This document discusses the applications of differential equations in RL and RC electrical circuit problems. It provides examples of RL circuits consisting of resistors and inductors and RC circuits consisting of resistors and capacitors. The formation of differential equations for these circuits is described based on Kirchhoff's laws and the voltage-current relationships for each component. An example problem demonstrates solving a first-order differential equation to find the current or voltage in an RL or RC circuit over time.

1. applications of
differential equations in
rl-rc electrical circuit
problems
PRESENTED BY :-
B-TECH (CSE) [Group -5]
PROJECT - 1
Digbijaya Mohapatra - 230301120235
Aarchi Kumari - 230301120236
Abhishek Dash – 230301120237
Omm Prakash Mishra – 230301120239
Aditya Roshan Pradhan - 230301120240
Swetalina Pradhan - 230301120241
DIFFERENTIAL EQUATION AND LINEAR ALGEBRA

2. A resistor–inductor circuit (RL circuit), or RL filter or RL network,
is an electric circuit composed of resistors and inductors driven
by a voltage or current source.
A first-order RL circuit is composed of one resistor and one
inductor, either in series driven by a voltage source or in parallel
driven by a current source. It is one of the simplest analogue
infinite impulse response electronic filters.
A LR Series Circuit consists basically of an inductor of
inductance, L connected in series with a resistor of resistance, R.
Applications of differential equations in RL electrical circuit
problems:-

3. The applications of RL circuit include the following :
• RF Amplifiers.
• Communication Systems.
• Filtering Circuits . Processing of Signal.
• Oscillator Circuits.
• Magnification of Current or Voltage.
• Variable Tunes Circuits.
• Radio Wave Transmitters.

4. An RL circuit with a switch to
turn current on and off. When
in position 1, the battery,
resistor, and inductor are in
series and a current is
established. In position 2, the
battery is removed and the
current eventually stops
because of energy loss in the
resistor.
A graph of current growth
versus time when the
switch is moved to position
1. Application of DE.
A graph of current growth
versus time when the switch
is moved to position 1
RL CIRCUIT

5. Variable voltage across the resistor:
Vr = Ir
Variable voltage across the inductor:
VL = L di/dt
Kirchhoff's voltage law:
Ri + L di/dt = V
Application of DE in a RL circuit

6. The formation of differential equation for an electric
circuit depends upon the following laws :
i) i = dq/dt
ii) Voltage drop across resistance (R) = RI
iii) Voltage drop across inductance (L) = L di/dt
iv) Voltage drop across capacitance (C) = q/c
Kirchhoff’s law: The algebraic sum of the voltage drop around
any closed circuit is equal to resultant emf in the circuit.
Current law: At a junction current coming is equal to current
going.

7. Example:
An RL circuit has an emf of 5 V, a resistance of 50 Ω, an inductance of 1 H, and no
initial current. Find the current in the circuit at any time t. Distinguish between
the transient and steady-state current.
+ / =
/ +50 =5
1st order DE : Y’ + P(x)y = Q(x)
IF: ^∫ ( ) = ^∫50
Therefore: ^∫50 = ∫5 ^∫50 )
^50 = ∫5 ^50
^50 =1/10 ^50 + C

8. Applications of differential equations in RC electrical circuit
problems:-
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is
an electric circuit composed of resistors and capacitors driven by a
voltage or current source.
A first-order RC circuit is composed of one resistor and one capacitor,
either in series driven by a voltage source or in parallel driven by a
current source. It is one of the simplest analogue infinite impulse
response electronic filters.
An RC Series Circuit consists basically of a capacitor of capacitance, C
connected in series with a resistor of resistance, R.

9. • Signal Filtering.
• Time Constant Calculations.
• Differentiator and Integrator Circuits.
• Time Circuits.
• Waveform Shaping.
• Low-Pass and High-Pass Filters.
The applications of RC circuit include the following:

10. RC CIRCUT
An RC circuit with a switch to charge
and discharge the capacitor. When
in position 1, the battery, resistor,
and capacitor are in series, and a
charge accumulates on the
capacitor. In position 2, the battery
is removed, and the capacitor
eventually discharges through the
resistor.
A graph of charge
growth versus
time when the
switch is moved to
position 1.
A graph of charge
decay versus time
when the switch is
moved to position 2.

11. Application of DE in a RC circuit
Variable voltage across the resistor:
Vr = iR
Variable voltage across the inductor:
VC = (1/C) ∫i dt
Kirchhoff's voltage law:
Ri + (1/C) ∫i dt = V

12. The formation of differential equation for an
electric circuit depends upon the following laws:-
i) i = C * dV/dt
ii) Voltage drops across resistance (R) = RI
iii) Voltage drops across capacitance (C) = (1/C) ∫i dt
iv) Voltage drops across inductance (L) = L di/dt
Kirchhoff’s law: The algebraic sum of the voltage drops around any closed circuit is
equal to resultant emf in the circuit.
Current law: Sum of incoming current entering at a junction is equal to sum of
current leaving the junction.

13. Example:
An RC circuit has an emf of 5 V, a resistance of 50 Ω, a capacitance of 1 F, and no initial
charge. Find the voltage across the capacitor at any time t. Distinguish between the
transient and steady-state voltage.
Vemf−VR−VC=0
→ Since, I=C⋅ dVc/dt
→ Vemf−(C⋅dVc/dt)⋅R−Vc=0
→ dVc /(Vemf−Vc) =dt/RC
→ −ln ∣Vemf −VC∣=t/RC+C1
Where C1 is the constant of integration
→ ∣Vemf −VC∣=e^(−t/RC−C1)
Since the initial charge is 0 (no initial charge), C1=0, and we can drop the absolute value.
→ Vemf – Vc = e^(-t/RC)
→ Vc(t)=Vemf−e^(−t/RC)
→ VC(t)=5 – e^(-t/50 ⋅ 1)
→ VC(t)=5 – e^(-t/50)