**Introduction**

Resistors are usually connected in a circuit in various ways, and the two most basic ways are series and parallel. This article will mainly introduce these two connection methods, including their definitions, formulas, circuit diagrams, examples and identification methods. In addition, the article also introduces Ohm's law and Kirchhoff's law, which are very important in understanding the series and parallel connections of resistors.

You may need these two calculators in reading this arrticle:

② Parallel and Series Resistance Calculator

The following video explains the basics of resistors in series and parallel, which can promote your understanding towards this article. But it does not matter so much if you skip this video since the article explains in detail and is comprehensive.

Resistors in series and parallel - deriving the formula

**Catalog**

5.2 Kirchhoff's First Law (Nodal Current Law) |

**I Series Connection of Resistors**

(1) Circuit characteristics

Figure1. Resistors in series

The figure shows the series connection of n resistors, and the voltage and current reference directions are related. The circuit characteristics are derived from Kirchhoff’s law:

(A) The resistors are connected in sequence. According to KCL, the current flowing through the resistors is the same;

(B) According to KVL, the total voltage of the circuit is equal to the sum of the voltages of the series resistors, namely:

(2) Equivalent resistance

Figure2. Equivalent resistance circuit

Substituting Ohm's law into the voltage expression, we get:

The above formula illustrates that the series circuit of multiple resistors in Figure (a) and the circuit of single resistor in Figure (b) have the same VCR, which is equivalent to each other.

The equivalent resistance is:

In conclusion:

- The resistors are connected in series, and the equivalent resistance is equal to the sum of the sub-resistances;
- The equivalent resistance is greater than any one of the series resistance.
- The partial pressure of series resistance

If the total voltage across the series resistor is known, what is the divided voltage on each resistor?

From figure (a) and figure (b) we know:

Meet

In conclusion:

Resistors are connected in series, and the voltage on each sub-resistor is proportional to the resistance value. The higher the resistance value, the higher the voltage. Therefore, the series circuit can be used as a voltage divider circuit.

Example 1: Calculate the voltage across the two series resistors as shown in the figure.

Figure3. Circuit of Example1

Solution: From the partial pressure formula of series resistance:

(Note the direction of U_{2})

(3) Power

The power of each resistor is:

So

Total power:

Draw conclusions from the above formulas:

- When resistors are connected in series, the power consumed by each resistor is proportional to the size of the resistor, that is, the larger the resistance, the larger the power consumed;
- The power consumed by the equivalent resistance is equal to the sum of the power consumed by each series resistor.

**II Parallel Connection **of Resistors

(1) Circuit characteristics

Figure4. Parallel circuit characteristics

The figure shows the parallel connection of n resistors, and the voltage and current reference directions are related. The circuit characteristics are derived from Kirchhoff's law:

(a) The two ends of each resistor are connected together. According to KVL, the two ends of each resistor are at the same voltage;

(b) According to KCL, the total current of the circuit is equal to the sum of the currents flowing through the parallel resistors, namely:

(2) Equivalent resistance

Figure5. Equivalent resistance in parallel connection

Substituting Ohm's law into the current expression, we get:

G =1/R is the conductance

The above formula illustrates that the parallel circuit of multiple resistors in Figure (a) and the circuit of single resistor in Figure (b) have the same VCR, which is equivalent to each other.

The equivalent conductance is:

Therefore,

Namely,

The most commonly used formula to find the equivalent resistance when two resistors are connected in parallel:

In conclusion:

- The resistors are connected in parallel, and the equivalent conductance is equal to the sum of the conductances and greater than the partial conductance;
- The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the sub-resistances, and the equivalent resistance is less than any parallel sub-resistance.
- Current distribution of parallel resistance

If the total current of the parallel resistance circuit is known, find the current on each sub-resistance and call it a shunt. From figure (a) and figure (b) we know:

Namely,

Meet

For two resistors in parallel, there are:

Conclusion: When the resistors are connected in parallel, the current on each sub-resistor is inversely proportional to the resistance value, and the current divided by the larger resistance value is smaller. Therefore, the parallel resistor circuit can be used as a shunt circuit.

(3) Power

The power of each resistor is:

So

Total power:

Draw conclusions from the above formulas:

- When resistors are connected in parallel, the power consumed by each resistor is inversely proportional to the size of the resistor, that is, the larger the resistance, the smaller the power consumed;
- The power consumed by the equivalent resistor is equal to the sum of the power consumed by each parallel connected resistor.

consumed by each series resistor.

**III Resistor Combination(Mixed Resistor Circuit)**

A circuit with resistors connected in series and connected in parallel is called a resistor combination or mixed resistor circuit. The part where the resistors are connected in series has the characteristics of a resistor series circuit, and the part where the resistors are connected in parallel has the characteristics of a resistor parallel circuit.

Example 2: The circuit is shown in the figure, please calculate the voltage and current of each branch.

Figure6. Example circuit 2

Solution: This is a resistor series and parallel circuit. First find the equivalent resistance Reg = 11W, and the current and voltage of each branch are:

The general steps for solving series and parallel circuits can be obtained from the above examples:

⚫ Find the equivalent resistance or equivalent conductance;

⚫ Apply Ohm's law to find the total voltage or total current;

⚫ Apply Ohm's law or voltage division and shunt formula to find the current and voltage on each resistor.

Therefore, the key issue in analyzing series-parallel circuits is to distinguish the relationship between series and parallel circuits.

To determine the series-parallel relationship of the circuit, the following 4 points should be mastered:

⚫ Look at the structural characteristics of the circuit. If two resistors are connected end-to-end, they are connected in series;

⚫ Look at the relationship between voltage and current. If the current flowing through the two resistors is the same current, it is connected in series; if the two electrical groups bear the same voltage, it is connected in parallel.

⚫ Equivalent to deformation of the circuit. For example, the left branch can be twisted to the right, the upper branch can be turned down, the curved branch can be straightened, etc.; the short circuit in the circuit can be compressed and extended at will; the multi-point grounding can be connected by a short circuit . Generally, if it is really a problem with a resistor series circuit, it can be distinguished.

⚫ Find the equipotential point. For circuits with symmetrical characteristics, if two points can be judged to be equipotential points, according to the concept of circuit equivalence, one is to use short wires to connect the equipotential points; the other is to break the branch that connects the equipotential points. Open (because there is no current in the branch), thus obtain the series-parallel relationship of the resistance.

## IV Ohm's Law

### 4.1 What is Ohm's Law?

(1) The content of Ohm's law

When there is a potential difference between the two ends of the conductor, an electric field appears inside the conductor, and the charge moves in a directional motion under the force of the electric field to generate current. German physicist Ohm summed up Ohm's law in 1826 through a large number of experiments: Under steady conditions, the intensity of the current passing through a section of conductor is proportional to the voltage across the conductor.

(2) Mathematical expression of Ohm's law

Note: The unit of the physical quantity in the formula: the unit of I is ampere (A), the unit of U is volt (V), and the unit of R is ohm (Ω).

The proportional coefficient R in the formula is determined by the properties of the conductor and is called the resistance of the conductor. Unit: Ohm (Ω). The reciprocal of resistance is called conductance and is represented by G, that is

Unit: Siemens (S).

(3) Understanding and explanation of Ohm's law

● Applicable conditions of Ohm's law: applicable to pure resistance circuits (that is, when working with electrical appliances, the consumed electrical energy is completely converted into internal energy.)

● I, U and R in the formula must correspond to the same conductor or the same circuit. If it is in different time, different conductor or different section of circuit, I, U, and R can not be mixed, therefore, the three physical quantities should be marked with angles in order to distinguish under normal circumstances.

● For the same conductor (that is, R does not change), I and U are proportional; for the same power source (that is, U does not change), I and R are inversely proportional.

● R=ρL/S is the definition of resistance, which means that the resistance of a conductor is determined by the material, length and cross-sectional area of the conductor itself. In addition, resistance is also related to factors such as temperature.

● The formula transformed from Ohm's law is a measure of resistance. It indicates that the resistance of a conductor can be given by U/I, that is, the ratio of R to U and I is related, but the magnitude of R itself is related to the applied voltage U and the passing current Factors such as the size of I are irrelevant.

● Knowing any two quantities among I, U and R, you can find another quantity.

● Issues that need special attention and re-emphasis: I, U and R in the formula must be in the same circuit; when using the formula to calculate, the unit of each physical quantity must be unified.

The above explanations are all part of Ohm’s law, which only applies to pure resistance circuits.

(4) Pure resistance circuit

A pure resistance circuit is a circuit with only resistance elements in addition to the power supply, or inductance and capacitance elements, but their influence on the circuit is negligible. The voltage and current have the same frequency and phase.

The resistance converts all the energy obtained from the power supply into internal energy. This kind of circuit is called a pure resistance circuit. Here is a brief explanation from the energy point of view.

Basically, as long as there is no conversion of electric energy other than internal energy, this circuit is a pure resistance circuit.

### 4.2 What is Closed Circuit Ohm's Law?

In an AC circuit, Ohm's law also holds, but the resistance R should be changed to impedance Z, that is, I = U/Z. If the circuit is closed and contains a power supply, it is called a full circuit, as shown in the figure below. The dotted line in the figure is the power supply, which is called an internal circuit. The circuit outside the power supply is called an external circuit. Since the power supply has internal resistance, the current not only has a voltage drop when passing through an external circuit, but also has an internal voltage drop when passing through an internal circuit. In the whole circuit, the current intensity is proportional to the electromotive force E of the power supply, and inversely proportional to the resistance (R+r) of the whole circuit (including the inner circuit and the outer circuit). This is the Ohm's law of the whole circuit, expressed by the formula:

Where I- the current in the circuit, A; E- the electromotive force of the power supply, V; R- the resistance of the external circuit, Ω; r- the resistance of the internal circuit, Ω.

From the above formula, in the circuit shown in the figure below, E=IR+Ir=Uouter+Uinner.

Figure7. The simplest closed circuit

In the formula, U external = IR-external circuit voltage; U internal = Ir-internal circuit voltage.

It should be noted that, since the internal resistance of the power supply itself and the internal resistance of the connecting wires are generally not large, the calculation results that are ignored in the calculation are basically correct. But sometimes it is necessary to calculate the internal voltage drop of the power supply, and to accurately calculate the current of the whole circuit, it is necessary to use the whole circuit Ohm's law. For example, in the figure below, if E=10V, r=0.1Ω, R=1kΩ, then:

Figure8. An application example of Ohm's law of closed circuit

① When S is connected to the 1 position, the circuit is in the open state,

Ammeter reading

The reading of the voltmeter is U=IR=0.01×1000=10 (V), or U=E-Ir=10-0.01×0.1≈10 (V).

②When S is connected to the 2 position, the circuit is in an open state, so the reading of the ammeter is 0; the reading of the voltmeter is U=E=10(V).

③When S is connected to the 3 position, the circuit is in a short-circuit state, the reading of the ammeter is I=E/r=10/0.1=100(A)A; the reading of the voltmeter U=0(V).

### 4.3 The Key Points of Studying Ohm's Law

Ohm's law is an important basic law in electricity. It is a law that is summarized and summarized through experiments. To master this law, we must pay attention to the following points:

(1) Ohm's law applies to the entire circuit or a part of the circuit from the positive pole to the negative pole of the power supply, and it is a pure resistance circuit.

(2) The current I "passing through" in Ohm's law, the voltage U at "both ends" and the resistance R of the "conductor" are all corresponding physical quantities on the same conductor or the same circuit. The above relationship does not exist between the current, voltage, and resistance of different conductors. Therefore, when using the formula I=U/R, the current, voltage, and resistance of the same conductor or the same circuit must be substituted into the calculation, and the three correspond one to one.

(3) There is simultaneity among the three physical quantities in Ohm’s law. Even on the same part of the circuit, the closing or opening of the switch and the movement of the sliding position of the sliding varistor will cause the change of the circuit, which will lead to the current in the circuit. , Voltage, resistance changes, so the three quantities in the formula I=U/R are the same time value.

(4) The difference between I=U/R and R=U/I:

Ohm's law expression I=U/R means that the current in the conductor is related to the voltage across the conductor and the resistance in the conductor. When the resistance R is constant, the current I in the conductor is proportional to the voltage U across the conductor; when the voltage U across the conductor is constant, the current I in the conductor is inversely proportional to the resistance R of the conductor.

R=U/I is derived from Ohm’s law expression. It means that the resistance value of a certain section of conductor is equal to the ratio of the voltage across the section of the conductor to the current passing through it. This ratio R is the property of the conductor itself and cannot be understood as R is directly proportional to U and inversely proportional to I. This is also the difference between physics and mathematics.

(5) Ohm's law reflects the causal relationship between current intensity and voltage, and the restrictive relationship between current intensity and resistance under certain conditions. That is, when the resistance is constant, the current intensity is proportional to the voltage across the conductor; when the voltage is constant, the current intensity is inversely proportional to the resistance of the conductor. When establishing a proportional relationship, we must pay attention to its conditions. Ohm's law states that the current intensity through a conductor is determined by two factors, the voltage across the conductor and the resistance of the conductor.

## V Kirchhoff's Law

Kirchhoff's law includes the first law and the second law. They are the basic laws that are indispensable for the analysis and calculation of complex circuits.

### 5.1 Concepts

• Branch

A two-terminal element connected in a circuit is a branch. Usually a certain current flows through the branch. (This definition is not universal. For example, if two components are connected in series and then connected in a circuit, it can only be regarded as a branch.)

• Node

The connection point between the branch and the branch is called a node. Usually the current diverges at the junction.

• Loop loop

A closed path formed by branches is called a loop.

Figure9. 6 elements, 6 branches, 4 nodes, 3 independent circuits

### 5.2 Kirchhoff's First Law (Nodal Current Law)

The textual expression of KCL: For any node, the algebraic sum of the current flowing into (or out of) the node is equal to zero.

Its mathematical expression:

The regulation of current positive and negative: Generally, the current flowing into the node is positive, and the current flowing out of the node is negative.

The physical meaning of KCL: conservation of charge

Note: KCL is not only applicable to a node, but also to a part of the circuit, as shown in the shaded part of the above figure:i_{3}＝i_{6}

### 5.3 Kirchhoff's Second Law (Law of Loop Voltage)

KVL’s literal expression: In any closed loop of the circuit, go around a circle in a certain direction, and the algebraic sum of the voltage of each segment is zero.

That is: or. When applying the law of loop voltage, the electromotive force is often written on the left side of the equation, and the voltage is written on the right side of the equation.

The method for determining the sign of each electromotive force and voltage in the second expression is as follows:

① First select the current direction of each branch.

② Any choice of the detour direction along the loop (clockwise or counterclockwise).

③ If the direction of the current flowing through the resistor is the same as the detour direction, the voltage drop on the resistor is positive, otherwise, it is negative.

④ If the direction of the electromotive force is the same as the direction of the orbit, the electromotive force is positive, otherwise, it is negative.

The physical meaning of KVL: energy conservation.

### 5.4 Application Note of Kirchhoff's Law

• Kirchhoff’s law is a general law that the circuit should satisfy, and has nothing to do with the specific properties of the components;

• Kirchhoff’s law applies to any lumped circuit, that is, nonlinear, time-varying circuits, etc.;

• Application steps:

A. Divide the branch roads and number them;

B. Specify the branch current and voltage reference direction, and generally need to be associated;

C. Select the appropriate node according to the meaning of the question, and apply KCL;

D. Or choose the appropriate circuit according to the meaning of the question, apply KVL, and pay attention to independence.

Example: Use KVL to derive the relationship between the total resistance and the sub-resistance and the voltage division formula in the series resistance circuit.

Apply KVL according to the current and voltage reference direction and the detour direction of the loop calibrated in the figure:

-u+u_{1}+u_{2}+…+u_{n} = 0 or u=u_{1}+u_{2}+…+u_{n}

Because the voltage and current of each resistor obey Ohm's law: u_{k}=iR_{k}, there are:

u = i × R_{1} + i× R_{2} +...... +i× R_{n}

= i× ( R_{1}+R_{2}+…+R_{n})

= i R_{e}

among them:

R_{e}=R_{1}+R_{2}+…+R_{n}, which is the total resistance or equivalent resistance.

u_{k} = iR_{k}=( u/R_{e} ) R_{k, }which is the voltage division formula of the series circuit

## VI Series and Parallel Circuit Identification Methods

Method 1: Current flow method

(1) Starting from the positive pole of the power supply, use arrows to mark the path of the current along the connected wires, and finally return to the negative pole of the power supply;

(2) Observe whether the current has a shunt and confluence point:

If there is only one path for the current in the circuit, the components are connected in series (as shown in Figure a below);

If there is a shunt point and a confluence point in the circuit, that is, the direction of the current is greater than one path, the components between the shunt point and the confluence point are connected in parallel (as shown in Figure b below)

Figure10. Current flow method

Method 2: Demolition method

Remove any electrical appliances:

If the other electrical appliance cannot work, the two electrical appliances are connected in series (as shown in Figure a below)

If the other consumer still works without being affected, the two consumers are connected in parallel (as shown in Figure b below)

Figure11. Demolition method

Method 3: Node Method

For other non-intuitive non-series circuits, the situation is more complicated and needs to be judged according to several steps:

The first step is to mark nodes. That is, use different letters (or symbols) to mark all nodes of the circuit. As shown in the following figure (a), the four points A, B, C, and D are all nodes in the circuit.

The second step is to merge the nodes. According to the characteristics of the nodes, some of the nodes you have marked may be equivalent to the same node. The letters (or symbols) belonging to the same node must be changed to the same letter (or symbol), as shown in the circuit shown in Figure (a) Point A and point C are the same node, C should be changed to A, point B and point D should be the same node, D should be rewritten as B, that is to say, the circuit shown in Figure (a) essentially has two nodes A and B .

Figure12. Node Method

The third step is to determine the connection mode of the circuit. There are usually two ways to judge:

Method one:

Direct judgment: as shown in the figure (a) above, both ends of the resistors R1, R2 and R3 are independently connected to nodes A and B, so R1, R2 and R3 are connected in parallel.

Method Two:

Drawing judgment: that is, draw the intuitive equivalent circuit diagram of the original diagram. The specific drawing method of the intuitive equivalent circuit diagram of the circuit diagram in Figure (a) is: first determine the two points A and B on the paper, and then combine the original diagrams A and B. The components between the two points B are independently connected to the newly determined points A and B, as shown in the above figure (b), that is, the equivalent circuit diagram of figure (a) is figure (b).

Warm reminder: The "node method" is generally used to identify irregular and more complex circuits, which has certain difficulties. There are many ways to identify series and parallel circuits, but you can choose the most suitable method according to your own understanding of the method when using it.

## VII Quiz

The voltage dropped across the 300 ohm resistor is

A. 6V

B.9V

C.2V

D.30V

Answer: A

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