4.2 Series and Parallel Circuits

4.2 Series and Parallel Circuits

Electronic circuits are configured in various topologies to direct current, drop voltages, and distribute power. The two primary configurations are series circuits and parallel circuits. Biomedical technicians must analyze these circuits to diagnose system failures, measure sensor outputs, and understand how power is distributed within complex medical systems. Additionally, they must apply Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) to solve complex circuit paths.

Series Circuits

A series circuit provides only a single path for electric current to flow. All components in a series circuit are connected end-to-end, forming a continuous loop.

Characteristics of Series Circuits

1. Constant Current: Because there is only one path for the electrons, the current flowing through every component in a series circuit is identical. $I_{total} = I_1 = I_2 = I_3 = \dots = I_n$
2. Cumulative Resistance: The total or equivalent resistance ($R_{total}$) of a series circuit is the sum of the individual resistances. Adding more resistors in series increases the total resistance, which decreases the total current. $R_{total} = R_1 + R_2 + R_3 + \dots + R_n$
3. Additive Voltage Drops: The total voltage supplied by the source is distributed across the series components. The voltage drop across each component is proportional to its resistance (according to Ohm's Law, $V = I \times R$). The sum of these individual voltage drops must equal the total source voltage: $V_{total} = V_1 + V_2 + V_3 + \dots + V_n$

Troubleshooting Series Circuits

If any single component in a series circuit opens (fails such that current cannot flow), the entire circuit is broken, and current stops flowing completely. An example of this is an older string of holiday lights where one burnt-out bulb darkens the entire string. In medical equipment, a blown fuse or a tripped thermal cutout switch placed in series with a power line will halt all current to protect the device.

---

Parallel Circuits

A parallel circuit contains multiple branches, providing more than one path for current to flow. The components are connected across the same common electrical nodes.

Characteristics of Parallel Circuits

1. Constant Voltage: The voltage across each branch of a parallel circuit is identical and equal to the source voltage, as they are all connected to the same electrical nodes. $V_{total} = V_1 = V_2 = V_3 = \dots = V_n$
2. Additive Branch Currents: The total current leaving the source divides among the parallel branches. The current flowing through each branch depends on that branch's resistance. The sum of all branch currents equals the total current drawn from the source: $I_{total} = I_1 + I_2 + I_3 + \dots + I_n$
3. Reciprocal Total Resistance: The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances: $\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n}$ For two resistors in parallel, this simplifies to the product-over-sum formula: $R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}$ Note: The total equivalent resistance of parallel resistors is always less than the resistance of the smallest individual resistor in the parallel network. This is because adding branches provides additional pathways for current, reducing the overall restriction to flow.

Troubleshooting Parallel Circuits

In a parallel circuit, if one branch opens, current continues to flow through the remaining functional branches. The voltage across the remaining branches remains constant, but the total current drawn from the source will decrease. In medical facilities, electrical outlets and devices are connected in parallel. If you unplug a patient monitor from a wall outlet, other devices connected to the same circuit continue to operate normally.

---

Kirchhoff's Laws

For complex circuits that cannot be simplified by basic series or parallel rules alone, German physicist Gustav Kirchhoff formulated two fundamental laws:

Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (KVL) is based on the law of conservation of energy. It states that the algebraic sum of all voltages around any closed loop in a circuit must equal zero. In practical terms, this means that the sum of the voltage drops across the components in a loop must equal the total voltage supplied by the source in that same loop:

$\sum V = 0 \quad \text{or} \quad V_{source} = V_{drop1} + V_{drop2} + \dots + V_{dropN}$

Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (KCL) is based on the law of conservation of electric charge. It states that the total current entering any electrical junction (or node) must exactly equal the total current leaving that junction:

$\sum I_{in} = \sum I_{out}$

For example, if $10\text{ A}$ enters a node that splits into three parallel paths, and two of those paths carry $3\text{ A}$ and $5\text{ A}$ respectively, KCL dictates that the third path must carry $2\text{ A}$ ($10\text{ A} - 3\text{ A} - 5\text{ A} = 2\text{ A}$).

---

Voltage Divider Circuits

A voltage divider is a simple series circuit of two or more resistors that scales down an input voltage to a lower, proportional output voltage. This is one of the most common sub-circuits found in biomedical sensors, microcontroller inputs, and reference voltage generators.

The Voltage Divider Formula

For a two-resistor voltage divider, the output voltage ($V_{out}$) measured across the second resistor ($R_2$) is:

$V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}$

Worked Example: Sensor Signal Conditioning

A temperature sensor outputting a $12\text{ V}$ signal needs to be interfaced with a patient monitor that can only accept a maximum of $8\text{ V}$. A technician designs a voltage divider using $R_1 = 2\text{ k}\Omega$ and $R_2 = 4\text{ k}\Omega$. Let's calculate the output voltage to verify if it is safe.

1. Identify variables:
   • $V_{in} = 12\text{ V}$
   • $R_1 = 2\text{ k}\Omega$
   • $R_2 = 4\text{ k}\Omega$
2. Apply the formula:
   • $V_{out} = 12\text{ V} \times \frac{4\text{ k}\Omega}{2\text{ k}\Omega + 4\text{ k}\Omega}$
   • $V_{out} = 12\text{ V} \times \frac{4,000}{6,000} = 12\text{ V} \times \frac{2}{3}$
   • $V_{out} = 8\text{ V}$
3. Conclusion: The voltage divider successfully scales the $12\text{ V}$ signal down to exactly $8\text{ V}$, making it safe for the patient monitor input.