. Figure P 28.70 shows a circuit model for the transmission...

$.$ Figure $\mathrm{P} 28.70$ shows a circuit model for the transmission of an electrical signal, such as cable TV, to a large number of subscribers. Each subscriber connects a load resistance $R_{L}$ between the transmission line and the ground. The ground is assumed to be at zero potential and able to carry any current between any ground connections with negligible resistance. The resistance of the transmission line itself between the connection points of different subscribers is modeled as the constant resistance $R_{T} .$ Show that the equivalent resistance across the signal source is $$R_{\mathrm{eq}}=\frac{1}{2}\left[\left(4 R_{T} R_{L}+R_{T}^{2}\right)^{1 / 2}+R_{T}\right]$$ Suggestion: Because the number of subscribers is large, the equivalent resistance would not change noticeably if the first subscriber cancelled his service. Consequently, the equivalent resistance of the section of the circuit to the right of the first load resistor is nearly equal to $R_{\mathrm{eq}}$ FIGURE CANT COPY

$.$ Figure $\mathrm{P} 28.70$ shows a circuit model for the transmission of an electrical signal, such as cable TV, to a large number of subscribers. Each subscriber connects a load resistance $R_{L}$ between the transmission line and the ground. The ground is assumed to be at zero potential and able to carry any current between any ground connections with negligible resistance. The resistance of the transmission line itself between the connection points of different subscribers is modeled as the constant resistance $R_{T} .$ Show that the equivalent resistance across the signal source is
$$R_{\mathrm{eq}}=\frac{1}{2}\left[\left(4 R_{T} R_{L}+R_{T}^{2}\right)^{1 / 2}+R_{T}\right]$$
Suggestion: Because the number of subscribers is large, the equivalent resistance would not change noticeably if the first subscriber cancelled his service. Consequently, the equivalent resistance of the section of the circuit to the right of the first load resistor is nearly equal to $R_{\mathrm{eq}}$
FIGURE CANT COPY

Key Concepts

Equivalent Resistance

This concept involves determining a single resistance value that can replace a complex network of resistors without changing the overall current-voltage relationship at the terminals. It is used to simplify analysis of electrical circuits by reducing a combination of resistors configured in series, parallel, or more complex arrangements into one effective resistor value.

Self-Similarity in Electrical Networks

Self-similarity refers to the property where a portion of an electrical network is structurally identical to the entire network. This allows the network to be analyzed using recursive or iterative techniques, by recognizing that removing a section leaves a remaining network that is essentially the same as the original configuration, thereby greatly simplifying the analysis.

Ladder Network Analysis

A ladder network is a type of repetitive electrical circuit that commonly appears in the analysis of transmission lines and filters. It is characterized by a repeating structure of series and shunt elements. Recognizing the ladder structure enables the use of recursive formulas to determine properties like equivalent resistance, especially when the structure is effectively infinite or very long.

Recursive Circuit Analysis

This technique involves setting up equations that relate the equivalent resistance of a network to the resistance of an additional section of the network. By assuming that a portion of the network has the same equivalent resistance as the entire network (thanks to self-similarity), one can derive an equation that, when solved, yields the overall equivalent resistance. This method is particularly useful for analyzing infinite or very large networks.

Transcript

00:01 Okay, so we have this circuit, okay, that it represents a circuit model for the transmission of an electrical signal, okay, and to which we have a large number of subscribers.

00:15 So each subscriber connects load resistance, that is this one, okay, to a transmission line that is represented here in rt, that is the transmission line resistance, okay? and we want to know what is the equivalent resistance of all of this.

00:33 Okay, so we have a hint, and the hint says that the equivalent resistance does not really change if the first subscriber cancels.

00:45 So what this means is that the resistance are, that it's to the left of the first load resistance.

00:57 It means all of this on the left is equal to the equivalent resistance.

01:04 Okay, so i can redraw this like this circuit.

01:09 Okay, so where i have hearing green, my first subscriber, okay? this is the first subscriber, and i have the equivalent resistance here.

01:18 And this means also that since the equivalent resistance does not really change if the first subscriber cancels, okay, all of this, the equivalent resistance of this circuit must be equal to the equivalent resistance that i'm looking for.

01:35 Okay, so what i have to do is just solve this circuit for our equivalent resistance.

01:46 Okay, so i can find this value.

01:48 And so i know that the equivalent resistance, okay, is one resistor in series, that is this.

01:56 This rt, okay, plus two resistors, that is this and this, in parallel.

02:04 Okay, and i know that the way i sum this is 1 over 1 over rl plus 1 over the equivalent resistance.

02:14 Okay, and i know that when i sum this up, i have that the equivalent resistance equals rt.

02:21 And i'm going to skip here, i'm just solving this part, okay, but you will find that this is rl times the equivalent resistance divided by rl plus req.

02:37 Okay? and what i'm going to do then is since i want to find for the equivalent resistance, i'm going to multiply all my equation for by this okay so you will find this is rl times our eq multiplying rt okay this rt and this is plus rl are q equal to rl plus rq okay time is the equivalent resistance and you do it you will see that sometimes cancelled each other out but you are going to find if you rearrange this that this is the equivalent resistance square okay minus rt at times the equivalent resistance okay minus rl rt equals zero and so remember you want to solve so for the equivalent resistance okay so this actually is a quadratic formula when i know that the solution is minus v and the square root minus or plus, the square root of v squared minus 4, ac divided by 2 times a, where a equals 1, that is the one i have here, okay, b equals minus rt, okay, and c equals minus rt, okay, and c equals minus are lrt...

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