11. Delta-to-Wye (Pi-to-Tee) Equivalent Circuits

In many circuits, resistors are neither in series nor in parallel, so the rules for series or parallel circuits described in previous chapters cannot be applied. For these circuits, it may be necessary to convert from one circuit form to another to simplify the solution. Two typical circuit configurations that often have these difficulties are the wye (Y) and delta (D ) circuits. They are also referred to as tee (T) and pi (P ) circuits, respectively.

Delta and wye circuits:

And the equations for converting from delta to wye:

 The equations can be presented in an alternate form based on the total resistance (Rd) of R₁, R₂, and R₃ (as though they were placed in series):

Rd = R₁+R₂+R₃

and:

R_A = (R₁*R₃) / Rd

R_B = (R₂*R₃) / Rd

R_C = (R₁*R₂) / Rd

 Wye and delta circuits:

And the equations for converting from wye to delta:

 

An alternate set of equations can be derived based on the total conductance (Gy) of R_A, R_B, and R_C (as though they were placed in parallel):

Gy = 1/R_A+1/R_B+1/R_C

and:

R₁ = R_B*R_C*Gy

R₂ = R_A*R_C*Gy

R₃ = R_A*R_B*Gy

sion system

. 12. Mesh-Current Method

The mesh current method is deduced from the Kirchhoff’s voltage law (KVL) and superposition theorem.

The following formulas are used to solve circuit.

where

Loop i resistance is the sum of resistances of all branches which contain in the given loop.

Loop i to Loop j resistance. If the directions of loops are opposite then resistance is negative. Otherwise, it's positive. The value equals the sum of branches resistances of all branches, which contains in both loops.

 Loop i – EMF

Number of loop equations equal number of KVL equations.

Loop Number = Branch Number –(Nodes Number –1) – Current sources Number

If the circuit contain current sources it is necessary to create additional loops. Branch with current source can be contained in one loop.

13. Node-Voltage Method

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Note all connected wire segments in the circuit..

Examples

The only unknown voltage in this circuit is V₁. There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node.

1. Current through resistor R₁: (V₁ - V_S) / R₁

2. Current through resistor R₂: V₁ / R₂

3. Current through current source I_S: -I_S

With Kirchhoff's current law, we get:

This equation can be solved in respect to V₁:

Finally, the unknown voltage can be solved by substituting numerical values for the symbols. Any unknown currents are easy to calculate after all the voltages in the circuit are known

14. Active and apparent power

Power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such asinductance and capacitance may result in periodic reversals of the direction of energy flow. The portion of power that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power. The portion of power due to stored energy, which returns to the source in each cycle, is known as reactive power.

15. Conversion between rectangular and polar formsTo convert between polar and rectangular coordinates, we make a right triangle to the point (x,y), like this: