Master Voltage Divider Rule (VDR): Step‑by‑Step Examples for Resistor, Inductor, and Capacitor Circuits

Voltage Division “VDR” for for Resistive, Inductive and Capacitive Circuits

What is Voltage Divider Rule?

In a circuit, when a number of elements are connected in series, input voltage divides across the elements. And in a circuit, when a number of elements are connected in parallel, the current divides across the elements.

Therefore, in a parallel circuit, the current divider rule is used and in a series circuit, the voltage divider rule is used to analyze and solve the circuit.

When two or more impedances are connected in series, the input voltage is divided into all impedances. To calculate the voltage across each element, the voltage divider rule is used. The voltage divider rule is the most important and simple rule in circuit analysis to calculate the individual voltage of any elements.

The voltage divider rule is also known as the potential divider rule. In some conditions, we require specific output voltage. But we don’t have that specific value of the source. In this condition, we make a series of passive elements and reduce the voltage level to a specific value. And here, the voltage divider rule is used to calculate the specific output voltage.

According to elements used in the circuit, the voltage divider rule can be classified into three types; resistive voltage divider, inductive voltage divider, and capacitive voltage divider. Now, we will prove the voltage divider rule for all these types of circuits.

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Voltage Divider Rule for Resistive Circuits

To understand the resistive voltage divider rule, we take a circuit having two resistors are connected in series with the voltage source.

As the resistors are connected in series, the current that passes through both resistors is the same. But the voltage is not the same for both resisters. The input voltage of the circuit divides into both resisters. And the value of individual voltage depends on the resistance.

As shown in the above figure, two resistors R₁ and R₂ are connected in series with the voltage source V_s. The total current supplied by the source is I ampere. As all elements are connected in series, it will make a single loop and the current that passes through all elements is the same (I amp).

The voltage across resistor R₁ is V_R1 and the voltage across resistor R₂ is V_R2. And the total supplied voltage divides between both resisters. Hence, the total voltage is a sum of V_R1 and V_R2.

V_S = V_R1 + V_R2 … (1)

According to Ohm’s law,

V_R1 = IR₁ + IR₂ … (2)

Therefore, from the equation-(1) and (2);

V_S = IR₁ + IR₂

V_S = I(R₁ + R₂)

Now, put the value of current I in the equation-(2);

V_R1 = IR₁

Similarly;

V_R2 = IR₂

Hence, the voltage divider rule for a resistive circuit is opposite to the current divider rule. Here, the voltage of the resistor is a ratio of multiplication of total voltage and that resistance to the total resistance.

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Solved Example of Resistive Circuit using VDR

Example-1

Find the voltage across each resistor using the voltage divider rule.

Here, three resistors (R₁, R₂, and R₃) are connected in series with 100V source voltage. The voltage across resistors R₁, R₂, and R₃ are V_R1, V_R2, and V_R3 respectively.

The voltage across resistor R₁;

V_R3 = 500 / 30

V_R3 = 16.67 V

The voltage across resistor R₂;

V_R3 = 100 / 30

V_R3 = 33.33 V

The voltage across resistor R₃;

V_R3 = 1500 / 30

V_R3 = 50 V

Total voltage V_T;

V_T =V_R1 + V_R2 + V_R3

V_T = 16.67 + 33.33 + 50

V_T = 100 V

V_T = V_S

Hence, it is proved that the total voltage= is similar to the supplied voltage.

Voltage Divider Rule for Inductive Circuits

When a circuit having more than two inductors is connected in series, the current that passes through the inductors is the same. But the source voltage is divided into all inductors. In this condition, the voltage across the individual inductor can be found by the inductor voltage divider rule.

Consider as shown in the above figure, two inductors (L₁ and L₂) are connected in series. And total current I pass through the inductor. The voltage across inductor L₁ is V_L1 and the voltage across inductor L₂ is V_L2. And the supply voltage is V_S. Now, we need to find the voltage V_L1 and V_L2 using the inductor voltage divider rule.

As we know the equation of voltage for inductor;

Where L_eq is the total inductance of the circuit. Here, two inductors are connected in series. Therefore, the equivalent inductance is a sum of both inductances.

L_eq = L₁ + L₂

From, equation-(3);

Now, the voltage across inductor L₁ is;

Similarly, the voltage across inductor L₂ is;

So, we can say that the voltage divider rule for an inductor is the same as the resistors.

Solved Example of Inductive Circuit using VDR

Example-2

Find the voltage across each inductor for the given circuit using the voltage divider rule.

Here, two inductors are connected in series with 100V, 60Hz source. The voltage across inductor L₁ is V_L1 and the voltage across inductor L₂ is V_L2.

To find the voltage across inductors, we need to find the reactive impedance of each inductor.

The reactive impedance across inductor L₁ is;

X_L1 = 2 π f L₁

X_L1 = 2 × 3.1415 × 60 x 10 × 10^-3

X_L1 = 3.769 Ω

The reactive impedance across inductor L₂ is;

X_L2 = 2 π f L₂

X_L2 = 2 × 3.1415 × 60 x 14 × 10^-3

X_L2 = 5.277 Ω

According to the voltage divider rule,

The voltage across inductor L₁ is;

V_L1 = 41.66 V

The voltage across inductor L₂ is;

V_L2 = 58.35 V

Total voltage V_T is;

V_T =V_L1 + V_L2

V_T = 41.66 + 58.35

V_T = 100 V

V_T = V_S

So, the total voltage is the same as supplied voltage.

Voltage Divider Rule for Capacitive Circuits

In a capacitor, the voltage divider rule is different compare to the inductor and resistor. To calculate the voltage divider rule for capacitors, let’s consider a circuit having two or more capacitors connected in series.

Here, two capacitors are connected in series with the source voltage V_S. The source voltage divides into two voltages; one voltage is across the capacitor C₁ and a second voltage is across the capacitor C₂.

The voltage across capacitor C₁ is V_C1 and the voltage across capacitor C₂ is V_C2. As shown in the above circuit diagram, both capacitors are connected in series. Therefore, the equivalent capacitance is;

The total charge delivered by the source is Q;

Q = C_eq V_S

The voltage across capacitor C₁ is;

V_C1 = Q₁ / C₁

The voltage across capacitor C₂ is;

V_C2 = Q₁ / C₂

So, from the calculation, we can say that the individual voltage across the capacitor is a ratio of multiplication of total source voltage and opposite capacitance to the total capacitance.

Solved Example of Capacitive Circuit using VDR

Example-3

Find the voltage across each capacitor for a given network using the voltage divider rule.

Here, two capacitors are connected in series with 100 V, 60 Hz source. The voltage across capacitor C₁ is V_C1 and the voltage across capacitor C₂ is V_C2.

To calculate the voltage across each capacitor, we need to find the capacitive impedance.

The capacitive impedance across C₁ is;

The capacitive impedance across C₂ is;

X_C2 = 1 / (2 π f C₂)

X_C2 = 1 / (2 π × 60 × 20 ×10^-6)

X_C2 = 10^-6 / 7539.822

X_C2 = 132.63 Ω

According to the voltage divider rule, the voltage across capacitor C₁ is;

V_C1 = 33.33 V

the voltage across capacitor C₂ is;

V_C2 = 66.67 V

The total voltage across capacitor V_T is;

V_T =V_C1 + V_C2

V_T = 33.33 + 66.67

V_T = 100 V

V_T = V_S

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