cannot charge fully each cycle, so Vout is attenuated. The capacitance slider sweeps the drive frequency relative to the cutoff, allowing direct observation of the lowpass attenuation behaviour on the oscilloscope.

Circuit Simulator � RC, RL, LC, AC, Filter & BJT Amplifier

Circuit Physics

First-order RC Circuit

An RC circuit is governed by a first-order linear ODE. For a step-function input V_s, the capacitor voltage follows an exponential approach with time constant t = RC:

dVc/dt = (Vin - Vc) / (R�C) Charging (Vin = V_s): Vc(t) = V_s � (1 - e^(-t/t)) Discharging: Vc(t) = V_s � e^(-t/t) Time constant: t = R�C (63.2% charge at t = t) Cutoff frequency: f_c = 1/(2pRC) (-3 dB point for AC)

First-order RL Circuit

dIL/dt = (Vin - R�IL) / L Transient current: IL(t) = (V_s/R) � (1 - e^(-t/t)) Time constant: t = L/R Cutoff frequency: f_c = R/(2pL) Voltage across L: VL = L � dIL/dt = V_s � e^(-t/t)

Series RLC � LC Oscillator

L � d�q/dt� + R � dq/dt + q/C = Vin(t) q = charge = C�Vc Natural frequency: ?0 = 1/v(LC) ? f0 = 1/(2pv(LC)) Damping ratio: ? = R/(2)�v(C/L) Quality factor: Q = ?0L/R = (1/R)�v(L/C) Underdamped (Q > 0.5): oscillating decay Critically damped (Q = 0.5): fastest non-oscillating return Overdamped (Q < 0.5): slow exponential return

AC RC Circuit � Impedance and Phase

BJT Common-Emitter Amplifier

DC biasing: Ic_Q = (Vcc - Vce_Q)/R_c (load line) Q-point: Vce_Q = Vcc/2 = 6 V, Ic_Q = 1.28 mA (Rc=4.7 kO, Vcc=12 V) Transconductance: gm = Ic_Q/Vt (Vt = 26 mV at 300 K) � 49 mA/V Small-signal gain: Av = -gm�Rc = -(with emitter degeneration: Av � -9.4) Output swing: �(Vcc/2 - Vsat) � �5.5 V (rails at supply/ground)

Preset Guide

Preset	Key Equation	Observable
🔋 RC Circuit	t = RC; f_c = 1/2pRC	Exponential charge/discharge; low f_c = slower response
🌀 RL Circuit	t = L/R; f_c = R/2pL	Current ramps up inductively; voltage spike at turn-off
📡 LC Oscillator	f0 = 1/2pvLC; Q = v(L/C)/R	Resonant oscillations; Q controls decay rate
⚡ AC Circuit	f = -arctan(1/?RC)	Vout lags Vin; phase & attenuation vs frequency
🔧 Filter Design	\|H(f)\| = 1/v(1+(f/f_c)�)	Below f_c: Vout�Vin; above f_c: attenuated
📡 BJT Amplifier	Av = -gm�Rc/(1+gm�Re)	Inverted, amplified output; rail clipping at �Vcc/2

Explore Circuit Theory

Read our deep-dive articles on circuit analysis and semiconductor physics.

Circuit Analysis ? Semiconductors ?

Curriculum Links

Level	Topic	Covered
GCSE Physics	Charge, current, voltage, resistance	RC/RL transients, Ohm's law
A-Level Physics	Capacitance, EM induction, AC circuits	RC time constant, LC resonance, impedance
A-Level Electronics	Transistor amplifiers, filters	BJT gain, f_c, Bode response
AP Physics C	Electromagnetic induction, AC circuits	RLC series, quality factor, resonance
IB Physics HL	Capacitance, Faraday, AC	Reactance, impedance, phase
University EE	Circuit theory, analogue electronics	Full ODE analysis, BJT small-signal model

Circuit Simulator #85