Chapter 34: Delta and Star-Delta Transformation
Date: 8/8/19
34.1 Introduction
Kirchhoff’s laws, mesh current analysis, nodal analysis, and the superposition theorem are means that can be applied to analyse circuits. Thevenin’s and Norton’s theorems can be used to analyse them as well and result in massive reductions of time spent. Star-delta and delta-star transformations may be applied to certain types of circuits to simplify them before the application of circuit theorems.
34.2 Star and Delta Networks
A delta connected network is sometimes referred to as a mesh connected network (often denoted by π or Δ).
A star connected network can be redrawn and is otherwise known as a T or Y connected network.
34.3 Delta to Star Transformations
It is possible to replace a delta connection with an equivalent star connection. This means that the impedance measured between each pair of terminals is the same as in the delta. The equivalent star network will have the same power factor and consume the same power as the delta network. A delta-star transformation may also be called a π to T transformation.
Transformation Formulas (Delta to Star)
The equivalent star impedances ($Z_1$, $Z_2$, $Z_3$) can be derived from the delta impedances ($Z_A$, $Z_B$, $Z_C$):
- Z1 = (ZA × ZB) / (ZA + ZB + ZC)
- Z2 = (ZB × ZC) / (ZA + ZB + ZC)
- Z3 = (ZA × ZC) / (ZA + ZB + ZC)
Summary: The impedance $Z_1$ is given by the product of the two impedances joined at node 1 divided by the sum of all impedances. Similarly, $Z_2$ and $Z_3$ are found using the product of the impedances at nodes 2 and 3, respectively, divided by the total sum.
34.4 Star to Delta Transformation
Conversely, it is possible to replace a star section with an equivalent delta section. This is sometimes called a T to π transformation.
Transformation Formulas (Star to Delta)
The equivalent delta impedances ($Z_A$, $Z_B$, $Z_C$) can be derived from the star impedances ($Z_1$, $Z_2$, $Z_3$):
- ZA = (Z1Z2 + Z2Z3 + Z3Z1) / Z2
- ZB = (Z1Z2 + Z2Z3 + Z3Z1) / Z3
- ZC = (Z1Z2 + Z2Z3 + Z3Z1) / Z1
Summary: The numerator is the sum of the products of the star impedances taken two at a time. The denominator for $Z_A$ (connected between terminals 1 and 3) is $Z_2$ (connected to terminal 2). The denominator for $Z_B$ (connected between terminals 1 and 2) is $Z_3$ (connected to terminal 3). The denominator for $Z_C$ (connected between terminals 2 and 3) is $Z_1$ (connected to terminal 1).
Chapter 35: Maximum Power Transfer Theorems and Impedance Matching
Date: 11/08/19
Why it is important to understand
This theorem states that maximum power transfer occurs when the impedance of the source equals the impedance of the load, and vice versa. It is also called Jacobi’s Law. A related concept is reflection-less impedance matching. This is where the source impedance, such as on a transmitter, matches the load impedance, such as an antenna, to avoid reflections in the transmission line.
In electronics, impedance matching is the practice of matching the input impedance of an electrical load (or the output impedance of its corresponding signal source) to maximize power transfer or minimize signal reflections from the load. Matching a load to a source for maximum power transfer is extremely important in microwaves, as well as all manner of low-frequency applications such as stereo sound systems, electrical generating plants, solar cells, and hybrid electrical cars. Impedance matching is very important where small signals are involved.