DC Circuits | Circuit Theorems

• 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.

• 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 v_s = 10 V gives i = 2 A. According to the linearity principle, v_s = 1 V will give i = 0.2 A. By the same token, i = 1 mA must be due to v_s = 5 mV.
• When current i₁ flows through R, the power is p₁ = Ri₁²; when current i₂ flows through R, the power is p₂ = Ri₂² . If current i₁ + i₂ flows through R, the power absorbed is . Thus, the relationship between power and voltage (or current) is nonlinear .

• 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 v₁ and v₂ are the contributions due to the 6-V voltage source and the 3-A current source, respectively.

To obtain v₁, we set the current source to zero, as shown:

Applying KVL to the loop gives

Thus,

We may also use voltage division to get v₁ by writing

To get v₂, we set the voltage source to zero, as shown:

Using current division,

Hence,

And we find

• 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_s-i_s relation, source transformation is not possible when 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: a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source V_Th in series with a resistor R_Th, where V_Th is the open-circuit voltage at the terminals and R_Th is the input or equivalent resistance at the terminals when the independent sources are turned off.
• 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.

Finding 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_Th : 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 V_Th in Fig.3(b), since the two circuits are equivalent.

• To find R_Th: 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 R_Th in Fig.3(b) because the two circuits are equivalent.

CASE 1 : If the network has no dependent sources, we turn off all independent sources. R_Th is the input resistance of the network looking between terminals a and b, as shown in Fig. 4(b).

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_o. Then, R_Th=v_o/i_o, as shown in Fig.5(a). Alternatively, we may insert a current source i_o at a-b as shown in Fig.5(b) and find the terminal voltage vo. Again, R_Th=v_o/i_o. Either of the two approaches will give the same result. We may assume any value of v_o and 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_L through the load and the voltage 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: 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_N, determine the short-circuit current flowing from terminal 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.

• 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 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_L = R_Th. When R_L is not equal to R_Th, we compute the power delivered to the load using the power equation (as stated above).