Analysis of Series, Parallel, and Mixed Electric Circuits

Posted on Dec 20, 2024 in Electronics

ITEM 55: Introduction to Electric Circuits

This topic covers the calculation of quantities in electric circuits arranged in series, parallel, and mixed configurations. Students will become familiar with the parameters of DC and AC circuits and understand the behavior of resistive, inductive, and capacitive impedances. After studying this subject, students will be able to differentiate between series, parallel, and mixed circuits in both DC and AC contexts.

Series Resistance Circuit

A series circuit consists of several resistors connected in a single line, through which the same current flows. Key features include:

• a) The current is the same through all resistors: I_T = I₁ = I₂ = I₃
• b) The voltage is not the same across all resistors. The total voltage is the sum of the partial voltages: V = V₁ + V₂ + V₃
• c) The total resistance equals the sum of all individual resistances: R_T = R₁ + R₂ + R₃ + … + R_n

Parallel Resistance Circuit

A parallel circuit is formed by several resistors placed side by side, all under the same voltage. Key features include:

• a) The inverse of the total resistance equals the sum of the inverses of the partial resistances: R_t = 1 / (1/R₁ + 1/R₂ + 1/R₃)
• b) The current is not the same through all resistors. The total current is the sum of the partial currents: I_T = I₁ + I₂ + I₃
• c) The total voltage is equal to the partial voltages. It is the same across all resistors: V_T = V₁ = V₂ = V₃

Mixed Resistance Circuit

A mixed circuit is one where resistors are connected in both series and parallel configurations. To calculate quantities, the same principles apply as for series and parallel circuits. The circuit is reduced to find the total resistance by combining series and parallel sections.

Series Circuit: Calculation of Magnitude

In both AC current and voltage, a sine wave is represented as: I = I_max * sin(ωt + φ). In a series circuit, since the current is the same, then: I = I_max * sin(ωt). This means that the voltage will be in phase with the current.

RL Series Circuit

An RL series circuit consists of a generator (G), a resistor (R), and an inductor (L). The total circuit resistance is concentrated in the resistor (R), and the total circuit reactance (X_L) is concentrated in the inductor (L).

1. The impedance (Z) of an AC circuit with resistance and inductance is the geometric sum of the ohmic resistance and inductive reactance: Z = √(R² + X_L²).
2. The impedance is the hypotenuse of the triangle formed by the resistance (R) and inductive reactance (X_L).
3. The values of resistance and inductive reactance as a function of impedance are: R = Z * Cos(φ), X_L = Z * Sin(φ), Tan(φ) = X_L / R.
4. The total voltage of the circuit will be advanced by an angle φ with respect to the current I_max. Then, e = e_max * sin(ωt + π/2).

RC Series Circuit

An RC series circuit consists of an ohmic resistance and a capacitor in series with capacitive reactance (X_C). In this circuit, all resistance is concentrated in R, and all capacitance is concentrated in C.

1. In contrast to the inductor, the voltage will lag 90° behind the current I_max. Then, e = e_max * sin(ωt – π/2).
2. The impedance is the geometric sum of the resistance and capacitive reactance: Z = √(R² + X_C²).
3. The impedance is the hypotenuse of the triangle formed by the resistance (R) and capacitive reactance (X_C). Reactance X_C = 1 / (ω * C).
4. The values of resistance and capacitive reactance as a function of impedance are: R = Z * Cos(φ), X_C = Z * Sin(φ), Tan(φ) = X_C / R.

Knowing the impedance, the total voltage is determined by Ohm’s Law: V = I * Z, and the current is I = V / Z.

RLC Series Circuit

An RLC series circuit consists of a generator (G), a resistor (R), an inductor (L), and a capacitor (C). Their sinusoidal voltage is the sum of three sinusoidal voltages:

1. The voltage of the inductor is ahead by 90 degrees (π/2) with respect to the current.
2. The voltage of the resistor is in phase (φ = 0) with respect to the current.
3. The voltage of the capacitor is delayed by 90° (π/2) with respect to the current.

The impedance (Z) of the AC circuit is the geometric sum of the ohmic resistance and combined reactance (X = X_L – X_C), where X_L = ωL, X_C = 1 / (ωC), and Z = √(R² + (X_L – X_C)²). Tan(φ) = X / R.

To determine the parameters, consider three cases:

1. If X_L > X_C, the circuit has an inductive character.
2. If X_L < X_C, the circuit has a capacitive character.
3. If X_L = X_C, the circuit is in resonance.

The impedance is the hypotenuse of the triangle formed by the resistance (R) and the inductive reactance (X_L) or capacitive reactance (X_C), as seen previously for RL and RC circuits. The values of resistance and reactance are calculated according to the case. Knowing the impedance, the total voltage is determined by Ohm’s Law: V = I * Z, and the current is I = V / Z.

Parallel Circuit: Calculation of Quantities

In a parallel circuit, since the voltage is the same, then: I = I_max * sin(ωt + φ). This means that the current is out of phase with respect to the voltage by an angle φ.

Parallel RL Circuit

A parallel RL circuit consists of a generator (G), a resistor (R), and an inductor (L).

1. The total current is the geometric sum of the partial currents, using maximum values: I_max = √(I_Rmax² + I_Lmax²).
2. For effective values: I = √(I_R² + I_L²).
3. Knowing the value of I, we get: I_R = I * Cos(φ), I_L = I * Sin(φ), Tan(φ) = I_L / I_R.
4. The total current of the circuit will lag behind the voltage by an angle φ. Then: I = I_max * sin(ωt – π/2).

Knowing that (I = V / R), we have for the inductor and resistor respectively: I_L = V / X_L, I_R = V / R. Substituting in tan(φ), we get: Tan(φ) = R / X_L, Cos(φ) = R / √(R² + X_L²).

The admittance (Y) is the hypotenuse of the triangle formed by conductance (G = 1 / R) and inductive susceptance (S = 1 / X_L). X_L = ω * L. From this, the impedance is Z = (R * X_L) / √(R² + X_L²). The total circuit current is I = V / Z.

Parallel RC Circuit

A parallel RC circuit consists of a generator (G), a resistor (R), and a capacitor (C).

1. The total current is the geometric sum of the partial currents, using maximum values: I_max = √(I_Cmax² + I_Rmax²).
2. The total current of the circuit will be advanced with respect to the voltage by an angle φ. Then: I = I_max * sin(ωt + π/2).
3. Substituting for effective values: I_R = I * Cos(φ), I_C = I * Sin(φ), and Tan(φ) = I_C / I_R.

Knowing that (I = V / R), we have for the capacitor and resistor respectively: I_C = V / X_C, I_R = V / R.

Substituting in tan(φ), we get: Tan(φ) = R / X_C, Cos(φ) = X_C / √(R² + X_C²).

The admittance (Y) is the hypotenuse of the triangle formed by conductance (G = 1 / R) and capacitive susceptance (S = 1 / X_C). X_C = 1 / (ω * C). From this, the impedance is Z = (R * X_C) / √(R² + X_C²). The total circuit current is I = V / Z.

Parallel RLC Circuit

A parallel RLC circuit consists of a generator (G), a resistor (R), a capacitor (C), and an inductor (L).

1. The total current is the geometric sum of the partial currents, using maximum values: I_max = √(I_Rmax² + (I_Cmax – I_Lmax)²).
2. The phase difference between voltage and current is given by: Tan(φ) = (I_L – I_C) / I_R, and the total circuit impedance is Z = V / I.

Mixed Circuits

Mixed circuits are formed by combinations of series and parallel connections. To solve them, first calculate the total impedance, then the partial currents, and finally the voltage drops across each element.