# DC Circuits | Circuit Theorems

## Linearity Property

- Linearity is the property of an element describing a linear relationship between cause and effect.
- It is a combination of the homogeneity property and the additivity property.
*Homogeneity property*requires that if the input (excitation) is multiplied by a constant, then the output (response) is multiplied by the same constant.- For a resistor, for example, Ohm’s law relates the input
*i*to the output*v*:*v=iR*. If*i*is increased by a constant*k*, then v increases correspondingly by*k*; that is

*Additivity property*requires that the response to a sum of inputs is the sum of the responses to each input applied separately. Thus for a resistor, if and, then applying gives- We say that a resistor is a
*linear element*because the voltage-current relationship satisfies both the homogeneity and the additivity properties. - In general, a circuit is linear if it is both additive and homogeneous.
- A
*linear circuit*is one whose output is linearly related (or directly proportional) to its input. A linear circuit consists of only linear elements, linear dependent sources, and independent sources.

To illustrate the linearity principle, consider the linear circuit:

- The circuit has no independent sources inside it, is excited by a voltage source
*v*_{s }(input), and is terminated by a load*R*. The current*i*through*R*can be taken as the output. - Suppose
gives**v**_{s}= 10 V**i = 2 A**will give*v*_{s}= 1 V. By the same token,**i = 0.2 A***i = 1 mA*.**v**_{s}= 5 mV - When current
*i*flows through_{1}*R*, the power is; when current i**p**_{1 }= Ri_{1}^{2}_{2}flows through*R*, the power is. If current**p**_{2}= Ri_{2}^{2}*i*+_{1}*i*flows through_{2}*R*, the power absorbed is . Thus, the relationship between power and voltage (or current) is**nonlinear**.

## Superposition Principle

The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

To apply the superposition principle, we must keep two things in mind:

Consider one independent source at a time while the others are turned off. Replace every voltage source by

*0 V*(short circuit), and every current source by*0 A*(open circuit) to obtain a simpler circuit.Dependent sources are left intact as they are controlled by circuit variables.

With these in mind, we apply the superposition principle in three steps:

**Step 1:** Turn off all independent sources except one source. Find the output (voltage or current) due to that active source.

**Step 2: **Repeat step 1 for each of the other independent sources.

**Step 3: **Find the total contribution by adding algebraically all the contributions due to the independent sources.

Using the superposition theorem to find *v* in the circuit below,

since there are two sources,

where v_{1} and v_{2} are the contributions due to the 6-V voltage source and the 3-A current source, respectively.

To obtain v_{1}, we set the current source to zero, as shown:

Applying KVL to the loop gives

Thus,

We may also use voltage division to get v_{1} by writing

To get v_{2}, we set the voltage source to zero, as shown:

Using current division,

Hence,

And we find

## Source Transformation

- A
*source transformation*is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa. Source transformation is another tool for simplifying circuits.

**Independent sources**

- Recall: An equivalent circuit is one whose
*v-i*characteristics are identical with the original circuit.

- If the sources are turned off, the resistance at terminals
*a-b*in both circuits is R. When terminals a-b are short-circuited, the short-circuit current from*a*to*b*is in circuit (a) and for circuit (b). Thus, in order for the two circuits to be equivalent. - Hence, source transformation requires that :

*Dependent sources*

- Source transformation also applies to dependent sources, by carefully handling the dependent variable.

- Keep the following points in mind when dealing with source transformation.
- Note from the circuits that the arrow of the current source is directed toward the positive terminal of the voltage source.
- From the
*v*relation, source transformation is not possible when_{s}-i_{s}*R=0*(ideal voltage source). For a non-ideal voltage source, R is not equal to zero. Similarly, an ideal current source with*infinite R*cannot be replaced by a finite voltage source.

- Source transformation does not affect the remaining part of the circuit. When applicable, source transformation eases circuit analysis.

## Thevenin’s Theorem

- Thevenin’s theorem: a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source
*V*in series with a resistor_{Th }*R*, where_{Th}*V*is the open-circuit voltage at the terminals and_{Th}*R*is the input or equivalent resistance at the terminals when the independent sources are turned off._{Th} - The Thevenin equivalent circuit was developed in 1883 by M. Leon Thevenin (1857–1926), a French telegraph engineer.
- The fixed part of the circuit is replaced by an equivalent circuit to avoid analyzing the circuit all over again when the variable elements change.
- The linear circuit Fig.3(a) can be replaced by Fig.3(b). The load may be a single resistor or another circuit.

F*inding **V _{Th }*

*and*

*R*

_{Th}*:*suppose two circuits are equivalent; that is, they have the same

*v-i*relation at their terminals.

- To find the Thevenin equivalent voltage
*V*: if the terminals a-b are made open-circuited, no current flows, so that the open-circuit voltage across the terminals a-b in Fig.3(a) must be equal to the voltage source_{Th }*V*in Fig.3(b), since the two circuits are equivalent._{Th }

- To find
*R*: with the load disconnected and a-b open-circuited, we turn off all independent sources. The input or equivalent resistance of the dead circuit at the terminals a-b in Fig.3(a) must be equal to_{Th}*R*in Fig.3(b) because the two circuits are equivalent._{Th}

*R*, there are two cases to consider.

_{Th}

** CASE 1** :

*If the network has no dependent sources,*we turn off all independent sources.

*R*is the input resistance of the network looking between terminals a and b, as shown in Fig. 4(b).

_{Th}** CASE 2** :

*If the network has dependent sources*, turn off all independent sources. Dependent sources are not to be turned off as they are controlled by circuit variables. Apply a voltage source vo at terminals

*a*and

*b*and determine the resulting current

*i*. Then,

_{o}*R*=

_{Th}*v*/

_{o}*i*, as shown in Fig.5(a). Alternatively, we may insert a current source

_{o}*i*at

_{o }*a-b*as shown in Fig.5(b) and find the terminal voltage vo. Again,

*R*=

_{Th}*v*/

_{o}*i*. Either of the two approaches will give the same result. We may assume any value of

_{o}*v*and

_{o }*i*.

_{o} *R _{Th}* can take a negative value. The negative resistance implies that the circuit is supplying power. This is possible in a circuit with dependent sources.

Consider a linear circuit terminated by a load *R _{L}*, as shown in Fig.6(a). The current

*I*through the load and the voltage

_{L }*V*

_{L }across the load are easily determined once the Thevenin equivalent of the circuit at the load’s terminals is obtained as shown in Fig.6(b).

From Fig.6(b), we obtain:

Note from Fig.6(b) that the Thevenin equivalent is a simple voltage divider, yielding *V _{L }*by mere inspection.

## Norton’s Theorem

Norton’s theorem: a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source I

_{N}in parallel with a resistor R_{N}, where I_{N}is the short-circuit current through the terminals and R_{N}is the input or equivalent resistance at the terminals when the independent sources are turned off.Thus, the circuit in Fig.7(a) can be replaced by the one in Fig.7(b).

E. L. Norton, an American engineer at Bell Telephone Laboratories, proposed the theorem in 1926.

The main concern is to get R

_{N}and I_{N}.We find R

_{N}in the same way we find R_{Th}. In fact, from what we know about source transformation, the Thevenin and Norton resistances are equal; that is,

- To find the Norton current
**I**, determine the short-circuit current flowing from terminal_{N}*a*to*b*in both circuits in Fig.7. It is evident that the short-circuit current in Fig.7(b) is I_{N}. This must be the same short-circuit current from terminal*a*to*b*in Fig.7(a), since the two circuits are equivalent. Thus,

Dependent and independent sources are treated the same way as in Thevenin’s theorem.

Observe the close relationship between Norton’ s and Thevenin’s theorems: R

_{N}= R_{Th}, and

- This is essentially source transformation. For this reason, source transformation is often called
**Thevenin-Norton**transformation. - Since V
_{Th}, I_{N}, and R_{Th}are related, to determine the Thevenin or Norton equivalent circuit requires that we find:- The open-circuit voltage v
_{oc}across terminals*a*and*b*. - The short-circuit current i
_{sc}at terminals a and b. - The equivalent or input resistance R
_{in}at terminals*a*and*b*when all independent sources are turned off.

- The open-circuit voltage v

- We can calculate any two of the three using the method that takes the least effort and use them to get the third using Ohm’s law. Also, since the open-circuit and short-circuit tests are sufficient to find any Thevenin or Norton equivalent of a circuit which contains at least one independent source.

## Maximum Power Transfer

*Maximum power*is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (R_{L}=R_{Th}).- The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load. We assume that we can adjust the load resistance R
_{L}. If the entire circuit is replaced by its Thevenin equivalent except for the load, as shown in Fig.9, the**power**delivered to the load is

- For a given circuit, V
_{Th}and R_{Th}are fixed. By varying R_{L}, the power delivered to the load varies as sketched in Fig.10.

- The power is small for small or large values of R
_{L}but maximum for some value of R_{L}between 0 and ∞. - Differentiating the power equation and setting the result equal to zero confirms that the maximum power occurs when RL is equal to RTh. This is known as the
*maximum power theorem*:**R**._{L}= R_{Th} - Substituting R
_{L}= R_{Th}to the power equation gives the maximum power transferred

- The equation above applies only when
**R**. When_{L}= R_{Th}**R**, we compute the power delivered to the load using the power equation (as stated above)._{L}is not equal to R_{Th}

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