RL Circuit Transfer Function Time Constant RL Circuit as Filter

A resistor and inductor are fundamental linear components, meaning they have a direct relationship between voltage and current. They are also passive components, consuming energy. When connected across a voltage supply, they form an RL circuit.

Types of RL Circuit

1. RL Series Circuit- When resistance and inductor are connected in series with voltage supply. The circuit is called series RL circuit.
2. RL Parallel Circuit- When resistance and inductor are connected in parallel with each other and is driven by voltage source, the circuit so obtained is called parallel RL circuit.

Transfer Function of Series RL Circuit

A transfer function is used to analyze an RL circuit. It is defined as the ratio of the output to the input of the system in the Laplace domain.

Consider a RL circuit in which resistor and inductor are connected in series with each other.
Let V_in be the input supply voltage,
V_L is the voltage across inductor, L,
V_R is the voltage across resistor,
and I is the current flowing through the circuit.

Now for finding transfer function apply voltage or potential divider rule. The voltage divider rule is a simplest rule used for determine the output voltage across any element in circuit.
It states that the voltage divided between the resistors is in direct proportion to their respective resistance.
Using voltage divider rule, the voltage across inductor V_L is:

The voltage across the resistor V_R is:

The transfer function, H_L for the inductor is:

Similarly, the transfer function, H_R for the resistor is,

Current
Since the circuit is in series so the current in resistor and inductor are the same and is given by:

Time Constant in RL Circuit

The time constant of an RL circuit is defined as the time it takes for the current to reach about 63.2% of its maximum value from its initial rate of rise.
The time constant of a series RL circuit equal to the ratio of value of inductor to the value of resistance:

Where,
T = time constant in seconds,
L = inductor in Henry,
R = resistance in ohms.

In RL circuit due to presence of inductor the current in the circuit does not build up at a steady rate because inductor has a property to oppose the change in current flowing through it. So rate of increase in current is initially rapid but it slows down as it approaches its maximum value. During each time constant, the current build up 63.2 % of its remaining distance. As shown in graph it takes 5 times constant to build up a current in RL circuit.

RL Circuit as Filter

Low Pass RL Filter

Consider a RL circuit is supplying with a voltage source of varying frequency and the circuit output voltage is taken across resistor R₁. The resistor R₁ is independent of frequency but the inductive reactance is directly proportional to frequency (as X_L = 2πfL). At low or zero (as in case of DC) frequency, the inductive reactance X_L is very small as compared to resistance because when frequency is low, inductive reactance is also low so, it act as a short circuit. As there is no voltage drop across inductor the output voltage is almost same as that of input voltage both in magnitude and the phase and it acts as low pass filter. Now when frequency is increases, inductive reactance, X_L also increases and this causes increase in magnitude of voltage drop across inductor and hence reduce the output voltage across resistor. This increase in inductive reactance creates a phase shift between input and output voltage.

High Pass RL Filter

In an RL circuit with a varying frequency voltage source, the output voltage is taken across the inductor L₁. At very low or zero frequency, the inductive impedance is zero, so the inductor acts as a short circuit, making the output voltage zero. As the frequency increases, the inductive reactance increases, causing more voltage to drop across the inductor, making it act as a high pass filter.